Dilations and Operator Models of W $${\mathcal {W}}$$ -Hypercontractions

Monojit Bhattacharjee (), B. Krishna Das (), Ramlal Debnath () and Samir Panja ()
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Monojit Bhattacharjee: Birla Institute of Technology and Science - Pilani, Department of Mathematics
B. Krishna Das: Indian Institute of Technology Bombay, Department of Mathematics
Ramlal Debnath: Indian Institute of Technology Bombay, Department of Mathematics
Samir Panja: Indian Institute of Technology Bombay, Department of Mathematics

A chapter in Multivariable Operator Theory, 2023, pp 119-154 from Springer

Abstract: Abstract We revisit the study of $$\omega $$ ω -hypercontractions corresponding to a single weight sequence $$\omega =\{\omega _k\}_{k\ge 0}$$ ω = { ω k } k ≥ 0 introduced by Olofsson ([22] J Oper Theory 74:249–280, 2015) and find an analogue of Nagy-Foias characteristic function in this setting. Explicit construction of characteristic functions is obtained and it is shown to be a complete unitary invariant. By considering a multi-weight sequence $${\mathcal {W}}$$ W and $${\mathcal {W}}$$ W -hypercontractions we extend Olofsson’s work ([22] J Oper Theory 74:249–280, 2015) in the multi-variable setting. Model for $${\mathcal {W}}$$ W -hypercontractions is obtained by finding their dilations on certain weighted Bergman spaces over the polydisc corresponding to the multi-weight sequence $${\mathcal {W}}$$ W . This recovers and provides a different proof of the earlier work of Curto and Vasilescu ([13] Indiana Univ Math J 42:791–810, 1993 and [14] Indiana Univ 12 Math J 44:727–746, 1995) for y-contractive multi-operators through a particular choice of multi-weight sequence.

Keywords: Hypercontraction; Weighted shift; Characteristic function; Bergman space over the polydisc; Commuting contractions; Bounded analytic functions; 47A13; 47A20; 47A45; 47A56; 46E22; 47B32; 32A36; 47B20 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/978-3-031-50535-5_6

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