Skip to main content Accessibility help
×
Home
Modern Approaches to the Invariant-Subspace Problem
  • Cited by 11
  • Export citation
  • Recommend to librarian
  • Buy the print book

Book description

One of the major unsolved problems in operator theory is the fifty-year-old invariant subspace problem, which asks whether every bounded linear operator on a Hilbert space has a nontrivial closed invariant subspace. This book presents some of the major results in the area, including many that were derived within the past few years and cannot be found in other books. Beginning with a preliminary chapter containing the necessary pure mathematical background, the authors present a variety of powerful techniques, including the use of the operator-valued Poisson kernel, various forms of the functional calculus, Hardy spaces, fixed point theorems, minimal vectors, universal operators and moment sequences. The subject is presented at a level accessible to postgraduate students, as well as established researchers. It will be of particular interest to those who study linear operators and also to those who work in other areas of pure mathematics.

Reviews

'I think this is a very useful book which will serve as a good source for a rich variety of methods that have been developed for proving positive results on the ISP. Moreover, there is much material in the book which is of interest beyond its application to the ISP. [It] should be of interest to analysts in general as well as being an essential source for study of the ISP.'

Sandy Davie Source: SIAM Review

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Send to Kindle
  • Send to Dropbox
  • Send to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
×

Contents

References
References
[1] Y.A., Abramovich and C. D., Aliprantis. An Invitation to Operator Theory. Graduate Studies in Mathematics, 50. American Mathematical Society, Providence, RI, 2002.
[2] Y.A., Abramovich, C. D., Aliprantis, and O., Burkinshaw. Invariant subspaces of operators on lp-spaces. J. Funct. Anal., 115(2):418–424, 1993.
[3] Y.A., Abramovich, C. D., Aliprantis, and O., Burkinshaw. Invariant subspace theorems for positive operators. J. Funct. Anal., 124(1):95–111, 1994.
[4] Y.A., Abramovich, C. D., Aliprantis, and O., Burkinshaw. Invariant subspaces for positive operators acting on a Banach space with basis. Proc. Amer. Math. Soc., 123(6):1773–1777, 1995.
[5] C.D., Aliprantis and O., Burkinshaw. Positive Operators. Springer, Dordrecht, 2006. Reprint of the 1985 original.
[6] G. R., Allan. Sums of idempotents and a lemma of N. J. Kalton. Studia Math., 121:185–192, 1996.
[7] D., Alpay, L., Baratchart, and J., Leblond. Some extremal problems linked with identification from partial frequency data. In R. F., Curtain, A., Bensoussan, and J.-L., Lions, eds., Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems (Sophia-Antipolis, 1992), pages 563–573. Springer, Berlin, 1993.
[8] C., Ambrozie and V., Müller. Invariant subspaces for polynomially bounded operators. J. Funct. Anal., 213(2):321–345, 2004.
[9] G., Androulakis. A note on the method of minimal vectors. In Trends in Banach Spaces and Operator Theory (Memphis, TN, 2001), pages 29–36. Contemporary Mathematics, 321. American Mathematical Society, Providence, RI, 2003.
[10] G., Androulakis and A., Flattot. Hyperinvariant subspaces for weighted composition operators on Lp ([0, 1]d). J. Operator Theory, in press.
[11] S., Ansari and P., Enflo. Extremal vectors and invariant subspaces. Trans. Amer. Math. Soc., 350(2):539–558, 1998.
[12] C., Apostol. Ultraweakly closed operator algebras. J. Operator Theory, 2:49–61, 1979.
[13] C., Apostol. Functional calculus and invariant subspaces. J. Operator Theory, 4:159–190, 1980.
[14] C., Apostol, H., Bercovici, C., Foias, and C., Pearcy. Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. I. J. Funct. Anal., 63(3):369–404, 1985.
[15] C., Apostol, H., Bercovici, C., Foias, and C., Pearcy. Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. II. Indiana Univ. Math. J., 34(4):845–855, 1985.
[16] C., Apostol, L. A., Fialkow, D. A., Herrero, and D., Voiculescu. Approximation of Hilbert Space Operators, Volume II. Research Notes in Mathematics, 102. Pitman (Advanced Publishing Program), Boston, MA, 1984.
[17] S. A., Argyros and R. G., Haydon. A hereditarily indecomposable L∞-space that solves the scalar-plus-compact problem. Preprint, 2009 arXiv:0903.3921.
[18] N., Aronszajn and K. T., Smith. Invariant subspaces of completely continuous operators. Ann. Math. (2), 60:345–350, 1954.
[19] A., Atzmon. An operator without invariant subspaces on a nuclear Fréchet space. Ann. Math., 117:660–694, 1983.
[20] A., Atzmon. On the existence of hyperinvariant subspaces. J. Operator Theory, 11(1):3–40, 1984.
[21] A., Atzmon and G., Godefroy. An application of the smooth variational principle to the existence of non-trivial invariant subspaces. C. R. Acad. Sci. Paris Sér. I Math., 332(2):151–156, 2001.
[22] A., Atzmon, G., Godefroy, and N. J., Kalton. Invariant subspaces and the exponential map. Positivity, 8(2):101–107, 2004.
[23] L., Baratchart and J., Leblond. Hardy approximation to Lp functions on subsets of the circle with 1 ≤ p < ∞. Constr. Approx., 14(1):41–56, 1998.
[24] S., Barclay. Banach spaces of analytic vector-valued functions. PhD thesis, University of Leeds, 2008.
[25] S., Barclay. A solution to the Douglas–Rudin problem for matrix-valued functions. Proc. London Math. Soc., 99:757–786, 2009.
[26] F., Bayart and É., Matheron. Dynamics of Linear Operators. Cambridge Tracts in Mathematics, 179. Cambridge University Press, Cambridge, 2009.
[27] B., Beauzamy. Introduction to Banach Spaces and Their Geometry. Notas de Matemática [Mathematical Notes], 86. North-Holland, Amsterdam, 1982.
[28] B., Beauzamy. Un opérateur sans sous-espace invariant: Simplification de l'exemple d'Enflo. Integr. Equat. Oper. Th., 8(3):314–384, 1985.
[29] B., Beauzamy. Introduction to Operator Theory and Invariant Subspaces. North-Holland Mathematical Library, 42. North-Holland, Amsterdam, 1988.
[30] R., Becker. Ordered Banach Spaces. Travaux en Cours [Works in Progress], 68. Hermann Éditeurs des Sciences et des Arts, Paris, 2008. With a preface by Gilles Godefroy.
[31] H., Bercovici. Factorization theorems and the structure of operators on Hilbert space. Ann. Math., 128:399–413, 1988.
[32] H., Bercovici. Operator Theory and Arithmetic in H∞. Mathematical Surveys and Monographs, 26. American Mathematical Society, Providence, RI, 1988.
[33] H., Bercovici. Notes on invariant subspaces. Bull. Amer. Math. Soc., 23(1):1–36, 1990.
[34] H., Bercovici, C., Foias, and C., Pearcy. Dilation theory and systems of simultaneous equations in the predual of an operator algebra, I. Michigan Math. J., 30:335–354, 1983.
[35] H., Bercovici, C., Foias, and C., Pearcy. Dual Algebras With Applications to Invariant Subspaces and Dilation Theory. CBMS Regional Conference Series in Mathematics, 56. American Mathematical Society, Providence, RI, 1985.
[36] H., Bercovici, C., Foias, and C., Pearcy. Two Banach space methods and dual operator algebras. J. Funct. Anal., 78:306–345, 1988.
[37] C. A., Berenstein and R., Gay. Complex Variables. Graduate Texts in Mathematics, 125. Springer, New York, 1991.
[38] C. A., Berger and J.G., Stampfli. Mapping theorems for the numerical range. Amer. J. Math., 89:1047–1055, 1967.
[39] A. R., Bernstein and A., Robinson. Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pac. J. Math., 16:421–431, 1966.
[40] E., Bishop and R.R., Phelps. A proof that every Banach space is subreflexive. Bull. Amer. Math. Soc., 67:97–98, 1961.
[41] E., Bishop and R.R., Phelps. The support functionals of a convex set. In Convexity: Proc. Sympos. Pure Math., Vol. VII, pages 27–35. American Mathematical Society, Providence, RI, 1963.
[42] D. P., Blecher and A. M., Davie. Invariant subspaces for an operator on L2 (Π) composed of a multiplication and a translation. J. Operator Theory, 23(1):115–123, 1990.
[43] R. P., Boas. Entire Functions. Academic Press, New York, 1954.
[44] B., Bollobás. An extension to the theorem of Bishop and Phelps. Bull. London Math. Soc., 2:181–182, 1970.
[45] F. F., Bonsall. Decompositions of functions as sums of elementary functions. Quart. J. Math. Oxford Ser. (2), 37(146):129–136, 1986.
[46] F. F., Bonsall and J., Duncan. Numerical Ranges, II. London Mathematical Society Lecture Notes Series, 10. Cambridge University Press, New York, 1973.
[47] P. S., Bourdon and A., Flattot. Images of minimal-vector sequences under weighted composition operators on L2(D). In J.A., Ball, V., Bolotnikov, J.W., Helton, L., Rodman, and I. M., Spitkovsky, eds., Topics in Operator Theory, Volume 1: Operators, Matrices and Analytic Functions, pages 39–52. Operator Theory: Advances and Applications, 202. Birkhäuser, Basel, 2010.
[48] J., Bourgain. A problem of Douglas and Rudin on factorization. Pac. J. Math., 121(1):47–50, 1986.
[49] L., Brown, A., Shields, and K., Zeller. On absolutely convergent exponential sums. Trans. Amer. Math. Soc., 96:162–183, 1960.
[50] S., Brown. Some invariant subspaces for subnormal operators. Integr. Equat. Oper. Th., 1:310–333, 1978.
[51] S., Brown. Contractions with spectral boundary. Integr. Equat. Oper. Th., 11:49–63, 1988.
[52] S., Brown and B., Chevreau. Toute contraction à calcul fonctionnel isométrique est réflexive. C. R. Acad. Sci. Paris Sér. I Math., 307:185–188, 1988.
[53] S., Brown, B., Chevreau, and C., Pearcy. Contractions with rich spectrum have invariant subspaces. J. Operator Theory, 1(1):123–136, 1979.
[54] S., Brown, B., Chevreau, and C., Pearcy. On the structure of contraction operators, II. J. Funct. Anal., 76(1):30–55, 1988.
[55] S. R., Caradus. Universal operators and invariant subspaces. Proc. Amer. Math. Soc., 23:526–527, 1969.
[56] L., Carleson. An interpolation problem for bounded analytic functions. Amer. J. Math., 80:921–930, 1958.
[57] G., Cassier and I., Chalendar. The group of the invariants of a finite Blaschke product. Complex Variables, 42:193–206, 2000.
[58] G., Cassier, I., Chalendar, and B., Chevreau. New examples of contractions illustrating membership and non-membership in the classes An,m. Acta Sci. Math. (Szeged), 64:707–731, 1998.
[59] G., Cassier, I., Chalendar, and B., Chevreau. Some mapping theorems for the classes An,m and boundary sets. Proc. London Math. Soc., 79(1):222–240, 1999.
[60] G., Cassier and T., Fack. Contractions in von Neumann algebras. J. Funct. Anal., 135(2):297–338, 1996.
[61] I., Chalendar. The operator-valued Poisson kernel and its application. Irish Math. Soc. Bull., 51:21–44, 2003.
[62] I., Chalendar and J., Esterle. L1-factorization for C00-contractions with isometric functional calculus. J. Funct. Anal., 154:174–194, 1998.
[63] I., Chalendar, A., Flattot, and N., Guillotin-Plantard. On the spectrum of multivariable weighted composition operators. Arch. Math. (Basel), 90(4) 353–359, 2008.
[64] I., Chalendar, A., Flattot, and J.R., Partington. The method of minimal vectors applied to weighted composition operators. In The Extended Field of Operator Theory, pages 89–105. Operator Theory: Advances and Applications, 171. Birkhäuser, Basel, 2007.
[65] I., Chalendar, E., Fricain, A. I., Popov, D., Timotin, and V. G., Troitsky. Finitely strictly singular operators between James spaces. J. Funct. Anal., 256(4):1258–1268, 2009.
[66] I., Chalendar and F., Jaeck. On the contractions in the classes An,m. J. Operator Theory, 38(2):265–296, 1997.
[67] I., Chalendar, J., Leblond, and J.R., Partington. Approximation problems in some holomorphic spaces, with applications. In Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), pages 143–168. Operator Theory: Advances and Applications, 129. Birkhäuser, Basel, 2001.
[68] I., Chalendar and J. R., Partington. L1 factorizations for some perturbations of the unilateral shift. C. R. Acad. Sci. Paris Sér. I Math., 332:115–119, 2001.
[69] I., Chalendar and J. R., Partington. Constrained approximation and invariant subspaces. J. Math. Anal. Appl., 280(1):176–187, 2003.
[70] I., Chalendar and J. R., Partington. On the structure of invariant subspaces for isometric composition operators on H2(D) and H2(ℂ+). Arch. Math. (Basel), 81(2):193–207, 2003.
[71] I., Chalendar and J. R., Partington. Spectral density for multiplication operators with applications to factorization of L1 functions. J. Operator Theory, 50(2):411–422, 2003.
[72] I., Chalendar and J. R., Partington. Convergence properties of minimal vectors for normal operators and weighted shifts. Proc. Amer. Math. Soc., 133(2):501–510, 2005.
[73] I., Chalendar and J. R., Partington. Variations on Lomonosov's theorem via the technique of minimal vectors. Acta Sci. Math. (Szeged), 71(3–4):603–617, 2005.
[74] I., Chalendar and J. R., Partington. Invariant subspaces for products of Bishop operators. Acta Sci. Math. (Szeged), 74(3–4):719–727, 2008.
[75] I., Chalendar, J. R., Partington, and E., Pozzi. Multivariable weighted composition operators: Point spectrum and cyclic vectors. In J. A., Ball, V., Bolotnikov, J.W., Helton, L., Rodman, and I. M., Spitkovsky, eds., Topics in Operator Theory, Volume 1: Operators, Matrices and Analytic Functions, pages 63–85. Operator Theory: Advances and Applications, 202. Birkhäuser, Basel, 2010.
[76] I., Chalendar, J. R., Partington, and M., Smith. Approximation in reflexive Banach spaces and applications to the invariant subspace problem. Proc. Amer. Math. Soc., 132(4):1133–1142, 2004.
[77] I., Chalendar, J. R., Partington, and R.C., Smith. L1 factorizations, moment problems and invariant subspaces. Studia Math., 167(2):183–194, 2005.
[78] B., Chevreau. Sur les contractions à calcul fonctionnel isométrique, II. J. Operator Theory, 20:269–293, 1988.
[79] B., Chevreau and C., Pearcy. Growth conditions on the resolvent and membership in the classes A and. J. Operator Theory, 16(2):375–385, 1986.
[80] B., Chevreau and C., Pearcy. On the structure of contraction operators, I. J. Funct. Anal., 76:1–29, 1988.
[81] V., Chkliar. Eigenfunctions of the hyperbolic composition operator. Integr. Equat. Oper. Th., 29(3):364–367, 1997.
[82] I., Colojoarǎ and C., Foiaş. Theory of Generalized Spectral Operators. Mathematics and its Applications, 9. Gordon and Breach Science Publishers, New York, 1968.
[83] J. B., Conway. The Theory of Subnormal Operators. Mathematical Surveys and Monographs, 36. American Mathematical Society, Providence, RI, 1991.
[84] J. B., Conway. A Course in Operator Theory. Graduate Studies in Mathematics, 21. American Mathematical Society, Providence, RI, 2000.
[85] C. C., Cowen and B.D., MacCluer. Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995.
[86] R. E., Curto and F. H., Vasilescu. Automorphism invariance of the operator-valued Poisson transform. Acta Sci. Math. (Szeged), 57:65–78, 1993.
[87] H. G., Dales. Banach Algebras and Automatic Continuity. London Mathematical Society Monographs, New Series, 24. Clarendon Press, Oxford University Press, New York, 2000.
[88] A. M., Davie. The approximation problem for Banach spaces. Bull. London Math. Soc., 5:261–266, 1973.
[89] A. M., Davie. Invariant subspaces for Bishop's operators. Bull. London Math. Soc., 6:343–348, 1974.
[90] L., de Branges and J., Rovnyak. Canonical models in quantum scattering theory. In Perturbation Theory and its Applications in Quantum Mechanics: Proceedings of an Advanced Seminar Conducted by The Mathematics Research Center, U.S. Army, and the Theoretical Chemistry Institute at the University of Wisconsin, Madison, WI, October 4–6, 1965, pages 295–392. Wiley, New York, 1966.
[91] R., Deville, G., Godefroy, and V., Zizler. A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions. J. Funct. Anal., 111(1):197–212, 1993.
[92] R., Deville, G., Godefroy, and V., Zizler. Smoothness and Renormings in Banach Spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, 64. Longman Scientific & Technical, Harlow, 1993.
[93] Y., Domar. On the analytic transform of bounded linear functionals on certain Banach algebras. Studia Math., 53(3):203–224, 1975.
[94] P., Enflo. A counterexample to the approximation problem in Banach spaces. Acta Math., 130:309–317, 1973.
[95] P., Enflo. On the invariant subspace problem in Banach spaces. In Séminaire Maurey–Schwartz (1975–1976): Espaces Lp, Applications Radonifiantes et Géométrie des Espaces de Banach, Exp. Nos. 14–15. Centre de Mathématiques, École de Polytechnique, Palaiseau, 1976.
[96] P., Enflo. On the invariant subspace problem in Banach spaces. Acta Math., 158:213–313, 1987.
[97] P., Enflo. Extremal vectors for a class of linear operators. In Functional Analysis and Economic Theory (Samos, 1996), pages 61–64. Springer, Berlin, 1998.
[98] P., Enflo and T., Hõim. Some results on extremal vectors and invariant subspaces. Proc. Amer. Math. Soc., 131(2):379–387, 2003.
[99] Z., Ercan and S., Onal. Invariant subspaces for positive operators acting on a Banach space with Markushevich basis. Positivity, 8(2):123–126, 2004.
[100] J., Eschmeier. Representations of H∞(G) and invariant subspaces. Math. Ann., 298(1):167–186, 1994.
[101] J., Eschmeier. Invariant subspaces for spherical contractions. Proc. London Math. Soc., 75:157–176, 1997.
[102] J., Esterle and M., Zarrabi. Local properties of powers of operators. Arch. Math. (Basel), 65:53–60, 1995.
[103] G., Exner and Il Bong, Jung. Dual operator algebras and contractions with finite defect indices. J. Operator Theory, 36(1):107–119, 1996.
[104] C., Fefferman. Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc., 77:587–588, 1971.
[105] C., Fefferman and E. M., Stein. Hp spaces of several variables. Acta Math., 129(3–4):137–193, 1972.
[106] M., Fekete. Über die verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Mathematische Zeitschrift, 17:228–249, 1923.
[107] A., Flattot. Hyperinvariant subspaces for Bishop-type operators. Acta Sci. Math. (Szeged), 74:687–716, 2008.
[108] J., Flores, P., Tradacete, and V. G., Troitsky. Invariant subspaces of positive strictly singular operators on Banach lattices. J. Math. Anal. Appl., 343(2):743–751, 2008.
[109] C., Foiaş. Sur certains théorèmes de J. von Neumann concernant les ensembles spectraux. Acta Sci. Math. (Szeged), 18:15–20, 1957.
[110] E., Gallardo-Gutiérrez and P., Gorkin. Minimal invariant subspaces for composition operators. J. Math. Pure. Appl., 95(3):245–259, 2011.
[111] E., Gallardo-Gutiérrez, P., Gorkin, and D., Suárez. Orbits of non-elliptic disc automorphisms on Hp. Preprint, 2010 arXiv:1002.3833.
[112] J. B., Garnett. Bounded Analytic Functions. Graduate Texts in Mathematics, 236. Springer, New York, 2007.
[113] R., Gellar. Operators commuting with a weighted shift. Proc. Amer. Math. Soc., 23:538–545, 1969.
[114] W. T., Gowers and B., Maurey. Banach spaces with small spaces of operators. Math. Ann., 307(4):543–568, 1997.
[115] S., Grivaux. Invariant subspaces for tridiagonal operators. Bull. Sci. Math., 126(8):681–691, 2002.
[116] P. R., Halmos. Spectra and spectral manifolds. Ann. Soc. Polon. Math., 25:43–49, 1952.
[117] P. R., Halmos. Invariant subspaces of polynomially compact operators. Pac. J. Math., 16:433–437, 1966.
[118] P. R., Halmos. A Hilbert Space Problem Book. Graduate Texts in Mathematics, 19, 2nd edition. Springer, New York, 1982.
[119] G. H., Hardy and E. M., Wright. An Introduction to the Theory of Numbers, 5th edition. Clarendon Press, Oxford University Press, New York, 1979.
[120] H., Hedenmalm. Maximal invariant subspaces in the Bergman space. Ark. Mat., 36(1):97–101, 1998.
[121] E., Heinz. Ein v. Neumannscher Satz über beschränkte Operatoren im Hilbertschen Raum. Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Ila., Math.- Phys.-Chem. Abt., 5–6, 1952.
[122] D. A., Herrero. Approximation of Hilbert Space Operators, Volume 1. Pitman Research Notes in Mathematics Series, 224, 2nd edition. Longman Scientific & Technical, Harlow, 1989.
[123] E., Hille and R. S., Phillips. Functional Analysis and Semi-Groups. American Mathematical Society Colloquium Publications, 31. American Mathematical Society, Providence, RI, 1957.
[124] J. A. R., Holbrook. On the power-bounded operators of Sz.-Nagy and Foiaş. Acta Sci. Math. (Szeged), 29:299–310, 1968.
[125] N. D., Hooker. Lomonosov's hyperinvariant subspace theorem for real spaces. Math. Proc. Cambridge Philos. Soc., 89(1):129–133, 1981.
[126] F., John and L., Nirenberg. On functions of bounded mean oscillation. Comm. Pure Appl. Math., 14:415–426, 1961.
[127] Il, Bong Jung. Dual operator algebras and the classes Am,n, I. J. Operator Theory, 27:309–323, 1992.
[128] S., Kakutani. Topological properties of the unit sphere of a Hilbert space. Proc. Imp. Acad. Tokyo, 19:269–271, 1943.
[129] N. J., Kalton. Sums of idempotents in Banach algebras. Can. Math. Bull., 31(4): 448–451, 1988.
[130] T., Kato. Perturbation theory for nullity, deficiency and other quantities of linear operators. J. Anal. Math., 6:261–322, 1958.
[131] H. J., Kim. Hyperinvariant subspace problem for quasinilpotent operators. Integr. Equat. Oper. Th., 61(1):103–120, 2008.
[132] H. J., Kim. Hyperinvariant subspaces for quasinilpotent operators on Hilbert spaces. J. Math. Anal. Appl., 350(1):262–270, 2009.
[133] S. V., Kisliakov. Quantitative aspect of correction theorems. Zap. Nauchn. Sem. LOMI, 92:182–191, 1979.
[134] S. V., Kisliakov. A sharp correction theorem. Studia Math., 113(2):177–196, 1995.
[135] G., Koenigs. Recherches sur les intégrales de certaines équations fonctionnelles. Ann. Sci. École Norm. Sup. (3), 1:3–41, 1884.
[136] P., Koosis. The Logarithmic Integral, I. Cambridge University Press, Cambridge, 1988.
[137] M. G., Kreĭn and P. Ja., Nudel'man. Approximation of functions in L2(ω1, ω2) by transmission functions of linear systems with minimal energy. Problemy Peredači Informacii, 11(2):37–60, 1975. English translation.
[138] H.-O., Kreiss. Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren. Nordisk Tidskr. Informations-Behandling, 2:153–181, 1962.
[139] T. L., Lance and M. I., Stessin. Multiplication invariant subspaces of Hardy spaces. Can. J. Math., 49(1):100–118, 1997.
[140] J., Leblond and J. R., Partington. Constrained approximation and interpolation in Hilbert function spaces. J. Math. Anal. Appl., 234:500–513, 1999.
[141] A., Lebow. A power-bounded operator that is not polynomially bounded. Michigan Math. J., 15:397–399, 1968.
[142] J., Lindenstrauss and L., Tzafriri. Classical Banach Spaces, I: Sequence Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 92. Springer, Berlin, 1977.
[143] J. E., Littlewood. On inequalities in the theory of functions. Proc. London Math. Soc. (2), 23:481–519, 1925.
[144] M. X., Liu. On hyperinvariant subspaces of contraction operators on a Banach space whose spectrum contains the unit circle. Acta Math. Sin. (Engl. Ser.), 24(9):1471–1474, 2008.
[145] V. I., Lomonosov. Invariant subspaces for operators commuting with compact operators. Funct. Anal. Appl., 7:213–214, 1973.
[146] V. I., Lomonosov. On real invariant subspaces of bounded operators with compact imaginary part. Proc. Amer. Math. Soc., 115(3):775–777, 1992.
[147] G. W., MacDonald. Invariant subspaces for Bishop-type operators. J. Funct. Anal., 91(2):287–311, 1990.
[148] G. W., MacDonald. Decomposable weighted rotations on the unit circle. J. Operator Theory, 35(2):205–221, 1996.
[149] J., Mashreghi and T., Ransford. Binomial sums and functions of exponential type. Bull. London Math. Soc., 37:15–24, 2005.
[150] V., Matache. On the minimal invariant subspaces of the hyperbolic composition operator. Proc. Amer. Math. Soc., 119(3):837–841, 1993.
[151] V., Matache. The eigenfunctions of a certain composition operator. In Studies on Composition Operators (Laramie, WY, 1996), pages 121–136. Contemporary Mathematics, 213. American Mathematical Society, Providence, RI, 1998.
[152] A. J., Michaels. Hilden's simple proof of Lomonosov's invariant subspace theorem. Adv. Math., 25(1):56–58, 1977.
[153] R., Mortini. Cyclic subspaces and eigenvectors of the hyperbolic composition operator. In Travaux Mathématiques, Fasc. VII, pages 69–79. Séminaire de Mathématique de Luxembourg. Centre Universitaire du Luxembourg, Luxembourg, 1995.
[154] N. K., Nikolski. Operators, Functions, and Systems: An Easy Reading, Volume 1. Mathematical Surveys and Monographs, 92. American Mathematical Society, Providence, RI, 2002. Translated from the French by A. Hartmann.
[155] N. K., Nikolski. Operators, Functions, and Systems: An Easy Reading, Volume 2. Mathematical Surveys and Monographs, 93. American Mathematical Society, Providence, RI, 2002. Translated from the French by A. Hartmann and revised by the author.
[156] E., Nordgren, P., Rosenthal, and F. S., Wintrobe. Invertible composition operators on Hp. J. Funct. Anal., 73:324–344, 1987.
[157] E. A., Nordgren. Composition operators. Can. J. Math., 20:442–449, 1968.
[158] A., Octavio and M., Kosiek. Representations of H∞(DN) and absolute continuity for N-tuples of contractions. Houston J. Math., 23(3):529–537, 1997.
[159] R. I., Ovsepian and A., Pełczyński. On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in L2. Studia Math., 54(2):149–159, 1975.
[160] J. R., Partington. Interpolation, Identification and Sampling. London Mathematical Society Monographs, 17. Clarendon Press, Oxford University Press, New York, 1997.
[161] J. R., Partington. Linear Operators and Linear Systems. London Mathematical Society Student Texts, 60. Cambridge University Press, Cambridge, 2004.
[162] J. R., Partington and E., Pozzi. Universal shifts and composition operators. Operators and Matrices, in press.
[163] V. I., Paulsen. Completely Bounded Maps and Dilations. Pitman Research Notes in Mathematics Series, 146. Longman Scientific & Technical, Harlow, 1986.
[164] A., Pełczyński. All separable Banach spaces admit for every ε > 0 fundamental total and bounded by 1 + ε biorthogonal sequences. Studia Math., 55(3):295–304, 1976.
[165] G., Pisier. A polynomially bounded operator on Hilbert space which is not similar to a contraction. J. Amer. Math. Soc., 10(2):351–369, 1997.
[166] G., Pólya and G., Szegő. Problems and Theorems in Analysis, I. Classics in Mathematics. Springer, Berlin, 1998. Reprint of 1978 English translation from the German by Dorothee Aeppli.
[167] H., Radjavi and P., Rosenthal. Invariant Subspaces. Springer, Berlin, 1973.
[168] C., Read. A solution to the invariant subspace problem. Bull. London Math. Soc., 16:337–401, 1984.
[169] C., Read. A solution to the invariant subspace problem on the space ℓ1. Bull. London Math. Soc., 17:305–317, 1985.
[170] C., Read. A short proof concerning the invariant subspace problem. J. London Math. Soc., 34(2):335–348, 1986.
[171] C., Read. The invariant subspace problem for a class of Banach spaces, II: Hypercyclic operators. Israel J. Math., 63(1):1–40, 1988.
[172] C., Read. Strictly singular operators and the invariant subspace problem. Studia Math., 132(3):203–226, 1999.
[173] O., Rejasse. Factorization and reflexivity results for polynomially bounded operators. J. Operator Theory, 60(2):219–238, 2008.
[174] W. C., Ridge. Approximate point spectrum of a weighted shift. Trans. Amer. Math. Soc., 147:349–356, 1970.
[175] F., Riesz and B. Sz.-, Nagy. Functional Analysis. Frederick Ungar Publishing, New York, 1955. Translated by Leo F. Boron.
[176] G.-C., Rota. Note on the invariant subspaces of linear operators. Rend. Circ. Mat. Palermo (2), 8:182–184, 1959.
[177] G.-C., Rota. On models for linear operators. Comm. Pure Appl. Math., 13:469–472, 1960.
[178] W., Rudin. Function Theory in the Unit Ball of ℂn. Springer, Berlin, 1980.
[179] W., Rudin. Real and complex analysis, 3rd edition. McGraw-Hill, New York, 1987.
[180] W., Rudin. Functional Analysis, 2nd edition. International Series in Pure and Applied Mathematics. McGraw-Hill, New York, 1991.
[181] H. H., Schaefer. Banach Lattices and Positive Operators. Die Grundlehren der Mathematischen Wissenschaften, 215. Springer, New York, 1974.
[182] M., Schreiber. Unitary dilations of operators. Duke Math. J., 23:579–594, 1956.
[183] H. S., Shapiro and A. L., Shields. On some interpolation problems for analytic functions. Amer. J. Math., 83:513–532, 1961.
[184] J. H., Shapiro. Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics. Springer, New York, 1993.
[185] A., Simonič. An extension of Lomonosov's techniques to non-compact operators. Trans. Amer. Math. Soc., 348(3):975–995, 1996.
[186] D., Slepian. On bandwidth. Proc. IEEE, 64(3):292–300, 1976.
[187] M., Smith. Constrained approximation in Banach spaces. Constr. Approx., 19(3): 465–476, 2003.
[188] M. N., Spijker, S., Tracogna, and B. D., Welfert. About the sharpness of the stability estimates in the Kreiss matrix theorem. Math. Comp., 72(242):697–713, 2003.
[189] B. Sz.-, Nagy. Sur les contractions de l'espace de Hilbert. Acta Sci. Math. (Szeged), 15:87–92, 1953.
[190] B. Sz.-, Nagy and C., Foiaş. On certain classes of power-bounded operators in Hilbert space. Acta Sci. Math. (Szeged), 27:17–25, 1966.
[191] B. Sz.-, Nagy and C., Foiaş. Harmonic Analysis of Operators on Hilbert Space. North-Holland, Amsterdam, 1970. Translated from the French and revised.
[192] F. H., Szafraniec. Some spectral properties of operator-valued representations of function algebras. Ann. Polon. Math., 25:187–194. 1971/72.
[193] J. E., Thomson. Invariant subspaces for algebras of subnormal operators. Proc. Amer. Math. Soc., 96(3):462–464, 1986.
[194] T. T., Trent. Maximal invariant subspaces for. Proc. Amer. Math. Soc., 132(8):2429–2432, 2004.
[195] V. G., Troitsky. Lomonosov's theorem cannot be extended to chains of four operators. Proc. Amer. Math. Soc., 128(2):521–525, 2000.
[196] V. G., Troitsky. Minimal vectors in arbitrary Banach spaces. Proc. Amer. Math. Soc., 132(4):1177–1180, 2004.
[197] G., Valiron. Sur l'iteration des fonctions holomorphes dans un demi-plan. Bull. Sci. Math. (2), 55:105–128, 1931.
[198] F. H., Vasilescu. An operator-valued Poisson kernel. J. Funct. Anal., 110(1):47–72, 1992.
[199] F. H., Vasilescu. Operator-valued Poisson kernels and standard models in several variables. In R. E., Curto and P.E.T., Jørgensen, editors, Algebraic Methods in Operator Theory, pages 37–46. Birkhäuser, Boston, MA, 1994.
[200] J. L., Walsh. The Location of Critical Points of Analytic and Harmonic Functions. American Mathematical Society Colloquium Publications, 34. American Mathematical Society, New York, NY, 1950.
[201] J., Wermer. The existence of invariant subspaces. Duke Math. J., 19:615–622, 1952.
[202] D. J., Westwood. On C00-contractions with dominating spectrum. J. Funct. Anal., 66(1):96–104, 1986.
[203] K., Yan. Invariant subspaces for joint subnormal operators. Chinese Ann. Math. A., 9:561–566, 1988.
[204] A. C., Zaanen. Introduction to Operator Theory in Riesz Spaces. Springer, Berlin, 1997.
[205] M., Zarrabi. On polynomially bounded operators acting on a Banach space. J. Funct. Anal., 225(1):147–166, 2005.
[206] C., Zenger. On convexity properties of the Bauer field of values of a matrix. Num. Math., 12:96–105, 1968.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.