Analysis Seminar
Tuesday, 30 Jan. 2:30 pm
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Venue: Ramanujan hall
Host: Sourav Pal
Speaker: Swapan Jana
Affiliation: IIT Bombay
Title: Positive definite functions and a generalized Naimark theorem
Abstract: We study the operator-valued positive definite functions on a group and characterize them in terms of positive block matrices. We present a generalization of a theorem of Naimark which states that the positive definite functions and unitary representations are in one-to-one correspondence. We discuss an alternative proof of Brehmer's positivity for doubly commuting operators. We classify all commuting unitary representations over a finite group and present a structure theorem. In the end, we show that the power of a positive-definite function and a unitary representation need not be a positive definite function and a unitary representation respectively. We also present a characterization of such a class of functions.