On Toeplitz operators with unimodular symbols: left invertibility and similarity to isometries

M. F. Gamal'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 38, Tome 376 (2010), pp. 5-24

Résumé

Toeplitz operators with unimodular symbols on the Hardy space $H^2$ on the unit circle are considered. It is shown that the left invertibility of a Toeplitz operator with symbol $e^{it}\mapsto\theta(e^{it})e^{it/2}$, $t\in(0,2\pi)$, where $\theta$ is an inner function, depends on $\theta$. Also, Toeplitz operators that are similar to isometries are studed. Bibl. – 28 titles.

Bibliographie

[1] H. Bercovici, Operator theory and arithmetic in $H^\infty$, Math. Surveys and Monographs, 26, AMS, 1988 | DOI | MR | Zbl

[2] H. Bercovici, “Notes on invariant subspaces”, Bull. Amer. Math. Soc., 23 (1990), 1–36 | DOI | MR | Zbl

[3] A. Böttcher, S. M. Grudsky, “Toeplitz operators with discontinuous symbols: phenomena beyond piecewise continuity”, Singular integral operators and related topics, Oper. Theory Adv. Appl., 90, Birkhäuser, Basel, 1996, 55–118 | MR | Zbl

[4] A. Böttcher, B. Silbermann, Analysis of Toeplitz operators, Springer, 1990 | MR | Zbl

[5] D. N. Clark, “On a similarity theory for rational Toeplitz operators”, J. Reine Angew. Math., 320 (1980), 6–31 | DOI | MR | Zbl

[6] D. N. Clark, “On Toeplitz operators with loops”, J. Operator Theory, 4 (1980), 37–54 | MR | Zbl

[7] D. N. Clark, “On Toeplitz operators with unimodular symbols”, Operators in indefinite metric spaces, scattering theory and other topics (Bucharest, 1985), Oper. Theory Adv. Appl., 24, Birkhäuser, Basel, 1987, 59–68 | MR

[8] D. N. Clark, “Perturbation and similarity of Toeplitz operators”, Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl., 48, Birkhäuser, Basel, 1990, 235–243 | MR

[9] D. N. Clark, J. H. Morrel, “On Toeplitz operators and similarity”, Amer. J. Math., 100 (1978), 973–986 | DOI | MR | Zbl

[10] P. L. Duren, Theory of $H^p$ spaces, Academic Press, 1970 | MR | Zbl

[11] V. B. Dybin, S. M. Grudsky, Introduction to the theory of Toeplitz operators with infinite index, Oper. Theory Adv. Appl., 137, Birkhäuser, Basel, 2002 | MR | Zbl

[12] M. F. Gamal, “Ob operatorakh Tëplitsa, podobnykh odnostoronnemu sdvigu”, Zap. nauchn. semin. POMI, 345, 2007, 85–104 | MR

[13] M. F. Gamal', “On Toeplitz operators similar to isometries”, J. Operator Theory, 59 (2008), 3–28 | MR | Zbl

[14] M. F. Gamal', “On contractions that are quasiaffine transforms of unilateral shifts”, Acta Sci. Math. (Szeged), 74 (2008), 757–767 | MR | Zbl

[15] R. Goor, “On Toeplitz operators which are contractions”, Proc. Amer. Math. Soc., 34 (1972), 191–192 | DOI | MR | Zbl

[16] V. V. Peller, “When is a function of a Toeplitz operator close to a Toeplitz operator?”, Toeplitz operators and spectral function theory, Oper. Theory Adv. Appl., 42, Birkhäuser, Basel, 1989, 59–85 | DOI | MR

[17] V. V. Peller, Hankel operators and their applications, Springer monographs in math., 2003 | DOI | MR | Zbl

[18] V. V. Peller, S. V. Khruschëv, “Operatory Gankelya, nailuchshie priblizheniya i statsionarnye gaussovskie protsessy”, Uspekhi mat. nauk, 37:1 (1982), 53–124 | MR | Zbl

[19] H. Radjavi, P. Rosenthal, Invariant subspaces, Springer, 1973 | MR | Zbl

[20] M. Rosenblum, J. Rovnyak, Hardy classes and operator theory, Oxford Math. Monogr., 1985 | MR | Zbl

[21] J. Rovnyak, “On the theory of unbounded Toeplitz operators”, Pacific J. Math., 31 (1969), 481–496 | DOI | MR | Zbl

[22] D. Sarason, “Approximation of piecewise continuous functions by quotients of bounded analytic functions”, Canadian J. Math., 24 (1972), 642–657 | DOI | MR | Zbl

[23] B. Sekefalvi-Nad, Ch. Foyash, Garmonicheskii analiz operatorov v gilbertovom prostranstve, Mir, M., 1970 | MR

[24] K. Takahashi, “Injection of unilateral shifts into contractions”, Acta Sci. Math. (Szeged), 57 (1993), 263–276 | MR | Zbl

[25] D. Wang, “Similarity theory of smooth Toeplitz operators”, J. Operator Theory, 12 (1984), 319–330 | MR

[26] D. V. Yakubovich, “Riemann surface models of Toeplitz operator”, Toeplitz operators and spectral function theory, Oper. Theory Adv. Appl., 42, Birkhäuser, Basel, 1989, 305–415 | DOI | MR

[27] D. V. Yakubovich, “K spektralnoi teorii operatorov Tëplitsa s gladkim simvolom”, Algebra i analiz, 3:4 (1991), 208–226 | MR | Zbl

[28] D. V. Yakubovich, “Dual piecewise analytic bundle shift models of linear operators”, J. Funct. Anal., 136 (1996), 294–330 | DOI | MR | Zbl