##
**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.

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.

Reviewer: V.F.Molchanov (Tambov)

### MSC:

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 |