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References

Published online by Cambridge University Press:  27 July 2021

Hwa-Long Gau  and
Pei Yuan Wu
Hwa-Long Gau
Affiliation:
National Central University, Taiwan
Pei Yuan Wu
Affiliation:
National Chiao Tung University, Taiwan
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Numerical Ranges of Hilbert Space Operators , pp. 456 - 479
Publisher: Cambridge University Press
Print publication year: 2021

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References

Abdollahi, A., The numerical range of a composition operator with conformal automor-phism symbol, Linear Algebra Appl., 408 (2005), 177–188.Google Scholar
Abu-Omar, A. and Kittaneh, F., Numerical radius inequalities for n×n operator matrices, Linear Algebra Appl., 468 (2015), 18–26.CrossRefGoogle Scholar
Abu-Omar, A. and Wu, P. Y., Scalar approximants of quadratic operators with applications, Oper. Matrices, 12 (2018), 253–262.Google Scholar
Agler, J., Geometric and topological properties of the numerical range, Indiana Univ. Math. J., 31 (1982), 767–777.Google Scholar
Agler, J., Hypercontractions and subnormality, J. Operator Theory, 13 (1985), 203–217.Google Scholar
Agler, J., Rational dilation on an annulus, Ann. of Math. (2), 121 (1985), 537–563.CrossRefGoogle Scholar
Agler, J., An abstract approach to model theory, in: Surveys of Some Recent Results in Operator Theory, vol. II, Conway, J. B. and Morrel, B. B., eds., Longman, Harlow, Essex, 1988, pp. 1–23.Google Scholar
Agler, J., Harland, J., and Raphael, B. J., Classical Function Theory, Operator Dilation Theory, and Machine Computation on Multiply-Connected Domains, Mem. Amer. Math. Soc., 191 (2008), no. 892.Google Scholar
Agler, J. and Young, N. J., A commutant lifting theorem for a domain in C 2 and spectral interpolation, J. Funct. Anal., 161 (1999), 452–477.Google Scholar
Agler, J. and Young, N. J., A model theory for Ɣ-contractions, J. Operator Theory, 49 (2003), 45–60.Google Scholar
Alpin, Y. A., Chien, M.-T., and Yeh, L., The numerical radius and bounds for zeros of a polynomial, Proc. Amer. Math. Soc., 131 (2003), 725–730.Google Scholar
Aluthge, A., On p-hyponormal operators for 0 < p < 1, Integral Equations Operator Theory, 13 (1990), 307–315.Google Scholar
Anderson, J. H., Derivations, Commutators, and the Essential Numerical Range, Ph.D. dissertation, Indiana University, 1971.Google Scholar
Anderson, J. H., Commutators of compact operators, J. Reine Angew. Math., 291 (1977), 128–132.Google Scholar
Anderson, J. H., Commutators in ideals of trace class operators II, Indiana Univ. Math. J., 35 (1986), 373–378.Google Scholar
Anderson, J. H., A secular equation for the eigenvalues of a diagonal matrix perturbation, Linear Algebra Appl., 246 (1996), 49–70.Google Scholar
Anderson, J. H. and Stampfli, J. G., Commutators and compressions, Israel J. Math., 10 (1971), 433–441.Google Scholar
Anderson, J. H. and Vaserstein, L. N., Commutators in ideals of trace class operators, Indiana Univ. Math. J., 35 (1986), 345–372.Google Scholar
Ando, T., On a pair of commutative contractions, Acta Sci. Math. (Szeged), 24 (1963), 88–90.Google Scholar
Ando, T., Distance to the set of thin operators, preprint, 1972.Google Scholar
Ando, T., Structure of operators with numerical radius one, Acta Sci. Math. (Szeged), 34 (1973), 11–15.Google Scholar
Ando, T., Aluthge transforms and the convex hull of the spectrum of a Hilbert space operator, Oper. Theory Adv. Appl., 160 (2005), 21–39.Google Scholar
Ando, T. and Okubo, K., Induced norms of the Schur multiplier operator, Linear Algebra Appl., 147 (1991), 181–199.CrossRefGoogle Scholar
Antezana, J., Massey, P., and Stojanoff, D., λ-Aluthge transforms and Schatten ideals, Linear Algebra Appl., 405 (2005), 177–199.Google Scholar
Antezana, J., Pujals, E. R., and Stojanoff, D., Iterated Aluthge transforms: a brief survey, Rev. Un. Mat. Argentina, 49 (2008), 29–41.Google Scholar
Antezana, J., Pujals, E. R., and Stojanoff, D., The iterated Aluthge transforms of a matrix converge, Adv. Math., 226 (2011), 1591–1620.Google Scholar
Apostol, C., Fialkow, L. A., Herrero, D. A., and Voiculescu, D., Approximation of Hilbert Space Operators, Pitman, Boston, 1984.Google Scholar
Apostol, T. M. and Mnatsakanian, M. A., Subtangents – an aid to visual calculus, Amer. Math. Monthly, 109 (2002), 525–533.Google Scholar
Arlinskiǐ, Yu. M. and Popov, A. B., On sectorial matrices, Linear Algebra Appl., 370 (2003), 133–146.Google Scholar
Arsene, G. and Gheondea, A., Completing matrix contractions, J. Operator Theory, 7 (1982), 179–189.Google Scholar
Arveson, W. B., Subalgebras of C ∗ -algebras, Acta Math., 123 (1969), 141–224.Google Scholar
Arveson, W. B., Subalgebras of C ∗ -algebras II, Acta Math., 128 (1972), 271–308.CrossRefGoogle Scholar
Arveson, W. B., Notes on extensions of C ∗ -algebras, Duke Math. J., 44 (1977), 329–355.Google Scholar
Audenaert, K. M. R., Variance bound, with an application to norm bounds for commutators, Linear Algebra Appl., 432 (2010), 1126–1143.Google Scholar
Auzinger, W., Sectorial operators and normalized numerical range, Appl. Numer. Math., 45 (2003), 367–388.CrossRefGoogle Scholar
Badea, C. and Cassier, G., Constrained von Neumann inequalities, Adv. Math., 166 (2002), 260–297.Google Scholar
Badea, C., Crouzeix, M., and Delyon, B., Convex domains and K-spectral sets, Math. Z., 252 (2006), 345–365.Google Scholar
Badea, C., Crouzeix, M., and Klaja, H., Spectral sets and operator radii, Bull. London Math. Soc., 50 (2018), 986–996.Google Scholar
Baghdad, A. and Kaadoud, M. C., On the maximal numerical range of a hyponormal operator, Oper. Matrices, 13 (2019), 1163–1171.Google Scholar
Barbier, E., Note sur le problème de l’aiguille et le jeu joint couvert, J. Math. Pures Appl. (2), 5 (1860), 273–286.Google Scholar
Barraa, M. and Boumazgour, M., Inner derivations and norm equality, Proc. Amer. Math. Soc., 130 (2002), 471–476.Google Scholar
Bebiano, N., Lemos, R., da Providência, J., and Soares, G., On generalized numerical ranges of operators on an indefinite inner product space, Linear Multilinear Algebra, 52 (2004), 203–233.Google Scholar
Beckermann, B. and Crouzeix, M., A lenticular version of a von Neumann inequality, Arch. Math. (Basel), 86 (2006), 352–355.Google Scholar
Beckermann, B. and Crouzeix, M., Operators with numerical range in a conic domain, Arch. Math. (Basel), 88 (2007), 547–559.Google Scholar
Berberian, S. K., Approximate proper vectors, Proc. Amer. Math. Soc., 13 (1962), 111–114.Google Scholar
Berberian, S. K., The numerical range of a normal operator, Duke Math. J., 31 (1964),479–483.Google Scholar
Berberian, S. K., A note on operators whose spectrum is a spectral set, Acta Sci. Math. (Szeged), 27 (1966), 201–203.Google Scholar
Berberian, S. K. and Orland, G. H., On the closure of the numerical range of an operator, Proc. Amer. Math. Soc., 18 (1967), 499–503.Google Scholar
Bercovici, H., Operator Theory and Arithmetic in H ∞, Amer. Math. Soc., Providence, 1988.Google Scholar
Bercovici, H., Numerical ranges of operators of class C0, Linear Multilinear Algebra, 50 (2002), 219–222.Google Scholar
Bercovici, H., Foiaş, C., and Tannenbaum, A., A relative Toeplitz–Hausdorff theorem, Oper. Theory Adv. Appl., 71 (1994), 29–32.Google Scholar
Bercovici, H., Foiaş, C., and Tannenbaum, A., On the structured singular value for operators on Hilbert space, in: Feedback Control, Nonlinear Systems, and Complexity, Francis, B. A. and Tannenbaum, A. R., eds., Springer, London, 1995, pp. 11–23.CrossRefGoogle Scholar
Bercovici, H. and Timotin, D., The numerical range of a contraction with finite defect numbers, J. Math. Anal. Appl., 417 (2014), 42–56.Google Scholar
Berg, I. D., An extension of the Weyl-von Neumann theorem to normal operators, Trans. Amer. Math. Soc., 160 (1971), 365–371.Google Scholar
Berg, I. D. and Sims, B., Denseness of operators which attain their numerical radius, J. Austral. Math. Soc. Ser. A, 36 (1984), 130–133.Google Scholar
Berger, C. A., Normal Dilations, Ph.D. dissertation, Cornell University, 1963.Google Scholar
Berger, C. A. and Stampfli, J. G., Mapping theorems for the numerical range, Amer. J. Math., 89 (1967), 1047–1055.Google Scholar
Berger, M., Geometry I, Springer, Berlin, 1987.Google Scholar
Bhatia, R., Horn, R. A., and Kittaneh, F., Normal approximants to binormal operators, Linear Algebra Appl., 147 (1991), 169–179.Google Scholar
Bhatia, R., Kahan, W., and Li, R.-C., Pinchings and norms of scaled triangular matrices, Linear Multilinear Algebra, 50 (2002), 15–21.Google Scholar
Binding, P., Farenick, D. R., and Li, C.-K., A dilation and norm in several variable operator theory, Canad. J. Math., 47 (1995), 449–461.Google Scholar
Birbonshi, R., Spitkovsky, I. M., and Srivastava, P. D., A note on Anderson’s theorem in the infinite-dimensional setting, J. Math. Anal. Appl., 461 (2018), 349–353.Google Scholar
Bohnenblust, H. F. and Karlin, S., Geometrical properties of the unit sphere of Banach algebras, Ann. of Math. (2), 62 (1955), 217–229.Google Scholar
Bonnesen, T., Beviser for nogle saetninger om konvekse kurver, Nyt Tidsskr. Mat., 31 (1920), 47–54.Google Scholar
Bonnesen, T. and Fenchel, W., Theory of Convex Bodies, BCS Associates, Moscow, Idaho, 1987.Google Scholar
Bonsall, F. F. and Duncan, J., Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, Cambridge University Press, London, 1971.Google Scholar
Bonsall, F. F. and Duncan, J., Numerical Ranges II, Cambridge University Press, London, 1973.Google Scholar
Bonsall, F. F. and Duncan, J., Numerical ranges, in: Studies in Functional Analysis, Bartle, R. G., ed., Math. Asso. Amer., Washington, DC, 1980, pp. 1–49.Google Scholar
Bos, H. J. M., Kers, C., Oort, F., and Raven, D. W., Poncelet’s closure theorem, Exposition. Math., 5 (1987), 289–364.Google Scholar
Bouali, S. and Ech-chad, M., Generalized numerical range, Int. J. Math. Anal., 9 (2015), 631–636.Google Scholar
Bouldin, R., The numerical range of a product, J. Math. Anal. Appl., 32 (1970), 459–467.Google Scholar
Bouldin, R., The numerical range of a product, II, J. Math. Anal. Appl., 33 (1971), 212–219.Google Scholar
Bouldin, R., Numerical range for certain classes of operators, Proc. Amer. Math. Soc., 34 (1972), 203–206.Google Scholar
Bouldin, R., The essential minimum modulus, Indiana Univ. Math. J., 30 (1981), 513–517.Google Scholar
Bourbaki, N., Fonctions d’une Variable Réelle, Hermann, Paris, 1958.Google Scholar
Bourdon, P. S. and Shapiro, J. H., The numerical ranges of automorphic composition operators, J. Math. Anal. Appl., 251 (2000), 839–854.Google Scholar
Bourdon, P. S. and Shapiro, J. H., When is zero in the numerical range of a composition operator?, Integral Equations Operator Theory, 44 (2002), 410–441.Google Scholar
Bourin, J.-C., Compressions and pinchings, J. Operator Theory, 50 (2003), 211–220.Google Scholar
Bourin, J.-C., Total dilations, Linear Algebra Appl., 368 (2003), 159–169.Google Scholar
Brickman, L., On the field of values of a matrix, Proc. Amer. Math. Soc., 12 (1961), 61–66.Google Scholar
Brieskorn, E. and Knörrer, H., Plane Algebraic Curves, Birkhäuser, Basel, 1986.Google Scholar
Brown, A. and Halmos, P. R., Algebraic properties of Toeplitz operators, J. Reine Angew. Math., 231 (1963), 89–102.Google Scholar
Brown, A., Halmos, P. R., and Shields, A. L., Cesàro operators, Acta Sci. Math. (Szeged), 26 (1965), 125–137.Google Scholar
Brown, A. and Pearcy, C., Structure of commutators of operators, Ann. of Math. (2), 82 (1965), 112–127.Google Scholar
Brown, E. S. and Spitkovsky, I. M., On matrices with elliptical numerical ranges, Linear Multilinear Algebra, 52 (2004), 177–193.Google Scholar
Brown, E. S. and Spitkovsky, I. M., On flat portions on the boundary of the numerical range, Linear Algebra Appl., 390 (2004), 75–109.Google Scholar
Brown, J. W. and Churchill, R. V., Complex Variables and Applications, 7th ed., McGraw-Hill, Singapore, 2003.Google Scholar
Brown, L. G., Traces of commutators of Schatten–von Neumann class operators, J. Reine Angew. Math., 451 (1994), 171–174.Google Scholar
Bunce, J. W., Shorted operators and the structure of operators with numerical radius one, Integral Equations Operator Theory, 11 (1988), 287–291.Google Scholar
Bunce, J. W. and Salinas, N., Completely positive maps on C ∗ -algebras and the left matricial spectra of an operator, Duke Math. J., 43 (1976), 747–774.Google Scholar
Cain, B. E., Operators which remain convergent when multiplied by certain Hermitian operators, Linear Algebra Appl., 297 (1999), 57–61.CrossRefGoogle Scholar
Cain, B. E., DeAlba, L. M., Hogben, L., and Johnson, C. R., Multiplicative perturbations of stable and convergent operators, Linear Algebra Appl., 268 (1998), 151–169.Google Scholar
Calbeck, W., Elliptic numerical ranges of 3 × 3 companion matrices, Linear Algebra Appl., 428 (2008), 2715–2722.Google Scholar
Calkin, J., Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. (2), 42 (1941), 839–873.Google Scholar
Camenga, K. A., Deaett, L., Rault, P. X., Sendova, T., Spitkovsky, I. M., and Yates, R. B. J., Singularities of base polynomials and Gau–Wu numbers, Linear Algebra Appl., 581 (2019), 112–127.Google Scholar
Camenga, K. A., Rault, P. X., Sendova, T., and Spitkovski, I. M., On the Gau–Wu number for some classes of matrices, Linear Algebra Appl., 444 (2014), 254–262.Google Scholar
Carey, R. W. and Pincus, J. D., Unitary equivalence modulo the trace class for self-adjoint operators, Amer. J. Math., 98 (1976), 481–514.Google Scholar
Cassier, G. and Fack, T., Contractions in von Neumann algebras, J. Funct. Anal., 135 (1996), 297–338.Google Scholar
Cassier, G. and Zerouali, E. H., Operator matrices in class Cρ, Linear Algebra Appl., 420 (2007), 361–376.Google Scholar
Caston, L., Savova, M., Spitkovsky, I., and Zobin, N., On eigenvalues and boundary curvature of the numerical range, Linear Algebra Appl., 322 (2001), 129–140.Google Scholar
Causey, R. L., On Closest Normal Matrices, Ph.D. dissertation, Stanford University, 1964.Google Scholar
Chalendar, I., Gorkin, P., and Partington, J. R., Numerical ranges of restricted shifts and unitary dilations, Oper. Matrices, 3 (2009), 271–281.Google Scholar
Chalendar, I., Gorkin, P., and Partinton, J. R., Determination of inner functions by their value sets on the circle, Comput. Methods Funct. Theory, 11 (2011), 353–373.Google Scholar
Chan, J.-T. and Chan, K., An observation about normaloid operators, Oper. Matrices, 11 (2017), 885–890.Google Scholar
Chan, J.-T., Li, C.-K., and Poon, Y.-T., Closedness of the k-numerical range, Linear Multilinear Algebra, to appear.Google Scholar
Chang, C.-T., Gau, H.-L., and Wang, K.-Z., Equality of higher-rank numerical ranges of matrices, Linear Multilinear Algebra, 62 (2014), 626–638.Google Scholar
Chang, C.-T., Gau, H.-L., Wang, K.-Z., and Wu, P. Y., Numerical ranges and Geršgorin discs, Linear Algebra Appl., 438 (2013), 1170–1192.Google Scholar
Chang, C.-C., Gau, H.-L., Wang, Y.-S., Wu, S.-C., and Yeh, Y.-T., Matrices with defect index one, Oper. Matrices, 7 (2013), 865–885.Google Scholar
Cheung, W.-S. and Li, C.-K., Elementary proofs for some results on the circular symmetry of the numerical range, Linear Multilinear Algebra, 61 (2013), 596–602.Google Scholar
Cheung, W.-S., Li, C.-K., and Rodman, L., Operators with numerical range in a given closed half plane, Taiwanese J. Math., 11 (2007), 471–481.Google Scholar
Cheung, W.-S. and Tam, T.-Y., Multiplicity and numerical range, preprint, 2007.Google Scholar
Cheung, W.-S. and Tsing, N.-K., The C-numerical range of matrices is star-shaped, Linear Multilinear Algebra, 41 (1996), 245–250.Google Scholar
Chien, M.-T., On the numerical range of tridiagonal operators, Linear Algebra Appl., 246 (1996), 203–214.Google Scholar
Chien, M.-T. and Nakazato, H., Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices, Linear Multilinear Algebra, 56 (2008), 143–162.Google Scholar
Chien, M.-T. and Tam, B.-S., Circularity of the numerical range, Linear Algebra Appl., 201 (1994), 113–133.Google Scholar
Chien, M.-T., Yeh, L., Yeh, Y.-T., and Lin, F.-Z., On geometric properties of the numerical range, Linear Algebra Appl., 274 (1998), 389–410.Google Scholar
Chkliar, V., Numerical radii of simple powers, Linear Algebra Appl., 265 (1997), 119–121.Google Scholar
Chō, M., Jung, I. B., and Lee, W. Y., On Aluthge transform of p-hyponormal operators, Integral Equations Operator Theory, 53 (2005), 321–329.Google Scholar
Choi, M.-D., Completely positive linear maps on complex matrices, Linear Algebra Appl., 10 (1975), 285–290.Google Scholar
Choi, M.-D., Kribs, D. W., and Zyczkowski, K., Higher-rank numerical ranges and compression problems, Linear Algebra Appl., 418 (2006), 828–839.Google Scholar
Choi, M.-D. and Li, C.-K., Numerical ranges and dilations, Linear Multilinear Algebra, 47 (2000), 35–48.Google Scholar
Choi, M.-D. and Li, C.-K., Constraint unitary dilations and numerical ranges, J. Operator Theory, 46 (2001), 435–447.Google Scholar
Choi, M.-D. and Li, C.-K., Numerical ranges of the powers of an operator, J. Math. Anal. Appl., 365 (2010), 458–466.Google Scholar
Chuan, W.-F., The unitary equivalence of compact operators, Glasgow Math. J., 26 (1985), 145–149.Google Scholar
Chui, C. K., Smith, P. W., Smith, R. R., and Ward, J. D., L-ideals and numerical range preservation, Illinois J. Math., 21 (1977), 365–373.Google Scholar
Cima, J. A., Thomson, J., and Wogen, W. R., On some properties of composition operators, Indiana Univ. Math. J., 24 (1974), 215–220.Google Scholar
Clancey, K. F., Examples of nonnormal seminormal operators whose spectra are not spectral sets, Proc. Amer. Math. Soc., 24 (1970), 797–800.Google Scholar
Clark, D. N., One dimensional perturbations of restricted shifts, J. D’Analyse Math., 25 (1972), 169–191.Google Scholar
Coburn, L. A., Weyl’s theorem for nonnormal operators, Michigan Math. J., 13 (1966), 285–288.Google Scholar
Conway, J. B., A Course in Functional Analysis, 2nd ed., Springer, New York, 1990.Google Scholar
Conway, J. B., The Theory of Subnormal Operators, Amer. Math. Soc., Providence, 1991.Google Scholar
Coolidge, J. L., A Treatise on Algebraic Plane Curves, Dover, New York, 1959.Google Scholar
Cowen, C. C. and MacCluer, B. D., Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.Google Scholar
Crabb, M. J., The powers of an operator of numerical radius one, Michigan Math. J., 18 (1971), 252–256.Google Scholar
Crabb, M. J., Duncan, J., and McGregor, C. M., Mapping theorems and the numerical radius, Proc. London Math. Soc., 25 (1972), 486–502.Google Scholar
Crabb, M. J., Duncan, J., and McGregor, C. M., Characterizations of commutativity for C ∗ -algebras, Glasgow Math. J., 15 (1974), 172–175.Google Scholar
Crawford, C. R., A stable generalized eigenvalue problem, SIAM J. Numer. Anal., 13 (1976), 854–860.Google Scholar
Crouzeix, M., Bounds for analytic functions of matrices, Integral Equations Operator Theory, 48 (2004), 461–477.Google Scholar
Crouzeix, M., Operators with numerical range in a parabola, Arch. Math. (Basel), 82 (2004), 517–527.Google Scholar
Crouzeix, M., Numerical range and functional calculus in Hilbert space, J. Funct. Anal., 244 (2007), 668–690.Google Scholar
Crouzeix, M., Spectral sets and 3 × 3 nilpotent matrices, in: Topics in Functional and Harmonic Analysis, Badea, C., Li, D., and Petkova, V., eds., Theta Ser. Adv. Math., 14, Theta, Bucharest, 2013, pp. 27–42.Google Scholar
Crouzeix, M. and Delyon, B., Some estimates for analytic functions of strip or sectorial operators, Arch. Math. (Basel), 81 (2003), 559–566.Google Scholar
Crouzeix, M. and Greenbaum, A., Spectral sets: numerical range and beyond, SIAM J.Matrix Anal. Appl., 40 (2019), 1087–1101.Google Scholar
Crouzeix, M. and Palencia, C., The numerical range is a (1 + √2)-spectral set, SIAM J. Matrix Anal. Appl., 38 (2017), 649–655.Google Scholar
Daepp, U., Gorkin, P., and Mortini, R., Ellipses and finite Blaschke products, Amer. Math. Monthly, 109 (2002) 785–795.Google Scholar
Daepp, U., Gorkin, P., Shaffer, A., and Vass, K., Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other, Amer. Math. Soc., Providence, 2018.Google Scholar
Daepp, U., Gorkin, P., and Voss, K., Poncelet’s theorem, Sendov’s conjecture, and Blaschke products, J. Math. Anal. Appl., 365 (2010), 93–102.Google Scholar
Davidson, K. R. and Holbrook, J. A. R., Numerical radii of zero-one matrices, Michigan Math. J., 35 (1988), 261–267.Google Scholar
Davidson, K. R., Paulsen, V. I., and Woerdeman, H. J., Complete spectral sets and numerical range, Proc. Amer. Math. Soc., 146 (2018), 1189–1195.Google Scholar
Davis, C., The shell of a Hilbert-space operator, Acta Sci. Math. (Szeged), 29 (1968), 69–86.Google Scholar
Davis, C., The shell of a Hilbert-space operator. II, Acta Sci. Math. (Szeged), 31 (1970), 301–318.Google Scholar
Delyon, B. and Delyon, F., Generalization of von Neumann’s spectral sets and integral representation of operators, Bull. Soc. Math. France, 127 (1999), 25–41.Google Scholar
DePrima, C. R. and Richard, B. K., A characterization of the positive cone of B(H), Indiana Univ. Math. J., 23 (1973), 163–172.Google Scholar
Dixmier, J., Les fonctionelles linéaires sur l’ensemble des oṕerateurs bornés d’un espace de Hilbert, Ann. of Math. (2), 51 (1950), 387–408.Google Scholar
Djoković, D. Ž., Unitary similarity of projectors, Aequationes Math., 42 (1991), 220– 224.Google Scholar
Dombrowski, J., Spectral properties of real parts of weighted shift operators, Indiana Univ. Math. J., 29 (1980), 249–259.Google Scholar
Donoghue, W. F. Jr., On the numerical range of a bounded operator, Michigan Math. J., 4 (1957), 261–263.Google Scholar
Douglas, R. G., On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), 413–415.Google Scholar
Douglas, R. G. and Pearcy, C., A characterization of thin operators, Acta. Sci. Math. (Szeged), 29 (1968), 295–297.Google Scholar
Dragomir, S. S., Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces, Linear Algebra Appl., 419 (2006), 256–264.Google Scholar
Dragomir, S. S., Reverse inequalities for the numerical radius of linear operators in Hilbert spaces, Bull. Austral. Math. Soc., 73 (2006), 255–262.Google Scholar
Dragomir, S. S., Inequalities for the norm and the numerical radius of linear operators in Hilbert space, Demonstr. Math., 40 (2007), 411–417.Google Scholar
Dragomir, S. S., Inequalities for the numerical radius, the norm and the maximum of the real part of bounded linear operators in Hilbert spaces, Linear Algebra Appl., 428 (2008), 2980–2994.Google Scholar
Dritschel, M. A. and McCullough, S., The failure of rational dilation on a triply connected domain, J. Amer. Math. Soc., 18 (2005), 873–918.Google Scholar
Dritschel, M. A., McCullough, S., and Woerdeman, H. J., Model theory for ρ-contractions, ρ ≤ 2, J. Operator Theory, 41 (1999), 321–350.Google Scholar
Dritschel, M. A. and Woerdeman, H. J., Model Theory and Linear Extreme Points in the Numerical Radius Unit Ball, Mem. Amer. Math. Soc., 129 (1997), no. 615.Google Scholar
Drnovšek, R. and Peperko, A., Inequalities for the Hadamard weighted geometric mean of positive kernel operators on Banach function spaces, Positivity, 10 (2006), 613–626.Google Scholar
Drury, S. W., Symbolic calculus of operators with unit numerical radius, Linear Algebra Appl., 428 (2008), 2061–2069.Google Scholar
Du, H.-K., Dou, Y.-N., Zhang, H.-Y., and Shen, L.-M., A generalization of Gudder–Nagy’s theorem with numerical ranges of operators, J. Math. Phys., 52 (2011), 023501, 9pp.Google Scholar
Duarte, A. L., O Teorema de Tarski e suas Aplicações em Teoria de Matrizes, Ph.D. dissertation, Universidade de Coimbra, 1993.Google Scholar
Durszt, E., On the numerical range of normal operators, Acta Sci. Math. (Szeged), 25 (1964), 262–265.Google Scholar
Eckstein, G., A short proof of a theorem of Sz.-Nagy and Foiaş, Rev. Roumaine Math. Pures Appl., 18 (1973), 349–350.Google Scholar
Eckstein, G. and Rácz, A., Weighted shifts of class Cρ, Acta Sci. Math. (Szeged), 35 (1973), 187–194.Google Scholar
Ed-dari, E., On the numerical index of Banach spaces, Linear Algebra Appl., 403 (2005), 86–96.Google Scholar
Eiermann, M., Fields of values and iterative methods, Linear Algebra Appl., 180 (1993), 167–197.Google Scholar
Embry, M. R., The numerical range of an operator, Pacific J. Math., 32 (1970), 647–650.Google Scholar
Embry, M. R., Similarities involving normal operators on Hilbert space, Pacific J. Math., 35 (1970), 331–336.Google Scholar
Embry, M. R., Classifying special operators by means of subsets associated with the numerical range, Pacific J. Math., 38 (1971), 61–65.Google Scholar
Eoff, C. M. and Kalton, N. J., Zeros of entire functions and characteristic determinants, Complex Variables Theory Appl., 14 (1990), 53–63.Google Scholar
Fan, K., Analytic functions of a proper contraction, Math. Z., 160 (1978), 275–290.Google Scholar
Fan, K., A remark on orthogonality of eigenvectors, Linear Multilinear Algebra, 23 (1988), 283–284.Google Scholar
Fan, K. and Pall, G., Imbedding conditions for Hermitian and normal matrices, Canad. J. Math., 9 (1957), 298–304.Google Scholar
Fan, P., On the diagonal of an operator, Trans. Amer. Math. Soc., 283 (1984), 239–251.Google Scholar
Fan, P., Remark on “Tridiagonal matrix representations of cyclic self-adjoint operators,” Proc. Amer. Math. Soc., 98 (1986), 85–88.Google Scholar
Fan, P. and Fong, C. K., Which operators are the self-commutators of compact operators?, Proc. Amer. Math. Soc., 80 (1980), 58–60.Google Scholar
Fan, P. and Fong, C. K., Operators similar to zero diagonal operators, Proc. Roy. Irish Acad. Sect. A, 87 (1988), 147–153.Google Scholar
Fan, P. and Fong, C. K., An intrinsic characterization for zero-diagonal operators, Proc. Amer. Math. Soc., 121 (1994), 803–805.Google Scholar
Fan, P., Fong, C. K., and Herrero, D. A., On zero-diagonal operators and traces, Proc. Amer. Math. Soc., 99 (1987), 445–451.Google Scholar
Farenick, D. R., The fundamental group of a unitary orbit, Linear Multilinear Algebra, 26 (1990), 1–4.Google Scholar
Farenick, D. R., C ∗ -convexity and matricial ranges, Canad. J. Math., 44 (1992), 280– 297.Google Scholar
Farenick, D. R., Matricial extensions of the numerical range: a brief survey, Linear Multilinear Algebra, 34 (1993), 197–211.Google Scholar
Farenick, D. R., Normal compressions and restrictions to 2-dimensional subspaces, and the Berberian spectral extension, Linear Multilinear Algebra, 37 (1994), 131–138.Google Scholar
Farenick, D. R., Arveson’s criterion for unitary similarity, Linear Algebra Appl., 435 (2011), 769–777.Google Scholar
Farenick, D. R. and Morenz, P. B., C ∗ -extreme points of some compact C ∗ -convex sets, Proc. Amer. Math. Soc., 118 (1993), 765–775.Google Scholar
Farid, F. O., On a conjecture of Hubner, Proc. Indian Acad. Sci. Math. Sci., 109 (1999), 373–378.Google Scholar
Fialkow, L., A note on the operator X → AX − XB, Trans. Amer. Math. Soc., 243 (1978), 147–168.Google Scholar
Fialkow, L., A note on norm ideals and the operator X → AX − XB, Israel J. Math., 32 (1979), 331–348.Google Scholar
Fiedler, M., Geometry of the numerical range of matrices, Linear Algebra Appl., 37 (1981), 81–96.Google Scholar
Fiedler, M., Numerical range of matrices and Levinger’s theorem, Linear Algebra Appl., 220 (1995), 171–180.Google Scholar
Fillmore, P. A., On similarity and the diagonal of a matrix, Amer. Math. Monthly, 76 (1969), 167–169.Google Scholar
Fillmore, P. A., Fong, C. K., and Sourour, A. R., Real parts of quasi-nilpotent operators, Proc. Edinburgh Math. Soc. (2), 22 (1979), 263–269.Google Scholar
Fillmore, P. A., Stampfli, J. G., and Williams, J. P., On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math. (Szeged), 33 (1972), 179–192.Google Scholar
Fillmore, P. A. and Williams, J. P., On operator ranges, Adv. Math., 7 (1971), 254–281.Google Scholar
Fillmore, P. A. and Williams, J. P., Some convexity theorems for matrices, Glasgow Math. J., 12 (1971), 110–117.Google Scholar
Finet, C., Martín, M., and Payá, R., Numerical index and renorming, Proc. Amer. Math. Soc., 131 (2003), 871–877.Google Scholar
Foiaş, C., Unele aplicatii ale mutimilor spectrale, Studii si Cercelari Math., 10 (1959), 365–401.Google Scholar
Foiaş, C., Jung, I. B., Ko, E., and Pearcy, C., Complete contractivity of maps associated with the Aluthge and Duggal transforms, Pacific J. Math., 209 (2003), 249–259.Google Scholar
Foiaş, C. and Williams, J. P., Some remarks on the Volterra operator, Proc. Amer. Math. Soc., 31 (1972), 177–184.Google Scholar
Folland, G. B., Introduction to Partial Differential Equations, Princeton University Press, Princeton, 1995.Google Scholar
Fong, C. K., Diagonals of nilpotent operators, Proc. Edinburgh Math. Soc. (2), 29 (1986), 221–224.Google Scholar
Fong, C. K., Norm estimates related to self-commutators, Linear Algebra Appl., 74 (1986), 151–156.Google Scholar
Fong, C. K. and Sourour, A. R., Sums and products of quasi-nilpotent operators, Proc. Roy. Soc. Edinburgh Sect. A, 99 (1984), 193–200.Google Scholar
Fong, C. K. and Wu, P. Y., Diagonal operators: dilation, sum and product, Acta Sci. Math. (Szeged), 57 (1993), 125–138.Google Scholar
Fong, C. K. and Wu, P. Y., Band-diagonal operators, Linear Algebra Appl., 248 (1996), 185–204.Google Scholar
Fujii, J. I., Fujii, M., Izumino, S., Kubo, F., and Nakamoto, R., Strang’s inequality, Math. Japon., 37 (1992), 479–486.Google Scholar
Fujii, M. and Kubo, F., Buzano’s inequality and bounds for roots of algebraic equations, Proc. Amer. Math. Soc., 117 (1991), 359–362.Google Scholar
Furtano, S. and Johnson, C. R., Spectral variation under congruence for a nonsingular matrix with 0 on the boundary of its field of values, Linear Algebra Appl., 359 (2003), 67–78.Google Scholar
Furuta, T., Some characterizations of convexoid operators, Rev. Roumaine Math. Pures Appl., 18 (1973), 893–900.Google Scholar
Furuta, T., Relations between generalized growth conditions and several classes of convexoid operators, Canad. J. Math., 29 (1977), 1010–1030.Google Scholar
Furuta, T., Applications of polar decompositions of idempotent and 2-nilpotent operators, Linear Multilinear Algebra, 56 (2008), 69–79.Google Scholar
Furuta, T. and Nakamoto, R., Certain numerical radius contraction operators, Proc. Amer. Math. Soc., 29 (1971), 521–524.Google Scholar
Furuta, T. and Nakamoto, R., On the numerical range of an operator, Proc. Japan Acad., 47 (1971), 279–284.Google Scholar
Furuta, T. and Takeda, Z., A characterization of spectraloid operators and its generalization, Proc. Japan Acad., 43 (1967), 599–604.Google Scholar
Gaaya, H., On the numerical radius of the truncated adjoint shift, Extracta Math., 25 (2010), 165–182.Google Scholar
Gaaya, H., A sharpened Schwarz–Pick operatorial inequality for nilpotent operators, Indiana Univ. Math. J., 61 (2012), 223–248.Google Scholar
Gaaya, H., On the higher rank numerical range of the shift operator, J. Math. Sci. Adv. Appl., 13 (2012), 1–19.Google Scholar
Gau, H.-L., Numerical Range of S(φ), Ph.D. dissertation, National Chiao Tung University, 1998.Google Scholar
Gau, H.-L., Elliptic numerical ranges of 4 × 4 matrices, Taiwanese J. Math., 10 (2006), 117–128.Google Scholar
Gau, H.-L., Numerical ranges of reducible companion matrices, Linear Algebra Appl., 432 (2010), 1310–1321.Google Scholar
Gau, H.-L., Huang, C.-Y., and Wu, P. Y., Numerical radius inequalities for square-zero and idempotent operators, Oper. Matrices, 2 (2008), 137–141.Google Scholar
Gau, H.-L. and Lu, Y.-H., Extremality of numerical radii of tensor products of matrices, Linear Algebra Appl., 565 (2019), 82–98.Google Scholar
Gau, H.-L., Li, C.-K., Poon, Y.-T., and Sze, N.-S., Higher rank numerical ranges of normal matrices, SIAM J. Matrix Anal. Appl., 32 (2011), 23–43.Google Scholar
Gau, H.-L., Li, C.-K., and Wu, P. Y., Higher-rank numerical ranges and dilations, J. Operator Theory, 63 (2010), 181–189.Google Scholar
Gau, H.-L., Tsai, M.-C., and Wang, H.-C., Weighted shift matrices: unitary equivalence, reducibility and numerical ranges, Linear Algebra App., 438 (2013), 498–513.Google Scholar
Gau, H.-L. and Wang, K.-Z., Matrix powers with circular numerical range, Linear Algebra App., 603 (2020), 190–211.Google Scholar
Gau, H.-L., Wang, K.-Z., and Wu, P. Y., Numerical radii for tensor products of operators, Integral Equations Operator Theory, 78 (2014), 375–382.Google Scholar
Gau, H.-L., Wang, K.-Z., and Wu, P. Y., Zero-dilation index of a finite matrix, Linear Algebra Appl., 440 (2014), 111–124.Google Scholar
Gau, H.-L., Wang, K.-Z., and Wu, P. Y., Numerical radii for tensor products of matrices, Linear Multilinear Algebra, 63 (2015), 1916–1936.Google Scholar
Gau, H.-L., Wang, K.-Z., and Wu, P. Y., Circular numerical ranges of partial isometries, Linear Multilinear Algebra, 64 (2016), 14–35.Google Scholar
Gau, H.-L., Wang, K.-Z., and Wu, P. Y., Numerical ranges of row stochastic matrices, Linear Algebra Appl., 506 (2016), 478–505.Google Scholar
Gau, H.-L., Wang, K.-Z., and Wu, P. Y., Constant norms and numerical radii of matrix powers, Oper. Matrices, 13 (2019), 1035–1054.Google Scholar
Gau, H.-L., Wang, K.-Z., and Wu, P. Y., Equality of numerical ranges of matrix powers, Linear Algebra Appl., 578 (2019), 95–110.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical range of S(φ), Linear Multilinear Algebra, 45 (1998), 49–73.Google Scholar
Gau, H.-L. and Wu, P. Y., Dilation to inflations of S(φ), Linear Multilinear Algebra, 45 (1998), 109–123.Google Scholar
Gau, H.-L. and Wu, P. Y., Lucas’ theorem refined, Linear Multilinear Algebra, 45 (1999), 359–373.Google Scholar
Gau, H.-L. and Wu, P. Y., Finite Blaschke products of contractions, Linear Algebra Appl., 368 (2003), 359–370.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical range and Poncelet property, Taiwanese J. Math., 7 (2003), 173–193.Google Scholar
Gau, H.-L. and Wu, P. Y., Condition for the numerical range to contain an elliptic disc, Linear Algebra Appl., 364 (2003), 213–222.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical range circumscribed by two polygons, Linear Algebra Appl., 382 (2004), 155–170.Google Scholar
Gau, H.-L. and Wu, P. Y., Companion matrices: reducibility, numerical ranges and similarity to contractions, Linear Algebra Appl., 383 (2004), 127–142.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical range of a normal compression, Linear Multilinear Algebra, 52 (2004), 195–201.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical range of a normal compression II, Linear Algebra Appl., 390 (2004), 121–136.Google Scholar
Gau, H.-L. and Wu, P. Y., Anderson’s theorem for compact operators, Proc. Amer. Math. Soc., 134 (2006), 3159–3162.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical ranges of companion matrices, Linear Algebra Appl., 421 (2007), 202–218.Google Scholar
Gau, H.-L. and Wu, P. Y., Line segments and elliptic arcs on the boundary of a numerical range, Linear Multilinear Algebra, 56 (2008), 131–142.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical ranges of nilpotent operators, Linear Algebra Appl., 429 (2008), 716–726.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical ranges of KMS matrices, Acta Sci. Math. (Szeged), 79 (2013), 583–610.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical ranges and compressions of S n -matrices, Oper. Matrices, 7 (2013), 465–476.Google Scholar
Gau, H.-L. and Wu, P. Y., Higher-rank numerical ranges and Kippenhahn polynomials, Linear Algebra Appl., 438 (2013), 3054–3061.Google Scholar
Gau, H.-L. and Wu, P. Y., Excursions in numerical ranges, Bull. Inst. Math. Acad. Sin. (N. S.), 9 (2014), 351–370.Google Scholar
Gau, H.-L. and Wu, P. Y., Zero-dilation indices of KMS matrices, Ann. Funct. Anal., 5 (2014), 30–35.Google Scholar
Gau, H.-L. and Wu, P. Y., Extremality of numerical radii of matrix products, Linear Algebra Appl., 501 (2016), 17–36.Google Scholar
Gau, H.-L. and Wu, P. Y., Zero-dilation index of Sn -matrix and companion matrix, Electronic J. Linear Algebra, 31 (2016), 666–678.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical radius of Hadamard product of matrices, Linear Algebra Appl., 504 (2016), 292–308.Google Scholar
Gau, H.-L. and Wu, P. Y., Numerical ranges of products of two positive contractions, J. Math. Anal. Appl., 455 (2017), 939–946.Google Scholar
Gau, H.-L. and Wu, P. Y., Operators with real parts at least −1/2, Linear Multilinear Algebra, 65 (2017), 1988–1999.Google Scholar
Gau, H.-L. and Wu, P. Y., Lower bounds for the numerical radius, Oper. Matrices, 11 (2017), 999–1014.Google Scholar
Gau, H.-L. and Wu, P. Y., Equality of three numerical radius inequalities, Linear Algebra Appl., 544 (2018), 51–67.Google Scholar
Gevorgyan, L., On operators with large self-commutators, Oper. Matrices, 4 (2010), 119–125.Google Scholar
Gevorgyan, L., Characterization of spectraloid and normaloid operators, Dokl. Nat. Akad. Nauk Armen., 113 (2013), 231–239.Google Scholar
Gevorgyan, L., On the transcendental radius of the Volterra integration operator, Ann. Funct. Anal., 6 (2015), no. 1, 54–58.Google Scholar
Gilfeather, F., Seminormal operators, Michigan Math. J., 18 (1971), 235–242.Google Scholar
Goldberg, M. and Straus, E. G., On a theorem by Mirman, Linear Multilinear Algebra, 5 (1977), 77–78.Google Scholar
Goldberg, M. and Straus, E. G., Elementary inclusion relations for generalized numerical ranges, Linear Algebra Appl., 18 (1977), 1–24.Google Scholar
Goldberg, M. and Straus, E. G., Norm properties of C-numerical radii, Linear Algebra Appl., 24 (1979), 113–131.Google Scholar
Goldberg, M. and Tadmor, E., On the numerical radius and its applications, Linear Algebra Appl., 42 (1982), 263–284.Google Scholar
Goldberg, M., Tadmor, E., and Zwas, G., The numerical radius and spectral matrices, Linear Multilinear Algebra, 2 (1975), 317–326.Google Scholar
Gomilko, A., Wróbel, I., and Zemánek, J., Numerical ranges in a strip, in: Operator Theory 20, Davidson, K. R., Gaşpar, D., Strătilă, Ş., Timotin, D., and Vasilescu, F.-H., eds., Theta Ser. Adv. Math., 6, Theta, Bucharest, 2006, pp. 111–121.Google Scholar
Greenbaum, A. and Overton, M. L., Numerical investigation of Crouzeix’s conjecture, Linear Algebra Appl., 542 (2018), 225–245.Google Scholar
Gudder, S. and Nagy, G., Sequential quantum measurements, J. Math. Phys., 42 (2001), 5212.Google Scholar
Gustafson, K. E. and Rao, D. K. M., Numerical Range, The Field of Values of Linear Operators and Matrices, Springer, New York, 1997.Google Scholar
Gutkin, E., The Toeplitz–Hausdorff theorem revisited: relating linear algebra and geometry, Math. Intelligencer, 26 (2004), no. 1, 8–14.Google Scholar
Guyker, J., On reducing subspaces of composition operators, Acta Sci. Math. (Szeged), 53 (1989), 369–376.Google Scholar
Haagerup, U. and de la Harpe, P., The numerical radius of a nilpotent operator on a Hilbert space, Proc. Amer. Math. Soc., 115 (1992), 371–379.Google Scholar
Hadwin, D. W., Completely positive maps and approximate equivalence, Indiana Univ. Math. J., 36 (1987), 211–228.Google Scholar
Hadwin, D. W., Harrison, K. J., and Ward, J. A., Numerical ranges and matrix completions, Linear Algebra Appl., 315 (2000), 145–154.Google Scholar
Halmos, P. R., Numerical ranges and normal dilations, Acta Sci. Math. (Szeged), 25 (1964), 1–5.Google Scholar
Halmos, P. R., Two subspaces, Trans. Amer. Math. Soc., 144 (1969), 381–389.Google Scholar
Halmos, P. R., Limits of shifts, Acta Sci. Math. (Szeged), 34 (1973), 131–139.Google Scholar
Halmos, P. R., A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982.Google Scholar
Hamed, A. N. and Spitkovsky, I. M., On the maximal numerical range of some matrices, Electron. J. Linear Algebra, 34 (2018), 288–303.Google Scholar
Hara, T., The structure of the core of a numerical contraction, Integral Equations Operator Theory, 23 (1995), 179–204.Google Scholar
Harris, T. R., Mazzella, M., Patton, L. J., Renfrew, D., and Spitkovsky, I. M., Numerical ranges of cube roots of the identity, Linear Algebra Appl., 435 (2011), 2639–2657.Google Scholar
Harte, R., Kim, Y. O., and Lee, W. Y., Spectral pictures of AB and BA, Proc. Amer. Math. Soc., 134 (2006), 105–110.Google Scholar
Hartman, J., A hyponormal weighted shift whose spectrum is not a spectral set, J. Operator Theory, 8 (1982), 401–403.Google Scholar
Hausdorff, F., Der Wertevorrat einer Bilinearform, Math. Z., 3 (1919), 314–316.Google Scholar
Heatherly, H., A characterization of infinite dimension for vector spaces, Math. Mag., 6 (1988), 239–242.Google Scholar
Heinz, E., Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann., 123 (1951), 415–438.Google Scholar
Helton, J. and Howe, R., Traces of commutators of integral operators, Acta Math., 135 (1975), 271–305.Google Scholar
Helton, J. W. and Spitkovsky, I. M., The possible shapes of numerical ranges, Oper. Matrices, 6 (2012), 607–611.Google Scholar
Helton, J. W. and Vinnikov, V., Linear matrix inequality representation of sets, Comm. Pure Appl. Math., 60 (2007), 654–674.Google Scholar
Henrion, D., Detecting rigid convexity of bivariate polynomials, Linear Algebra Appl., 432 (2010), 1218–1233.Google Scholar
Herrero, D. A., What Fσ sets can be numerical ranges of operators?, Rev. Un. Mat. Argentina, 31 (1983), 40–43.Google Scholar
Herrero, D. A., The diagonal entries of a Hilbert space operator, Rocky Mountain J. Math., 21 (1991), 857–864.Google Scholar
Herzog, G. and Schmoeger, C., A note on Gâteaux differentials of hermitian elements in Banach algebras, Demonstr. Math., 41 (2008), 909–913.Google Scholar
Hildebrandt, S., Über den numerischen Wertebereich eines operators, Math. Ann., 163 (1966), 230–247.Google Scholar
Hoffman, K., Banach Spaces of Analytic Functions, Prentice Hall, Englewood Cliffs, 1962.Google Scholar
Holbrook, J. A. R., Multiplicative properties of the numerical radius in operator theory, J. Reine Angew. Math., 237 (1969), 166–174.Google Scholar
Holbrook, J. A. R., Operators similar to contractions, Acta Sci. Math. (Szeged), 34 (1973), 163–168.Google Scholar
Holbrook, J. A. R., Inequalities of von Neumann type for small matrices, in: Function Spaces, Jarosz, K., ed., Marcel Dekker, New York, 1992, pp. 189–193.Google Scholar
Holbrook, J. A. R., Schur norms and the multivariate von Neumann inequality, Oper. Theory Adv. Appl., 127 (2001), 375–386.Google Scholar
Holbrook, J. A. R. and Schoch, J.-P., Theory vs. experiment: multiplicative inequalities for the numerical radius of commuting matrices, Oper. Theory Adv. Appl., 202 (2010), 273–284.Google Scholar
Hopenwasser, A., Moore, R. L., and Paulsen, V. I., C ∗ -extreme points, Trans. Amer. Math. Soc., 266 (1981), 291–307.Google Scholar
Horn, R. A. and Johnson, C. R., Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1994.Google Scholar
Horn, R. A. and Johnson, C. R., Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013.Google Scholar
Hou, J. and Di, Q., Maps preserving numerical ranges of operator products, Proc. Amer. Math. Soc., 134 (2006), 1435–1446.Google Scholar
Hou, J. C. and Du, H. K., Norm inequalities of positive operator matrices, Integral Equations Operator Theory, 22 (1995), 281–294.Google Scholar
Hübner, M., Spectrum where the boundary of the numerical range is not round, Rocky Mountain J. Math., 25 (1995), 1351–1355.Google Scholar
Hung, C.-N., The Numerical Range and the Core of Hilbert-Space Operators, Ph.D. dissertation, University of Toronto, 2004.Google Scholar
Ikramov, Kh. D., Congruent reduction of a matrix to quasidiagonal form, Dokl. Akad. Nauk, 382 (2002), 589–592; English translation in Dokl. Math., 65 (2002), 80–82.Google Scholar
Isaev, G. A. and Kuliev, T. Yu., Analytical properties of the boundary of the numerical range of operators in Hilbert space, Soviet Math. Dokl., 34 (1987), 569–572.Google Scholar
Issos, J. N., The Field of Values of Non-Negative Irreducible Matrices, Ph.D. dissertation, Auburn University, 1966.Google Scholar
Ito, M., Nakazato, H., Okubo, K., and Yamazaki, T., On generalized numerical range of the Aluthge transformation, Linear Algebra Appl., 370 (2003), 147–161.Google Scholar
Ji, G., Liu, N., and Li, Z., Essential numerical range and maximal numerical range of the Aluthge transform, Linear Multilinear Algebra, 55 (2007), 315–322.Google Scholar
Johnson, C. R., A Geršgorin inclusion set for the field of values of a finite matrix, Proc. Amer. Math. Soc., 41 (1973), 57–60.Google Scholar
Johnson, C. R., Functional characterization of the field of values and the convex hull of the spectrum, Proc. Amer. Math. Soc., 61 (1976), 201–204.Google Scholar
Johnson, C. R., Normality and the numerical range, Linear Algebra Appl., 15 (1976), 89–94.Google Scholar
Johnson, C. R., An inclusion region for the field of values of a doubly stochastic matrix based on its graph, Aequationes Math., 17 (1978), 305–310.Google Scholar
Johnson, C. R., Numerical ranges of principal submatrices, Linear Algebra Appl., 37 (1981), 23–34.Google Scholar
Johnson, C. R. and Spitkovsky, I. M., Factorization of operators with angularly constrained spectra, Oper. Theory Adv. Appl., 62 (1993), 125–143.Google Scholar
Johnson, C. R., Spitkovsky, I., and Gottlieb, S., Inequalities involving the numerical radius, Linear Multilinear Algebra, 37 (1994), 13–24.Google Scholar
Jung, I. B., Ko, E., and Pearcy, C., Aluthge transforms of operators, Integral Equations Operator Theory, 37 (2000), 437–448.Google Scholar
Kaadoud, M. C., Domaine numérique de l’opérateur produit M2,A,B et de la dérivation généralisée δ2,A,B, Extracta Math., 17 (2002), 59–68.Google Scholar
Kaadoud, M. C., Géométrie du spectre dans une algèbre de Banach et domaine numérique, Studia Math., 162 (2004), 1–14.Google Scholar
Kac, M., Murdock, W. L., and Szegő, G., On the eigenvalues of certain Hermitian forms, J. Rational Mech. Anal., 2 (1953), 767–800.Google Scholar
Kadison, R. V. and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras, vol. I, Academic Press, New York, 1983.Google Scholar
Kadison, R. V. and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras, vol. II, Academic Press, Orlando, 1986.Google Scholar
Kahan, W., Every n × n matrix Z with real spectrum satisfies ‖Z − Z ∗ ‖ ≤ ‖Z + Z ∗ ‖(log2 n + 0.038), Proc. Amer. Math. Soc., 39 (1973), 235–241.Google Scholar
Kalton, N. J., Trace-class operators and commutators, J. Funct. Anal., 86 (1989), 41–74.Google Scholar
Kato, T., Some mapping theorems for the numerical range, Proc. Japan Acad., 41 (1965), 652–655.Google Scholar
Kato, T., Perturbation Theory for Linear Operators, Springer, Berlin, 1995.Google Scholar
Keeler, D. S., Rodman, L., and Spitkovsky, I. M., The numerical range of 3 × 3 matrices, Linear Algebra Appl., 252 (1997), 115–139.Google Scholar
Kérchy, L., Uniqueness of the numerical range of truncated shifts, Acta Sci. Math. (Szeged), 83 (2017), 243–261.Google Scholar
Kérchy, L., Numerical ranges of model operators and Clark measures, J. Math. Anal. Appl., 491 (2020), 124281.Google Scholar
Khalil, R., On a theorem of Weiss, Arch. Math., 65 (1995), 69–70.Google Scholar
King, J. L., Three problems in search of a measure, Amer. Math. Monthly, 101 (1994), 609–628.Google Scholar
Kippenhahn, R., Über den Wertevorrat einer Matrix, Math. Nachr., 6 (1951), 193–228; English translation: P. F. Zachlin and M. E. Hochstenbach, On the numerical range of a matrix, Linear Multilinear Algebra, 56 (2008), 185–225.Google Scholar
Kirwan, F., Complex Algebraic Curves, Cambridge University Press, Cambridge, 1992.Google Scholar
Kittaneh, F., On zero-trace commutators, Bull. Austral. Math. Soc., 34 (1986), 119–126.Google Scholar
Kittaneh, F., Some trace class commutators of trace zero, Proc. Amer. Math. Soc., 113 (1991), 655–661.Google Scholar
Kittaneh, F., Commutator inequalities associated with the polar decomposition, Proc. Amer. Math. Soc., 130 (2002), 1279–1283.Google Scholar
Kittaneh, F., Norm inequalities for sums of positive operators, J. Operator Theory, 48 (2002), 95–103.Google Scholar
Kittaneh, F., A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math., 158 (2003), 11–17.Google Scholar
Kittaneh, F., Norm inequalities for sums and differences of positive operators, Linear Algebra Appl., 383 (2004), 85–91.Google Scholar
Kittaneh, F., Numerical radius inequalities for Hilbert space operators, Studia Math., 168 (2005), 73–80.Google Scholar
Klaja, H., Mashreghi, J., and Ransford, T., On mapping theorems for numerical range, Proc. Amer. Math. Soc., 144 (2016), 3009–3018.Google Scholar
Klein, E. M., The numerical range of a Toeplitz operator, Proc. Amer. Math. Soc., 35 (1972), 101–103.Google Scholar
Knowles, G. J., Numerical ranges and reduction theory, J. London Math. Soc., 29 (1984), 164–170.Google Scholar
Komal, B. S. and Sharma, S., Numerical ranges of composition operators on ℓ 2, Hokkaido Math. J., 35 (2006), 1–12.Google Scholar
Kriete, T. L. and Trutt, D., The Cesàro operator in ℓ2 is subnormal, Amer. J. Math., 93 (1971), 215–225.Google Scholar
Kuliev, T. Yu., Filtered products, direct integrals, and convex analysis of numerical ranges, Math. Notes, 48 (1990), 653–658.Google Scholar
Kutkut, M., Weak closure of the unitary orbit of contractions, Acta Math. Hungar., 46 (1985), 255–263.Google Scholar
Kyle, J., Ws (T ) is convex, Pacific J. Math., 72 (1977), 483–485.Google Scholar
Kyle, J., Numerical ranges of derivations, Proc. Edinburgh Math. Soc. (2), 21 (1978), 33–39.Google Scholar
Lancaster, J. S., The boundary of the numerical range, Proc. Amer. Math. Soc., 49 (1975), 393–398.Google Scholar
Lancaster, P. and Tismenetsky, M., The Theory of Matrices with Applications, 2nd ed., Academic Press, New York, 1985.Google Scholar
Langer, H., Über die Wurzeln eines maximalen dissipativen Operators, Acta Math. Acad. Sci. Hunger., 13 (1962), 415–424.Google Scholar
Lauric, V. and Pearcy, C. M., Trace-class commutators with trace zero, Acta Sci. Math. (Szeged), 66 (2000), 341–349.Google Scholar
Lax, P. D., Differential equations, difference equations and matrix theory, Comm. Pure Appl. Math., 11 (1958), 175–194.Google Scholar
Lax, P. and Wendroff, B., Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math., 17 (1964), 381–398.Google Scholar
Lebow, A., On von Neumann’s theory of spectral sets, J. Math. Anal. Appl., 7 (1963), 64–90.Google Scholar
Lee, H.-Y., Diagonals and numerical ranges of direct sums of matrices, Linear Algebra Appl., 439 (2013), 2584–2597.Google Scholar
Lee, H.-Y., Gau–Wu numbers of nonnegative matrices, Linear Algebra Appl., 469 (2015), 594–608.Google Scholar
Lenard, A., The numerical range of a pair of projections, J. Funct. Anal., 10 (1972), 410–423.Google Scholar
Lewis, A. S., Parrilo, P. A., and Ramana, M. V., The Lax conjecture is true, Proc. Amer. Math. Soc., 133 (2005), 2495–2499.Google Scholar
Li, C.-K., The C-convex matrices, Linear Multilinear Algebra, 21 (1987), 303–312.Google Scholar
Li, C.-K., C-numerical ranges and C-numerical radii, Linear Multilinear Algebra, 37 (1994), 51–82.Google Scholar
Li, C.-K., A simple proof of the elliptical range theorem, Proc. Amer. Math. Soc., 124 (1996), 1985–1986.Google Scholar
Li, C.-K., Some convexity theorems for the generalized numerical ranges, Linear Multilinear Algebra, 40 (1996), 235–240.Google Scholar
Li, C.-K., A survey on linear preservers of numerical ranges and radii, Taiwanese J. Math., 5 (2001), 477–496.Google Scholar
Li, C.-K. and Mathias, R., Distances from a Hermitian pair to diagonalizable and nondiagonalizable Hermitian pairs, SIAM J. Matrix Anal. Appl., 28 (2006), 301–305.Google Scholar
Li, C.-K., Mehta, P. P., and Rodman, L., A generalized numerical range: the range of a constrained sesquilinear form, Linear Multilinear Algebra, 37 (1994), 25–49.Google Scholar
Li, C.-K. and Nakazato, H., Some results on the q-numerical range, Linear Multilinear Algebra, 43 (1998), 385–409.Google Scholar
Li, C.-K. and Poon, Y.-T., Convexity of the joint numerical range, SIAM J. Matirx Anal. Appl., 21 (1999), 668–678.Google Scholar
Li, C.-K. and Poon, Y.-T., The joint essential numerical range of operators: convexity and related results, Studia Math., 194 (2009), 91–104.Google Scholar
Li, C.-K. and Poon, Y.-T., Spectrum, numerical range and Davis–Wielandt shell of a normal operator, Glasgow Math. J., 51 (2009), 91–100.Google Scholar
Li, C.-K. and Poon, Y.-T., Generalized numerical ranges and quantum error correction, J. Operator Theory, 66 (2011), 335–351.Google Scholar
Li, C.-K. and Poon, Y.-T., Submultiplicativity of the numerical radius of commuting matrices of order two, J. Math. Anal. Appl., 475 (2019), 730–735.Google Scholar
Li, C.-K., Poon, Y.-T., and Sze, N.-S., Davis–Wielandt shells of operators, Oper. Matrices, 2 (2008), 341–355.Google Scholar
Li, C.-K., Poon, Y.-T., and Sze, N.-S., Higher rank numerical ranges and low rank permutations of quantum channels, J. Math. Anal. Appl., 348 (2008), 843–855.Google Scholar
Li, C.-K., Poon, Y.-T., and Sze, N.-S., Condition for the higher rank numerical range to be non-empty, Linear Multilinear Algebra, 57 (2009), 365–368.Google Scholar
Li, C.-K., Poon, Y.-T., and Sze, N.-S., Elliptical range theorems for generalized numerical ranges of quadratic operators, Rocky Mountain J. Math., 41 (2011), 813–832.Google Scholar
Li, C.-K., Spitkovsky, I., and Shukla, S., Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils, Linear Algebra Appl., 270 (1998), 323–349.Google Scholar
Li, C.-K. and Sze, N.-S., Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations, Proc. Amer. Math. Soc., 136 (2008), 3013–3023.Google Scholar
Li, C.-K., Tam, B.-S., and Wu, P. Y., The numerical range of a nonnegative matrix, Linear Algebra Appl., 350 (2002), 1–23.Google Scholar
Li, C.-K. and Tsing, N.-K., On the unitarily invariant norms and some related results, Linear Multilinear Algebra, 20 (1987), 107–119.Google Scholar
Li, C.-K. and Tsing, N.-K., Norms that are invariant under unitary similarities and the C-numerical radii, Linear Multilinear Algebra, 24 (1989), 209–222.Google Scholar
Li, C.-K. and Tsing, N.-K., Matrices with circular symmetry on their unitary orbits and C-numerical ranges, Proc. Amer. Math. Soc., 111 (1991), 19–28.Google Scholar
Li, C.-K. and Tsing, N.-K., On the kth matrix numerical range, Linear Multilinear Algebra, 28 (1991), 229–239.Google Scholar
Linden, H., Containment regions for zeros of polynomials from numerical ranges of companion matrices, Linear Algebra Appl., 350 (2002), 125–145.Google Scholar
Littlewood, J. E., On inequalities in the theory of functions, Proc. London Math. Soc., 23 (1925), 481–519.Google Scholar
Liu, X.-M., Yang, X.-B., and Du, H.-K., Numerical range of the Aluthge transform, Acta Anal. Funct. Appl., 7 (2005), 193–197.Google Scholar
Loebl, R. I. and Paulsen, V. I., Some remarks on C ∗ -convexity, Linear Algebra Appl., 35 (1981), 63–78.Google Scholar
Luecke, G. R., A class of operators on Hilbert space, Pacific J. Math., 41 (1972), 153–156.Google Scholar
Lumer, G., Semi-inner-product spaces, Trans. Amer. Math. Soc., 100 (1961), 29–43.Google Scholar
Lumer, G. and Rosenblum, M., Linear operator equations, Proc. Amer. Math. Soc., 10 (1959), 32–41.Google Scholar
Macaev, V. I. and Palant, Ju. A., On the powers of a bounded dissipative operator, Ukrain. Math. Zh., 14 (1962), 329–337.Google Scholar
Magajna, B., C ∗ -convexity and the numerical range, Canad. Math. Bull., 43 (2000), 193–207.Google Scholar
Makai, E. Jr. and Zemánek, J., The surjectivity radius, packing numbers and boundedness below of linear operators, Integral Equations Operator Theory, 6 (1983), 372–384.Google Scholar
Marcus, M. and Moyls, B. N., Field convexity of a square matrix, Proc. Amer. Math. Soc., 6 (1955), 981–985.Google Scholar
Marcus, M. and Sandy, M., Conditions for the generalized numerical range to be real, Linear Algebra Appl., 71 (1985), 219–239.Google Scholar
Marcus, M. and Sandy, M., Symmetry properties of higher numerical ranges, Linear Algebra Appl., 104 (1988), 141–164.Google Scholar
Marden, M., Geometry of Polynomials, 2nd ed., Amer. Math. Soc., Providence, 1966.Google Scholar
Maroulas, J., Psarrakos, P. J., and Tsatsomeros, M. J., Perron–Frobenius type results on the numerical range, Linear Algebra Appl., 348 (2002), 49–62.Google Scholar
Marshall, A. W. and Olkin, I., Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.Google Scholar
Martín, M., A survey on the numerical index of Banach space, Extracta Math., 15 (2000), 265–276.Google Scholar
Martín, M. and Payá, R., Numerical index of vector-valued function spaces, Studia Math., 142 (2000), 269–280.Google Scholar
Martínez-Finkelshtein, A., Simanek, B., and Simon, B., Poncelet’s theorem, paraorthog-onal polynomials and the numerical range of compressed multiplication operators, Adv. Math., 349 (2019), 992–1035.Google Scholar
Matache, V., Numerical ranges of composition operators, Linear Algebra Appl., 331 (2001), 61–74.Google Scholar
Matache, V., Numerical ranges of composition operators with inner symbols, Rocky Mountain J. Math., 42 (2012), 235–249.Google Scholar
Mathias, R., Matrix completions, norms, and Hadamard products, Proc. Amer. Math. Soc., 117 (1993), 905–918.Google Scholar
Meng, C.-H., A condition that a normal operator have a closed numerical range, Proc. Amer. Math. Soc., 8 (1957), 85–88.Google Scholar
Meng, C.-H., On the numerical range of an operator, Proc. Amer. Math. Soc., 14 (1963), 167–171.Google Scholar
Militzer, E., Patton, L. J., Spitkovsky, I. M., and Tsai, M.-C., Numerical ranges of 4-by-4 nilpotent matrices: flat portions on the boundary, Oper. Theory Adv. Appl., 259 (2017), 561–591.Google Scholar
Millman, R. S. and Parker, G. D., Elements of Differential Geometry, Prentice Hall, Englewood Cliffs, 1977.Google Scholar
Mirman, B., Numerical range and norm of a linear operator, Voronež Trudy Seminara po Funkcional’nomu Analizu, 10 (1968), 51–55.Google Scholar
Mirman, B., Numerical ranges and Poncelet curves, Linear Algebra Appl., 281 (1998), 59–85.Google Scholar
Mirman, B., UB-matrices and conditions for Poncelet polygon to be closed, Linear Algebra Appl., 360 (2003), 123–150.Google Scholar
Mirman, B., Borovikov, V., Ladyzhensky, L., and Vinograd, R., Numerical ranges, Poncelet curves, invariant measures, Linear Algebra Appl., 329 (2001), 61–75.Google Scholar
Mirman, B. and Shukla, P., A characterization of complex plane Poncelet curve, Linear Algebra Appl., 408 (2005), 86–119.Google Scholar
Moyls, B. N. and Marcus, M., Field convexity of a square matrix, Proc. Amer. Math. Soc., 6 (1955), 981–985.Google Scholar
Müller, V., The numerical radius of a commuting product, Michigan Math. J., 35 (1988), 255–260.Google Scholar
Munkres, J. R., Topology, 2nd ed., Prentice Hall, Upper Saddle River, 2000.Google Scholar
Murnaghan, F. D., On the field of values of a square matrix, Proc. Nat. Acad. Sci. U.S.A., 18 (1932), 246–248.Google Scholar
Naimark, M. A., Positive definite operator functions on a commutative group, Izv. Akad. Nauk SSSR Ser. Mat., 7 (1943), 237–244.Google Scholar
Naimark, M. A., On a representation of additive operator set functions, C. R. (Doklady) Acad. Sci. URSS, 41 (1943), 359–361.Google Scholar
Nakamura, Y., Numerical range and norm, Math. Japon., 27 (1982), 149–150.Google Scholar
Nakamura, Y., Saito, K.-S., and Sakaba, K., The numerical index of nonselfadjoint operator algebras, Proc. Amer. Math. Soc., 117 (1993), 1105–1107.Google Scholar
Nakazato, H., The C-numerical range of a 2 × 2 matrix, Sci. Rep. Hirosaki Univ., 41 (1994), 197–206.Google Scholar
Nakazato, H., On unitary invariants of 3 by 3 matrices, Sci. Rep. Hirosaki Univ., 42 (1995), 179–182.Google Scholar
Nakazato, H., Quartic curves associated to 4 by 4 matrices, Sci. Rep. Hirosaki Univ., 43 (1996), 209–221.Google Scholar
Nakazato, H., k-numerical range and the Minkowski sum, Bull. Fac. Sci. Technol. Hirosaki Univ., 9 (2006), 1–7.Google Scholar
Nakazato, H., Nishikawa, Y., and Takaguchi, M., On the boundary of the C-numerical range of a matrix, Linear Multilinear Algebra, 39 (1995), 231–240.Google Scholar
Nakazi, T. and Okubo, K., ρ-contraction and 2 × 2 matrix, Linear Algebra Appl., 283 (1998), 165–169.Google Scholar
Nakazi, T. and Takahashi, K., Two-dimensional representations of uniform algebras, Proc. Amer. Math. Soc., 123 (1995), 2777–2784.Google Scholar
Namba, M., Geometry of Projective Algebraic Curves, Marcel Dekker, New York, 1984.Google Scholar
Narcowich, F. J., Analytic properties of the boundary of the numerical range, Indiana Univ. Math. J., 29 (1980), 67–77.Google Scholar
Narcowich, F. J. and Ward, J. D., A Toeplitz–Hausdorff theorem for matrix ranges, J. Operator Theory, 6 (1981), 87–101.Google Scholar
Newburgh, J. D., The variation of spectra, Duke Math. J., 18 (1951), 165–176.Google Scholar
Nordgren, E. A., Composition operators, Canad. J. Math., 20 (1968), 442–449.Google Scholar
Nylen, P. and Tam, T.-Y., Numerical range of a doubly stochastic matrix, Linear Algebra Appl., 153 (1991), 161–176.Google Scholar
Okayasu, T. and Ueta, Y., Estimates for moduli of coefficients of positive trigonometric polynomials, Sci. Math. Jpn., 56 (2002), 115–122.Google Scholar
Okubo, K., On weakly unitarily invariant norm and the Aluthge transformation, Linear Algebra Appl., 371 (2003), 369–375.Google Scholar
Okubo, K. and Ando, T., Constants related to operators of class Cρ, Manuscripta Math., 16 (1975), 385–394.Google Scholar
Okubo, K. and Ando, T., Operator radii of commuting products, Proc. Amer. Math. Soc., 56 (1976), 203–210.Google Scholar
Okubo, K. and Spitkovsky, I., On the characterization of 2 × 2 ρ-contraction matrices, Linear Algebra Appl., 325 (2001), 177–189.Google Scholar
Orland, G. H., On a class of operators, Proc. Amer. Math. Soc., 15 (1964), 75–79.Google Scholar
Parker, W. V., Sets of complex numbers associated with a matrix, Duke Math. J., 15 (1948), 711–715.Google Scholar
Patel, S. M. and Yamazaki, T., On comparisons of norms and the numerical ranges of an operator with its generalized Aluthge transformation, Sci. Math. Jpn., 62 (2005), 349–361.Google Scholar
Paulsen, V. I., Completely Bounded Maps and Operator Algebras, Cambridge University Press, Cambridge, 2002.Google Scholar
Pearcy, C., An elementary proof of the power inequality for the numerical radius, Michigan Math. J., 13 (1966), 289–291.Google Scholar
Pearcy, C. and Topping, D., Sums of small numbers of idempotents, Michigan Math. J., 14 (1967), 453–465.Google Scholar
Pearcy, C. and Topping, D., On commutators in ideals of compact operators, Michigan Math. J., 18 (1971), 247–252.Google Scholar
Pellegrini, V., Numerical range preserving operators on a Banach algebra, Studia Math., 54 (1975), 143–147.Google Scholar
Phillips, J., Nearest normal approximation for certain operators, Proc. Amer. Math. Soc., 67 (1977), 236–240.Google Scholar
Pickering, J., Counterexamples to rational dilation on symmetric multiply connected domains, Complex Anal. Oper. Theory, 4 (2010), 55–95.Google Scholar
Pollack, F. M., Properties of the matrix range of an operator, Indiana Univ. Math. J., 22 (1972), 419–427.Google Scholar
Pollack, F. M., Numerical range and convex sets, Canad. Math. Bull., 17 (1974), 295–296.Google Scholar
Pólya, G. and Szegö, G., Problems and Theorems in Analysis, vol. II, Springer, Berlin, 1976.Google Scholar
Poon, Y.-T., Another proof of a result of Westwick, Linear Multilinear Algebra, 9 (1980), 35–37.Google Scholar
Poon, Y.-T., Generalized numerical ranges, joint positive definiteness and multiple eigenvalues, Proc. Amer. Math. Soc., 125 (1997), 1625–1634.Google Scholar
Pop, C., On a result of Haagerup and de la Harpe, Rev. Roumaine Math. Pures Appl., 43 (1998), 869–871.Google Scholar
Pták, V., A maximum problem for matrices, Linear Algebra Appl., 28 (1979), 193–204.Google Scholar
Pták, V., Lyapunov equations and Gram matrices, Linear Algebra Appl., 49 (1983), 33–55.Google Scholar
Putinar, M. and Sandberg, S., A skew normal dilation on the numerical range of an operator, Math. Ann., 331 (2005), 345–357.Google Scholar
Putnam, C. R., On the numerical ranges of commutators, J. London Math. Soc., 34 (1959), 23–26.Google Scholar
Putnam, C. R., On the spectra of semi-normal operators, Trans, Amer. Math. Soc., 119 (1965), 509–523.Google Scholar
Radjabalipour, M. and Radjavi, H., On the geometry of numerical ranges, Pacific J. Math., 61 (1975), 507–511.Google Scholar
Radjavi, H., Structure of A∗ A − AA∗, J. Math. Mech., 16 (1966), 19–26.Google Scholar
Ransford, T. and Schwenninger, F. L.,√ Remarks on the Crouzeix–Palencia proof that the numerical range is a (1 + 2)-spectral set, SIAM J. Matrix Anal. Appl., 39 (2018), 342–345.Google Scholar
Rao, D. K. M., Rango numérico de productos commutativos, Rev. Colombiana Mat., 27 (1993), 231–233.Google Scholar
Rellich, F., Perturbation Theory of Eigenvalue Problems, Gordon and Breach, New York, 1969.Google Scholar
Ridge, W. C., Numerical range of a weighted shift with periodic weights, Proc. Amer. Math. Soc., 55 (1976), 107–110.Google Scholar
Riesz, F. and Sz.-Nagy, B., Über Kontraktionen des Hilbertschen Raumes, Acta Sci. Math. (Szeged), 10 (1943), 202–205.Google Scholar
Riesz, F. and Sz.-Nagy, B., Functional Analysis, Frederick Ungar, New York, 1955.Google Scholar
Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, 1970.Google Scholar
Rodman, L. and Spitkovsky, I. M., 3 × 3 matrices with a flat portion on the boundary of the numerical range, Linear Algebra Appl., 397 (2005), 193–207.Google Scholar
Rodman, L. and Spitkovsky, I. M., On generalized numerical ranges of quadratic operators, Oper. Theory Adv. Appl., 179 (2007), 241–256.Google Scholar
Rodman, L. and Spitkovsky, I. M., On numerical ranges of rank-two operators, Integral Equations Operator Theory, 77 (2013), 441–448.Google Scholar
Rosenblum, M., On the operator equation BX − XA = Q, Duke Math. J., 23 (1956), 263–269.Google Scholar
Rubin, H. and Wesler, O., A note on convexity in Euclidean n-space, Proc. Amer. Math. Soc., 9 (1958), 522–523.Google Scholar
Rudin, W., Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.Google Scholar
Ryff, J. V., Subordinate H p functions, Duke Math. J., 33 (1966), 347–354.Google Scholar
Saitô, T., Numerical ranges of tensor products of operators, Tôhoku Math. J., 19 (1967), 98–100.Google Scholar
Salinas, N., Reducing essential eigenvalues, Duke Math. J., 40 (1973), 561–580.Google Scholar
Salinas, N. and Velasco, M. V., Normal essential eigenvalues in the boundary of the numerical range, Proc. Amer. Math. Soc., 129 (2001), 505–513.Google Scholar
Sano, T. and Uchiyama, A., Numerical radius and unitarity, Acta Sci. Math. (Szeged), 76 (2010), 581–584.Google Scholar
Sarason, D., On spectral sets having connected complement, Acta Sci. Math. (Szeged), 26 (1965), 289–299.Google Scholar
Sarason, D., A remark on the Volterra operator, J. Math. Anal. Appl., 12 (1965), 244– 246.Google Scholar
Sarason, D., Generalized interpolation in H ∞, Trans. Amer. Math. Soc., 127 (1967), 179–203.Google Scholar
Schäffer, J. J., On unitary dilations of contractions, Proc. Amer. Math. Soc., 6 (1955), 322.Google Scholar
Schechter, M., Principles of Functional Analysis, Academic Press, New York, 1971.Google Scholar
Schreiber, M., Numerical range and spectral sets, Michigan Math. J., 10 (1963), 283–288.Google Scholar
Seddik, A., The numerical range of elementary operators II, Linear Algebra Appl., 338 (2001), 239–244.Google Scholar
Seddik, A., The numerical range of elementary operators, Integral Equations Operator Theory, 43 (2002), 248–252.Google Scholar
Shapiro, D., Problem 1331, Math. Mag., 62 (1989), 275; H. Schmidt, Solution of Problem 1331, Math. Mag., 63 (1990), 278–279.Google Scholar
Shapiro, H., On a conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. I, Linear Algebra Appl., 43 (1982), 201–221.Google Scholar
Shapiro, H., A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II, Linear Algebra Appl., 45 (1982), 97–108.Google Scholar
Shapiro, H. and Taussky, O., Alternative proofs of a theorem of Moyls and Marcus on the numerical range of a square matrix, Linear Multilinear Algebra, 8 (1980), 337–340.Google Scholar
Shapiro, J. H., Composition Operators and Classical Function Theory, Springer, New York, 1993.Google Scholar
Shapiro, J. H., What do composition operators know about inner functions?, Monatsh. Math., 130 (2000), 57–70.Google Scholar
Shaw, S.-Y., On numerical ranges of generalized derivations and related properties, J. Austral. Math. Soc. Ser. A, 36 (1984), 134–142.Google Scholar
Shields, A. L., Weighted shift operators and analytic function theory, in: Topics in Operator Theory, Pearcy, C., ed., Amer. Math. Soc., Providence, 1974, pp. 49–128.Google Scholar
Shiu, E. S. W., Growth of numerical ranges of powers of Hilbert space operators, Michigan Math. J., 23 (1976), 155–160.Google Scholar
Shiu, E. S. W., Numerical ranges of products and tensor products, Tôhoku Math. J., 30 (1978), 257–262.Google Scholar
Shoda, K., Einige Sätze über Matrizen, Japan. J. Math., 13 (1936), 361–365.Google Scholar
Simoncini, V. and Szyld, D. B., On the field of values of oblique projections, Linear Algebra Appl., 433 (2010), 810–818.Google Scholar
Sims, B., On a connection between the numerical range and spectrum of an operator on a Hilbert space, J. London Math. Soc., 8 (1974), 57–59.Google Scholar
Sinkhorn, R., Power symmetric stochastic matrices, Linear Algebra Appl., 40 (1981), 225–228.Google Scholar
Smith, J. H., Commutators of nilpotent matrices, Linear Multilinear Algebra, 4 (1976), 17–19.Google Scholar
Sourour, A. R., A short proof of Radjavi’s theorem on self-commutators, Math. Ann., 239 (1979), 137–139.Google Scholar
Sourour, A. R., Antiinvariant subspaces of maximum dimension, Linear Algebra Appl., 74 (1986), 39–45.Google Scholar
Spitkovsky, I. M., On the non-round points of the boundary of the numerical range, Linear Multilinear Algebra, 47 (2000), 29–33.Google Scholar
Spitkovsky, I. M., A note on the maximal numerical range, Oper. Matrices, 13 (2019), 601–605.Google Scholar
Stampfli, J. G., Hyponormal operators and spectral density, Trans. Amer. Math. Soc., 117 (1965), 469–476.Google Scholar
Stampfli, J. G., Extreme points of the numerical range of a hyponormal operator, Michigan Math. J., 13 (1966), 87–89.Google Scholar
Stampfli, J. G., Minimal range theorems for operators with thin spectra, Pacific J. Math., 23 (1967), 601–612.Google Scholar
Stampfli, J. G., The norm of a derivation, Pacific J. Math., 33 (1970), 737–747.Google Scholar
Stampfli, J. G. and Williams, J. P., Growth conditions and the numerical range in a Banach algebra, Tôhoku Math. J., 20 (1968), 417–424.Google Scholar
Stone, M. H., Linear Transformations in Hilbert Space and Their Application to Analysis, Amer. Math. Soc., New York, 1932.Google Scholar
Stout, Q. F., Schur products of operators and the essential numerical range, Trans. Amer. Math. Soc., 264 (1981), 39–47.Google Scholar
Stout, Q. F., The numerical range of a weighted shift, Proc. Amer. Math. Soc., 88 (1983), 495–502.Google Scholar
Strang, W. G., Eigenvalues of Jordan products, Amer. Math. Monthly, 69 (1962), 37–40.Google Scholar
Suen, C.-Y., The minimum norm of certain completely positive maps, Proc. Amer. Math. Soc., 123 (1995), 2407–2416.Google Scholar
Szegő, G., Conformal mapping of the interior of an ellipse onto a circle, Amer. Math. Monthly, 57 (1950), 474–478.Google Scholar
Sz.-Nagy, B., Sur les contractions de l’espace de Hilbert, Acta Sci. Math. (Szeged), 15 (1953), 87–92.Google Scholar
Sz.-Nagy, B., Sur la norme des fonctions de certains opérateurs, Acta Math. Acad. Sci. Hungar., 20 (1969), 331–334.Google Scholar
Sz.-Nagy, B. and Foiaş, C., An application of dilation theory to hyponormal operators, Acta Sci. Math. (Szeged), 37 (1975), 155–159.Google Scholar
Sz.-Nagy, B., Foiaş, C., Bercovici, H., and Kérchy, L., Harmonic Analysis of Operators on Hilbert Space, 2nd ed., Springer, New York, 2010.Google Scholar
Tam, B.-S., Circularity of numerical ranges and block-shift matrices, Linear Algebra Appl., 37 (1994), 93–109.Google Scholar
Tam, B.-S., On matrices with cyclic structure, Linear Algebra Appl., 302/303 (1999), 377–410.Google Scholar
Tam, B.-S. and Yang, S., On matrices whose numerical ranges have circular or weak circular symmetry, Linear Algebra Appl., 302/303 (1999), 193–221.Google Scholar
Tam, T.-Y., On a conjecture of Ridge, Proc. Amer. Math. Soc., 125 (1997), 3581–3592.Google Scholar
Thompson, R. C., The upper numerical range of a quaternionic matrix is not a complex numerical range, Linear Algebra Appl., 254 (1997), 19–28.Google Scholar
Thompson, R. C. and Kuo, C.-C. T., Doubly stochastic, unitary, unimodular, and complex orthogonal power embeddings, Acta Sci. Math. (Szeged), 44 (1982), 345–357.Google Scholar
Thukral, J. K., The numerical range of a Toeplitz operator with harmonic symbol, J. Operator Theory, 34 (1995), 213–216.Google Scholar
Tillekeratne, K., Spatial numerical range of an operator, Proc. Cambridge Philos. Soc., 76 (1974), 515–520.Google Scholar
Toeplitz, O., Das algebraische Analogon zu einem Satze von Fejér, Math. Z., 2 (1918), 187–197.Google Scholar
Tóth, L. F., Regular Figures, Macmillan, New York, 1964.Google Scholar
Tsai, M.-C., Gau, H.-L., and Wang, H.-C., Refinements for numerical ranges of weighted shift matrices, Linear Multilinear Algebra, 62 (2014), 568–578.Google Scholar
Tsai, M.-C. and Wu, P. Y., Numerical ranges of weighted shift matrices, Linear Algebra Appl., 435 (2011), 243–254.Google Scholar
Tsing, N.-K., On the shape of the generalized numerical ranges, Linear Multilinear Algebra, 10 (1981), 173–182.Google Scholar
Tsing, N.-K., Diameter and minimal width of the numerical range, Linear Multilinear Algebra, 14 (1983), 179–185.Google Scholar
Tsing, N.-K., The constrained bilinear form and the C-numerical range, Linear Algebra Appl., 56 (1984), 195–206.Google Scholar
Tso, S.-H. and Wu, P. Y., Matricial ranges of quadratic operators, Rocky Mountain J. Math., 29 (1999), 1139–1152.Google Scholar
Valentine, F. A., Convex Sets, McGraw-Hill, New York, 1964.Google Scholar
Varopoulos, N., On an inequality of von Neumann and an application of the metric theory of tensor products to operator theory, J. Funct. Anal., 16 (1974), 83–100.Google Scholar
Voiculescu, D., Some results on norm-ideal perturbations of Hilbert space operators, J. Operator Theory, 2 (1979), 3–37.Google Scholar
Wadhwa, B. L., A hyponormal operator whose spectrum is not a spectral set, Proc. Amer. Math. Soc., 38 (1973), 83–85.Google Scholar
Wang, J.-H. and Wu, P. Y., Sums of square-zero operators, Studia Math., 99 (1991), 115–127.Google Scholar
Wang, K.-Z. and Wu, P. Y., Numerical ranges of radial Toeplitz operators on Bergman space, Integral Equations Operator Theory, 65 (2009), 581–591.Google Scholar
Wang, K.-Z. and Wu, P. Y., Numerical ranges of weighted shifts, J. Math. Anal. Appl., 381 (2011), 897–909.Google Scholar
Wang, K.-Z. and Wu, P. Y., Diagonals and numerical ranges of weighted shift matrices, Linear Algebra Appl., 438 (2013), 514–532.Google Scholar
Wang, K.-Z., Wu, P. Y., and Gau, H.-L., Crawford numbers of powers of a matrix, Linear Algebra Appl., 433 (2010), 2243–2254.Google Scholar
Wegge-Olsen, N. E., K-Theory and C ∗ -Algebras, Oxford University Press, New York, 1993.Google Scholar
Weiss, G., The Fuglede commutativity theorem modulo the Hilbert–Schmidt class and generating functions for matrix operators. I, Trans. Amer. Math. Soc., 246 (1978), 193–209.Google Scholar
Weiss, G., Commutators of Hilbert–Schmidt operators II, Integral Equations Operator Theory, 314 (1980), 574–600.Google Scholar
Weiss, G., Commutators of Hilbert–Schmidt operators I, Integral Equations Operator Theory, 9 (1986), 877–892.Google Scholar
Weiss, G., Classification of certain commutator ideals and the tensor product closure property, Integral Equations Operator Theory, 12 (1989), 99–128.Google Scholar
Westwick, R., A theorem on numerical range, Linear Multilinear Algebra, 2 (1975), 311–315.Google Scholar
Wielandt, H., On eigenvalues of sums of normal matrices, Pacific J. Math., 5 (1955), 633–638.Google Scholar
Wielandt, H., On the eigenvalues of A + B and AB, J. Res. Nat. Bur. Standards Sect. B, 77 (1973), 61–63.Google Scholar
Williams, J. P., Spectra of products and numerical ranges, J. Math. Anal. Appl., 17 (1967), 214–220.Google Scholar
Williams, J. P., Schwarz norms for operators, Pacific J. Math., 24 (1968), 181–188.Google Scholar
Williams, J. P., Similarity and the numerical range, J. Math. Anal. Appl., 26 (1969), 307–314.Google Scholar
Williams, J. P., Finite operators, Proc. Amer. Math. Soc., 26 (1970), 129–136.Google Scholar
Williams, J. P., On compressions of matrices, J. London Math. Soc. (2), 3 (1971), 526–530.Google Scholar
Williams, J. P., On commutativity and the numerical range in Banach algebras, J. Funct. Anal., 10 (1972), 326–329.Google Scholar
Williams, J. P., The numerical range and the essential numerical range, Proc. Amer. Math. Soc., 66 (1977), 185–186.Google Scholar
Williams, J. P. and Crimmins, T., On the numerical radius of a linear operator, Amer. Math. Monthly, 74 (1967), 832–833.Google Scholar
Wu, P. Y., Products of positive semidefinite matrices, Linear Algebra Appl., 111 (1988), 53–61.Google Scholar
Wu, P. Y., Unitary dilations and numerical ranges, J. Operator Theory, 38 (1997), 25–42.Google Scholar
Wu, P. Y., A numerical range characterization of Jordan blocks, Linear Multilinear Algebra, 43 (1998), 351–361.Google Scholar
Wu, P. Y., Polygons and numerical ranges, Amer. Math. Monthly, 107 (2000), 528–540.Google Scholar
Wu, P. Y., Numerical range of Aluthge transform of operators, Linear Algebra Appl., 357 (2002), 295–298.Google Scholar
Wu, P. Y., Polar decompositions of C0 (N) contractions, Integral Equations Operator Theory, 56 (2006), 559–569.Google Scholar
Wu, P. Y., Numerical ranges as circular discs, Appl. Math. Lett., 24 (2011), 2115–2117.Google Scholar
Wu, P. Y., Gau, H.-L., and Tsai, M. C., Numerical radius inequality for C0 contractions, Linear Algebra Appl., 430 (2009), 1509–1516.Google Scholar
Wu, P. Y. and Takahashi, K., Singular unitary dilations, Integral Equations Operator Theory, 33 (1999), 231–247.Google Scholar
Yamazaki, T., Parallelisms between Aluthge transformation and powers of operators, Acta Sci. Math. (Szeged), 67 (2001), 809–820.Google Scholar
Yamazaki, T., On numerical range of the Aluthge transformation, Linear Algebra Appl., 341 (2002), 111–117.Google Scholar
Yamazaki, T., An expression of spectral radius via Aluthge transformation, Proc. Amer. Math. Soc., 130 (2002), 1131–1137.Google Scholar
Yamazaki, T., On upper and lower bounds of the numerical radius and an equality condition, Studia Math., 178 (2007), 83–89.Google Scholar
Yoshino, T., Introduction to Operator Theory, Longman, Harlow, Essex, 1993.Google Scholar
Yosida, K., Functional Analysis, Springer, Berlin, 1974.Google Scholar
Young, N. J., Analytic programmes in matrix algebras, Proc. London Math. Soc. (3), 36 (1978), 226–242.Google Scholar
Zhang, H.-Y., Dou, Y.-N., Wang, M.-F., and Du, H.-K., On the boundary of numerical ranges of operators, Appl. Math. Lett., 24 (2011), 620–622.Google Scholar
Zhang, X., Hou, J. C., and He, K., Maps preserving numerical radius and cross norms of operator products, Linear Multilinear Algebra, 57 (2009), 523–534.Google Scholar
Zhu, S., Approximate unitary equivalence of normaloid type operators, Banach J. Math. Anal., 9 (2015), 173–193.Google Scholar
Zhu, S. and Li, C. G., Operators similar to normaloid operators, J. Korean Math. Soc., 48 (2011), 1203–1223.Google Scholar

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  • References
  • Hwa-Long Gau, National Central University, Taiwan, Pei Yuan Wu, National Chiao Tung University, Taiwan
  • Book: Numerical Ranges of Hilbert Space Operators
  • Online publication: 27 July 2021
  • Chapter DOI: https://doi.org/10.1017/9781108782296.012
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  • References
  • Hwa-Long Gau, National Central University, Taiwan, Pei Yuan Wu, National Chiao Tung University, Taiwan
  • Book: Numerical Ranges of Hilbert Space Operators
  • Online publication: 27 July 2021
  • Chapter DOI: https://doi.org/10.1017/9781108782296.012
Available formats
×

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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Hwa-Long Gau, National Central University, Taiwan, Pei Yuan Wu, National Chiao Tung University, Taiwan
  • Book: Numerical Ranges of Hilbert Space Operators
  • Online publication: 27 July 2021
  • Chapter DOI: https://doi.org/10.1017/9781108782296.012
Available formats
×