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. 030524 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, coextends) to a pure isometry. In this talk, we will first address the question: can a pair of commuting pure contractions dilate (i.e. simultaneously coextend) 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 wellknown Hilbert spaces of analytic functions.
