## Boundary values of Berezin symbols.(English)Zbl 0874.47013

Feintuch, A. (ed.) et al., Nonselfadjoint operators and related topics. Workshop on Operator theory and its applications, Beersheva, Israel, February 24-28, 1992. Basel: Birkhäuser Verlag. Oper. Theory, Adv. Appl. 73, 362-368 (1994).
Let $$\mathcal H$$ be a functional Hilbert space of functions defined on a set $$S$$ with reproducing kernel $$\{k_z\}_{z\in S}$$, so that $$k_z\in{\mathcal H}$$ and $$f(z)= (f,k_z)$$ for $$f\in{\mathcal H}$$ and $$z\in S$$. Given a bounded linear operator $$A$$ on $$\mathcal H$$, the function $$\widetilde A$$ defined on $$S$$ by $$\widetilde A(z)= (A\widehat k_z,\widehat k_z)$$, where $$\widehat k_z=|k_z|^{-1}k_z$$, is called the Berezin symbol of $$A$$. Also, $$\mathcal H$$ is said to be standard if the underlying set $$S$$ is a subset of a topological space, the boundary $$\partial S$$ of $$S$$ is non-empty and $$\widehat k_{z_n}\to 0$$ weakly in $$\mathcal H$$ whenever $$\{z_n\}$$ is a sequence in $$S$$ converging to a point in $$\partial S$$. The common functional Hilbert spaces, such as the Hardy space $$H^2$$ and the Bergman space, are standard. For a standard functional Hilbert space and a compact operator $$A$$ on $$\mathcal H$$, $$\widetilde A(z_n)\to 0$$ as $$n\to\infty$$ whenever $$\{z_n\}$$ is a sequence in $$S$$ converging to a point in $$\partial S$$. This is described by saying that the Berezin symbol of a compact operator on a standard functional Hilbert space vanishes on the boundary.
The present note was stimulated by the following question raised by C. A. Berger and L. A. Coburn: on the Hardy and Bergman spaces, must a bounded linear operator be compact if its Berezin symbol vanishes on the boundary of $$S$$? (In this case $$S$$ is the unit disc in the complex plane.)
Firstly, the authors present several counterexamples to this question. In a positive direction they show that, if the Berezin symbols of all unitary equivalents of an operator $$A$$ on a standard functional Hilbert space vanish on the boundary, then $$A$$ is compact. This result follows from a characterization of the essential numerical range of $$A$$ as the set of all cluster values of the Berezin symbols of all operators unitarily equivalent to $$A$$. In a similar vein, they show that the Berezin symbols of all operators unitarily equivalent to $$A$$ extend continuously to $$S\cup\partial S$$ if and only if $$A$$ is a translate by a scalar multiple of the identity of a compact operator.
For the entire collection see [Zbl 0798.00021].

### MSC:

 47B38 Linear operators on function spaces (general) 47B07 Linear operators defined by compactness properties 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47A12 Numerical range, numerical radius 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)