Berezin transforms and group representations. (English) Zbl 0919.43008

This paper gives an amplification of some general constructions of the paper by A. Unterberger and H. Upmeier [Commun. Math. Phys. 164, 563-597 (1994; Zbl 0843.32019)], devoted to the proof of a remarkable Berezin’s formula for eigenvalues of the Berezin transform for Hermitian symmetric spaces. See the paper by F. A. Berezin [Sov. Math., Dokl. 19, 786-789 (1978); translation from Dokl. Akad. Nauk SSR 241, 15-17 (1978; Zbl 0439.47038)].
A general definition of the Berezin transform consists of the following. Let \(X\) be a space (a manifold) with a measure \(\mu\). Let \(H\) be a closed subspace of \(L^2(X,d\mu)\) consisting of continuous functions. Assume \(H\) has a continuous reproducing kernel \(\kappa(x,y)\). Let \(X_0\) consist of \(x\) with \(\kappa(x,x)\neq 0\) and \(d\mu_0= \kappa(x,x)d\mu\). For a bounded operator \(A\) on \(H\) the bounded function \(\sigma(A)\) on \(X\) defined by \(\sigma(A)= (Ae_x, e_x)\), where \(e_x(y)= \kappa(y,x)(\kappa(x,x))^{-1/2}\), is called the Berezin covariant symbol of \(A\). Let \(P\) be the orthogonal projection of \(L^2(X, d\mu)\) onto \(H\). For every \(f\in L^\infty(X)\), the bounded operator \(\sigma^*(f)\) on \(H\) defined by \(\sigma^*(f)\xi= P(f\xi)\) (a Toeplitz operator) is called the Berezin operator with contravariant symbol \(f\). The Berezin transform \(B= \sigma\sigma^*\) acts on \(L^\infty(X)\). Let \(B_2(H)\) be the Hilbert space of Hilbert-Schmidt operators on \(H\). It turns out that \(\sigma\) maps \(B_2(H)\) into \(L^2(X_0, d\mu_0)\) and \(\sigma^*\) maps \(L^2(X_0, d\mu_0)\) into \(B_2(H)\), and they are adjoint to each other; the Berezin transform extends to a bounded operator on \(L^2(X_0, d\mu_0)\) with the norm \(\leq 1\).
The amplification is based on the observation that \(B_2(H)\) can be identified with the Hilbert space tensor product \(H\otimes\overline H\) (where \(\overline H\) denotes the Hilbert space conjugate-linearly isomorphic to \(H\)). It allows to consider \(\sigma\) as a “diagonalization” operator \(M\) defined firstly on the algebraic tensor product by \(\xi\otimes \overline\eta\to \kappa(x, x)^{-1} \xi(x)\overline{\eta(x)}\).
Let now \(\pi\) be a unitary representation of a locally compact group \(G\) on \(L^2(X, d\mu)\) with a cocycle. Let \(H\) be \(G\)-invariant. Then the quasiregular representation \(\rho\) of \(G\) on \(L^2(X_0, d\mu_0)\) is unitary, \(M\) gives an equivalence of \(\pi\otimes\overline \pi\) and \(\rho\) with corresponding restrictions, and the Berezin transform \(B\) is \(G\)-invariant with \(\text{Ker }B= (\text{range }M)^\perp\).
In the end of the paper two examples are given: (a) \(H\) is the classical Fock space of entire functions on \(\mathbb{C}^n\) with the Gaussian measure on \(\mathbb{C}^n\) and \(G\) is the Heisenberg group \(\mathbb{C}^n\cdot \mathbb{R}\); (b) \(X= \mathbb{R}^n\) with the Gaussian measure, \(G\) is a compact linear group acting on \(\mathbb{R}^n\) and \(H\) is an irreducible subspace of polynomials. It is proved in this case that \(B\) acts on the subspace of \(G\)-invariant functions as the orthogonal projection operator onto the one-dimensional subspace of constant functions, so that, in particular, \(\| B\|= 1\).
Reviewer’s note: The appearance of tensor products in the construction of the Berezin transform was observed first by Berezin himself, see, for example, loc. cit.


43A85 Harmonic analysis on homogeneous spaces
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
47N50 Applications of operator theory in the physical sciences
Full Text: EuDML