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Isometric dilation and Sarason’s commutant lifting theorem in several variables

Part of: Special classes of linear operators General theory of linear operators Spaces and algebras of analytic functions

Published online by Cambridge University Press:  25 February 2025

B. Krishna Das  and
Samir Panja
B. Krishna Das
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India e-mail: [email protected] [email protected]
Samir Panja*
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India e-mail: [email protected] [email protected]
*
e-mail: [email protected] [email protected]
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Abstract

The article deals with isometric dilation and commutant lifting for a class of n-tuples $(n\ge 3)$ of commuting contractions. We show that operator tuples in the class dilate to tuples of commuting isometries of BCL type. As a consequence of such an explicit dilation, we show that their von Neumann inequality holds on a one-dimensional variety of the closed unit polydisc. On the basis of such a dilation, we prove a commutant lifting theorem of Sarason’s type by establishing that every commutant can be lifted to the dilation space in a commuting and norm-preserving manner. This further leads us to find yet another class of n-tuples $(n\ge 3)$ of commuting contractions each of which possesses isometric dilation.

Keywords

Isometric dilationcommutant lifting theoremvon Neumann inequalityHardy spacecommuting contractionscommuting isometriesbounded analytic functions

MSC classification

Primary: 47A20: Dilations, extensions, compressions 47A13: Several-variable operator theory (spectral, Fredholm, etc.) 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
Secondary: 30H10: Hardy spaces
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 33
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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