Mathematics Seminar
Speaker: Srijan Sarkar, IISc
Date: Friday, 3 May 2024
Time: 2 PM
Venue: Lecture Hall 2
Analytic model for contractions on Hilbert spaces and Toeplitz operators
Srijan Sarkar
IISc.
03-05-24

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Abstract

A contraction T (that is, an operator with ||T|| ??? 1) on a Hilbert space H is said to be pure if the sequence ||T^{*n}h|| converges to 0 as n goes to infinity for all h in H. The fundamental analytic model of Sz.???Nagy???Foias shows that these contractions are unitarily equivalent to certain compressions of the shift operator on some Hilbert space (i.e. of the form PQMzPQ). This outcome arises from an important result: a pure contraction always dilates (or in other words, co-extends) to a pure isometry. In this talk, we will first address the question: can a pair of commuting pure contractions dilate (i.e. simultaneously co-extend) to pure isometries? We will identify certain pairs of commuting contractions that affirmatively answer this question.

Generally, pure contractive Toeplitz operators on Hilbert spaces of analytic functions play a vital role in connecting diverse fields such as operator theory and several complex variables. Despite this significance, a comprehensive understanding has been lacking regarding the conditions under which a Toeplitz operator transforms into a pure contraction. In the second part of this talk, we will find answers to this question for analytic Toeplitz operators acting on well-known Hilbert spaces of analytic functions.