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Compact and finite rank operators satisfying a Hankel type equationT 2X=XT *1

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Abstract

In 1997, V. Pták defined the notion of generalized Hankel operator as follows: Given two contractions\(\mathcal{B}(\mathcal{H}_1 )\) and\(\mathcal{B}(\mathcal{H}_2 )\), an operatorX:\(X:\mathcal{H}_1 \to \mathcal{H}_2 \) is said to be a generalized Hankel operator ifT 2 X=XT *1 andX satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations ofT 1 andT 2. The purpose behind this kind of generalization is to study which properties of classical Hankel operators depend on their characteristic intertwining relation rather than on the theory of analytic functions. Following this spirit, we give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Pták's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator, Hartman's characterization of compact Hankel operators and Kronecker's characterization of finite rank Hankel operators.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada II, Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41012, Sevilla, Spain

    Carmen H. Mancera & Pedro J. Paúl

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  1. Carmen H. Mancera
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  2. Pedro J. Paúl
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Dedicated to the memory of our master and friend Vlastimil Pták

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Mancera, C.H., Paúl, P.J. Compact and finite rank operators satisfying a Hankel type equationT 2X=XT *1 . Integr equ oper theory 39, 475–495 (2001). https://doi.org/10.1007/BF01203325

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