The essential spectrum of some Toeplitz operators

Clancey, Kevin F.; Morrel, Bernard B.

doi:10.1090/S0002-9939-1974-0341162-9

The essential spectrum of some Toeplitz operators
HTML articles powered by AMS MathViewer

by Kevin F. Clancey and Bernard B. Morrel PDF

Proc. Amer. Math. Soc. 44 (1974), 129-134 Request permission

Abstract:

The localization techniques of Douglas and Sarason are used to obtain the essential spectrum of the Toeplitz operator ${T_\varphi }$ for which $\varphi$ is the product of a continuous function and the characteristic function of a measurable subset of the unit circle. Examples are given of Toeplitz operators with one-dimensional self-commutator whose essential spectrum is the unit disk. Using an example of J. E. Brennan, the authors show the existence of a completely nonnormal, subnormal operator whose adjoint has no point spectrum.

References

James E. Brennan, Invariant subspaces and rational approximation, J. Functional Analysis 7 (1971), 285–310. MR 0423059, DOI 10.1016/0022-1236(71)90036-x
Richard W. Carey and Joel D. Pincus, An invariant for certain operator algebras, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 1952–1956. MR 344925, DOI 10.1073/pnas.71.5.1952
Kevin F. Clancey, The essential spectrum of a class of singular integral operators, Amer. J. Math. 96 (1974), 298–307. MR 361928, DOI 10.2307/2373634
Allen Devinatz, Toeplitz operators on $H^{2}$ spaces, Trans. Amer. Math. Soc. 112 (1964), 304–317. MR 163174, DOI 10.1090/S0002-9947-1964-0163174-9
Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893
R. G. Douglas and Donald Sarason, Fredholm Toeplitz operators, Proc. Amer. Math. Soc. 26 (1970), 117–120. MR 259639, DOI 10.1090/S0002-9939-1970-0259639-X
Philip Hartman, On unbounded Toeplitz matrices, Amer. J. Math. 85 (1963), 59–78. MR 150584, DOI 10.2307/2373186
Philip Hartman and Aurel Wintner, On the spectra of Toeplitz’s matrices, Amer. J. Math. 72 (1950), 359–366. MR 36936, DOI 10.2307/2372039
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
Marvin Rosenblum, A concrete spectral theory for self-adjoint Toeplitz operators, Amer. J. Math. 87 (1965), 709–718. MR 181901, DOI 10.2307/2373070
J. G. Stampfli, Hyponormal operators and spectral density, Trans. Amer. Math. Soc. 117 (1965), 469–476. MR 173161, DOI 10.1090/S0002-9947-1965-0173161-3