Reproducing kernel Hilbert spaces (RKHS) arise in various areas of mathematics and applied sciences, including approximation theory, statistics, machine learning, group representation theory, and complex analysis. In this talk, we introduce the formal definition of an RKHS and discuss some of its fundamental properties.
Given a bounded linear operator \( T \) acting on an RKHS \( \mathcal{H} \), the \(\textit{Berezin range}\) of \( T \) is defined as
\(\mathrm{Ber}(T) := \{ \langle T \hat{k}_x, \hat{k}_x \rangle_{\mathcal{H}} : x \in X \},\)
where \( \hat{k}_x \) denotes the normalized reproducing kernel of \( \mathcal{H} \) at \( x \in X \).
For comparison, the \(\textit{numerical range}\) of a bounded linear operator \( T \) on a Hilbert space \( \mathcal{H} \) is given by
\(W(T) := \{ \langle T u, u \rangle : \|u\| = 1 \}.\)
By the Toeplitz--Hausdorff theorem, \( W(T) \) is always convex. It is straightforward to verify that \( \mathrm{Ber}(T) \subseteq W(T) \); however, in general, the Berezin range need not be convex.
In this talk, we focus on characterizing the convexity of the Berezin range for multiplication operators and certain classes of composition operators on the Hardy--Hilbert space.
Dr. Shankar P is an Assistant Professor in the Department of Mathematics at CUSAT, Kerala. He obtained his M.Sc. from Pondicherry University and his Ph.D. from the Kerala School of Mathematics(KSoM), Kerala. Subsequently he served as a Postdoctoral Fellow at the Indian Statistical Institute(ISI), Bangalore. His research interests include functional analysis, operator algebras, and operator theory.
