Algebraic and spectral properties of dual Toeplitz operators

Stroethoff, Karel; Zheng, Dechao

doi:10.1090/S0002-9947-02-02954-9

Algebraic and spectral properties of dual Toeplitz operators
HTML articles powered by AMS MathViewer

by Karel Stroethoff and Dechao Zheng PDF

Trans. Amer. Math. Soc. 354 (2002), 2495-2520 Request permission

Abstract:

Dual Toeplitz operators on the orthogonal complement of the Bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement. In this paper we study algebraic and spectral properties of dual Toeplitz operators.

References

P. Ahern and Ž. Čučković, Products of Toeplitz Operators on the Bergman Space, Illinois J. Math. 45 (2001), 113–121.
Sheldon Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 1–50. MR 958569
Sheldon Axler and Željko Čučković, Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator Theory 14 (1991), no. 1, 1–12. MR 1079815, DOI 10.1007/BF01194925
Sheldon Axler and Dechao Zheng, The Berezin transform on the Toeplitz algebra, Studia Math. 127 (1998), no. 2, 113–136. MR 1488147, DOI 10.4064/sm-127-2-113-136
Sheldon Axler and Dechao Zheng, Compact operators via the Berezin transform, Indiana Univ. Math. J. 47 (1998), no. 2, 387–400. MR 1647896, DOI 10.1512/iumj.1998.47.1407
Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89–102. MR 160136, DOI 10.1007/978-1-4613-8208-9_{1}9
John Bunce, The joint spectrum of commuting nonnormal operators, Proc. Amer. Math. Soc. 29 (1971), 499–505. MR 283602, DOI 10.1090/S0002-9939-1971-0283602-7
Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
Pamela Gorkin and Dechao Zheng, Essentially commuting Toeplitz operators, Pacific J. Math. 190 (1999), no. 1, 87–109. MR 1722767, DOI 10.2140/pjm.1999.190.87
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74–111. MR 215102, DOI 10.2307/1970361
James W. Lark III, Spectral theorems for a class of Toeplitz operators on the Bergman space, Houston J. Math. 12 (1986), no. 3, 397–404. MR 869122
G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), no. 4, 595–611. MR 542947, DOI 10.1512/iumj.1979.28.28042
N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1
Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
Donald Sarason, Function theory on the unit circle, Virginia Polytechnic Institute and State University, Department of Mathematics, Blacksburg, Va., 1978. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19–23, 1978. MR 521811
Karel Stroethoff, Essentially commuting Toeplitz operators with harmonic symbols, Canad. J. Math. 45 (1993), no. 5, 1080–1093. MR 1239915, DOI 10.4153/CJM-1993-059-2
Karel Stroethoff, The Berezin transform and operators on spaces of analytic functions, Linear operators (Warsaw, 1994) Banach Center Publ., vol. 38, Polish Acad. Sci. Inst. Math., Warsaw, 1997, pp. 361–380. MR 1457018
Karel Stroethoff and De Chao Zheng, Toeplitz and Hankel operators on Bergman spaces, Trans. Amer. Math. Soc. 329 (1992), no. 2, 773–794. MR 1112549, DOI 10.1090/S0002-9947-1992-1112549-7
Karel Stroethoff and Dechao Zheng, Products of Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal. 169 (1999), no. 1, 289–313. MR 1726756, DOI 10.1006/jfan.1999.3489
Harold Widom, On the spectrum of a Toeplitz operator, Pacific J. Math. 14 (1964), 365–375. MR 163173
Ke He Zhu, Operator theory in function spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 139, Marcel Dekker, Inc., New York, 1990. MR 1074007