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On Compact Perturbations of Hankel Operators and Commutators of Toeplitz and Hankel Operators

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Abstract

Motivated by results in uniform algebras, a distance localization formula in C*-algebras is established in the framework of the Allan–Douglas localization principle, and is used to derive a locality result for products of Hankel operators as compact perturbations of Hankel operators. Using localization and certain QC functions, it is proved that the essential spectrum of the commutator of a Toeplitz and a Hankel operator is antipodal symmetric under a mild condition on the Hankel symbol function. Under the same condition the essential spectrum of a Hankel operator also exhibits this symmetry. Conjugates of interpolating Blaschke products and characteristic functions are constructed that satisfy the condition, while examples show the condition is only sufficient.

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  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Yi Yan

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  1. Yi Yan
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Correspondence to Yi Yan.

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This paper is part of the author’s Ph.D. thesis work under the guidance and encouragement of Professors Tyrone Duncan and Albert Sheu. The author is also grateful for the referee’s careful review and helpful suggestions.

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Yan, Y. On Compact Perturbations of Hankel Operators and Commutators of Toeplitz and Hankel Operators. Integr. Equ. Oper. Theory 92, 2 (2020). https://doi.org/10.1007/s00020-019-2557-8

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  • DOI: https://doi.org/10.1007/s00020-019-2557-8

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