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Range Spaces of Co-Analytic Toeplitz Operators

Part of: Function theory on the disc Spaces and algebras of analytic functions Linear function spaces and their duals

Published online by Cambridge University Press:  20 November 2018

Emmanuel Fricain ,
Andreas Hartmann  and
William T. Ross
Emmanuel Fricain
Affiliation:
Laboratoire Paul Painlevé, Université Lille 1, 59 655 Villeneuve d’Ascq Cédex, France, e-mail: emmanuel.fricain@math.univ-lille1.fr
Andreas Hartmann
Affiliation:
Institut de Mathématiques de Bordeaux, Université Bordeaux/Bordeaux INP/CNRS, 351 cours de la Libération 33405 Talence, France, e-mail: Andreas.Hartmann@math.u-bordeaux.fr
William T. Ross
Affiliation:
Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173, USA, e-mail: wross@richmond.edu
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Abstract

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In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.

Keywords

Toeplitz operatorHardy spacerange spacede Branges–Rovnyak spaceboundary behaviorkernel functionnon-extreme pointcorona pair

MSC classification

Secondary: 30J05: Inner functions 30H10: Hardy spaces 46E22: Hilbert spaces with reproducing kernels (=
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 6 , 01 December 2018 , pp. 1261 - 1283
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Ahern, P. R. and Clark, D. N., Radial limits and invariant subspaces. Amer. J. Math. 92 (1970), 332–342. http://dx.doi.org/10.2307/2373326Google Scholar
[2] Bolotnikov, V. and Kheifets, A., A higher order analogue of the Carathéodory-Julia theorem. J. Funct. Anal. 237 (2006), no. 1, 350–371. http://dx.doi.Org/10.1016/j.jfa.2006.03.016Google Scholar
[3] Crofoot, R. B., Multipliers between invariant subspaces of the backward shift. Pacific J. Math. 166 (1994), no. 2, 225–246. http://dx.doi.org/10.2140/pjm.1994.166.225Google Scholar
[4] de Branges, L. and Rovnyak, J., Canonical models in quantum scattering theory. In: Perturbation theory and its applications in quantum mechanics. Wiley, New York, 1966, pp. 295–392.Google Scholar
[5] de Branges, L. and Rovnyak, J., Square summablepower series. Holt, Rinehart and Winston, New York, 1966.Google Scholar
[6] Douglas, R. G., Shapiro, H. S., and Shields, A. L., Cyclic vectors and invariant subspaces for the backward shift operator. Ann. Inst. Fourier (Grenoble) 20 (1970), 37–76. http://dx.doi.Org/10.5802/aif.338Google Scholar
[7] Duren, P. L., Theory of Hp spaces. Academic Press, New York, 1970.Google Scholar
[8] El-Fallah, O., Elmadani, Y., and Kellay, K., Kernel estimate and capacity in Dirichlet type spaces. arxiv:1411.1036Google Scholar
[9] El-Fallah, O., Kellay, K., Mashreghi, J., and Ransford, T., A primer on the Dirichlet space. Cambridge Tracts in Mathematics, 203. Cambridge University Press, Cambridge, 2014.Google Scholar
[10] Fricain, E., Hartmann, A., and Ross, W. T., Concrete examples ofJ$f(b) spaces. Comput. Methods Funct. Theory 16 (2016), no. 2, 287–306. http://dx.doi.org/10.1007/s40315-015-0144-9Google Scholar
[11] Fricain, E. and Mashreghi, J., Boundary behavior of functions in the de Branges-Rovnyak spaces. Complex Anal. Oper. Theory 2 (2008), no. 1, 87–97. http://dx.doi.Org/10.1007/s11785-007-0028-8Google Scholar
[12] Fricain, E. and Mashreghi, J., Integral representation of the n-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel. Ann. Inst. Fourier (Grenoble) 58 (2008), no. 6, 2113–2135. http://dx.doi.org/10.5802/aif.2408Google Scholar
[13] Fricain, E. and Mashreghi, J., Theory of ℋ(b) spaces. Vol. 1-2. Cambridge University Press, 2016.Google Scholar
[14] Garcia, S. R., Mashreghi, J., and Ross, W. T., Introduction to model spaces and their operators. Cambridge Studies in Advanced Mathematics, 148. Cambridge University Press, Cambridge, 2016.Google Scholar
[15] Garcia, S. R., Mashreghi, J., and Ross, W. T., Real complex functions. In: Recent progress on operator theory and approximation in spaces of analytic functions. Contemp. Math., 679. Amer. Math. Soc, Providence, RI, 2016, pp. 91-128.Google Scholar
[16] Garcia, S. R. and Ross, W. T., Recent progress on truncated Toeplitz operators. In: Blaschke products and their applications. Fields Inst. Commun., 65. Springer, New York, 2013, pp. 275–319.Google Scholar
[17] Garcia, S. R. and Sarason, D., Real outer functions. Indiana Univ. Math. J. 52 (2003), no. 6, 1397–1412. http://dx.doi.org/10.1512/iumj.2003.52.2511Google Scholar
[18] Garnett, J., Bounded analytic functions. Graduate Texts in Mathematics, 236. Springer, New York, 2007.Google Scholar
[19] Guillot, D., Fine boundary behavior and invariant subspaces of harmonically weighted Dirichlet spaces. Complex Anal. Oper. Theory 6 (2012), no. 6,1211-1230. http://dx.doi.Org/10.1007/s11785-010-0124-zGoogle Scholar
[20] Hartmann, A. and Ross, W. T., Boundary values in range spaces of co-analytic truncated Toeplitz operators. Publ. Mat. 56 (2012), no. 1, 191–223. http://dx.doi.org/10.5565/PUBLMAT_56112_07Google Scholar
[21] Hartmann, A., Sarason, D., and Seip, K., Surjective Toeplitz operators. Acta Sci. Math. (Szeged) 70 (2004), no. 3-4, 609–621.Google Scholar
[22] Hartmann, A. and Seip, K., Extremal functions as divisors for kernels of toeplitz operators. J. Funct. Anal. 202 (2003), no. 2, 342–362. http://dx.doi.org/10.1016/S0022-1236(03)00074-0Google Scholar
[23] Hayashi, E., Classification of nearly invariant subspaces of the backward shift. Proc. Amer. Math. Soc. 110 (1990), no. 2, 441–448. http://dx.doi.org/10.1090/S0002-9939-1990-1019277-0Google Scholar
[24] Helson, H., Large analytic functions. II. In: Analysis and partial differential equations. Lecture Notes in Pure and Appl. Math., 122. Dekker, New York, 1990, pp. 217–220.Google Scholar
[25] Hitt, D., Invariant subspaces of H2 of an annulus. Pacific J. Math. 134 (1988), no. 1, 101–120. http://dx.doi.Org/10.2140/pjm.1 988.134.101Google Scholar
[26] Lanucha, B. and Nowak, M., De Branges-Rovnyak spaces and generalized Dirichlet spaces. Publ. Math. Debrecen 91 (2017), no. 1-2, 171–184. http://dx.doi.Org/10.5486/PMD.2017.7762Google Scholar
[27] Nikolski, N. K., Treatise on the shift operator. Springer-Verlag, Berlin, 1986.Google Scholar
[28] Nikolski, N. K., Operators, functions, and systems: an easy reading. Mathematical Surveys and Monographs, 92-93. American Mathematical Society, Providence, RI, 2002.Google Scholar
[29] Nikolski, N. K., Operators, functions, and systems: an easy reading. Mathematical Surveys and Monographs, 93. American Mathematical Society, Providence, RI, 2002.Google Scholar
[30] Paulsen, V. I. and Raghupathi, M., An introduction to the theory of reproducing kernel Hilbert spaces. Cambridge Studies in Advanced Mathematics, 152. Cambridge University Press, Cambridge, 2016.Google Scholar
[31] Richter, S., A representation theorem for cyclic analytic two-isometries. Trans. Amer. Math. Soc. 328 (1991), no. 1, 325–349. http://dx.doi.org/10.1090/S0002-9947-1991-1013337-1Google Scholar
[32] Sarason, D., Kernels of Toeplitz operators. In: Toeplitz operators and related topics. Oper. Theory Adv. Appl., 71. Birkhâuser, Basel, 1994, pp.153–164.Google Scholar
[33] Sarason, D., Sub-Hardy Hilbert spaces in the unit disk. University of Arkansas Lecture Notes in the Mathematical Sciences, 10. John Wiley and Sons, New York, 1994.Google Scholar
[34] Sarason, D., Unbounded Toeplitz operators. Integral Equations Operator Theory 61 (2008), no. 2, 281–298. http://dx.doi.org/10.1007/s00020-008-1588-3Google Scholar