Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


A note on the spectral area of Toeplitz operators

Authors: Cheng Chu and Dmitry Khavinson
Journal: Proc. Amer. Math. Soc. 144 (2016), 2533-2537
MSC (2010): Primary 30J99, 47B35
Published electronically: October 20, 2015
MathSciNet review: 3477069
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we show that for hyponormal Toeplitz operators, there exists a lower bound for the area of the spectrum. This extends the known estimate for the spectral area of Toeplitz operators with an analytic symbol.

References [Enhancements On Off] (What's this?)

  • [1] H. Alexander, B. A. Taylor, and J. L. Ullman, Areas of projections of analytic sets, Invent. Math. 16 (1972), 335-341. MR 0302935 (46 #2078)
  • [2] Sheldon Axler and Joel H. Shapiro, Putnam's theorem, Alexander's spectral area estimate, and VMO, Math. Ann. 271 (1985), no. 2, 161-183. MR 783550 (87b:30053),
  • [3] Steven R. Bell, Timothy Ferguson, and Erik Lundberg, Self-commutators of Toeplitz operators and isoperimetric inequalities, Math. Proc. R. Ir. Acad. 114A (2014), no. 2, 115-133. MR 3353499
  • [4] C. Bénéteau and D. Khavinson, Selected problems in classical function theory. To appear in CRM Proceedings and Lecture Notes.
  • [5] Catherine Bénéteau and Dmitry Khavinson, The isoperimetric inequality via approximation theory and free boundary problems, Comput. Methods Funct. Theory 6 (2006), no. 2, 253-274. MR 2291136 (2007i:30001),
  • [6] C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103 (1988), 809-812.
  • [7] Matthew Fleeman and Dmitry Khavinson, Extremal domains for self-commutators in the Bergman space, Complex Anal. Oper. Theory 9 (2015), no. 1, 99-111. MR 3300527,
  • [8] 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 (83g:30037)
  • [9] D. Khavinson, A note on Toeplitz operators, Banach spaces (Columbia, Mo., 1984) Lecture Notes in Math., vol. 1166, Springer, Berlin, 1985.
  • [10] J-F. Olsen and M. Reguera, On a sharp estimate for Hankel operators and Putnam's inequality, arXiv:1305.5193v2 (2014).
  • [11] C.R. Putnam, An inequality for the area of hypernormal spectra, Math. Z. 116 (1970), 323-330.
  • [12] W. Rudin, Functional analysis, McGraw-Hill, New York, 1991.
  • [13] Charles S. Stanton, Counting functions and majorization for Jensen measures, Pacific J. Math. 125 (1986), no. 2, 459-468. MR 863538 (88c:32002)
  • [14] C. Stanton, Isoperimetric inequalities and $ H^p$ estimates, Complex Var. Theory Appl. 12 (1989), no. 1-4, 17-21.
  • [15] H. Widom, On the spectrum of a Toeplitz operator, Pacific J. Math. 14 (1964), 365-375.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30J99, 47B35

Retrieve articles in all journals with MSC (2010): 30J99, 47B35

Additional Information

Cheng Chu
Affiliation: Department of Mathematics, Washington University in Saint Louis, Saint Louis, Missouri

Dmitry Khavinson
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida

Received by editor(s): March 23, 2015
Received by editor(s) in revised form: July 13, 2015
Published electronically: October 20, 2015
Communicated by: Pamela Gorkin
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society