Invariant Berezin transforms (Q2704079)

Let \(D\) be a domain in \(\mathbf C^n\), \(\mu\) a measure on~\(D\), and \(\mathcal H\) the Bergman space of all holomorphic functions in \(L^2(D,\mu)\). Assume that \(\mathcal H\) admits a reproducing kernel \(\kappa(x,y)\). Recall that the \textsl{Berezin transform} \(B\) is the integral operator on \(D\) defined by \(Bf(y)= \kappa(y,y)^{-1} \int_D f(x) |\kappa(x,y)|^2 d\mu(x)\). More generally, the same construction makes sense even for any reproducing kernel Hilbert subspace \(\mathcal H\) of \(L^2(X,\mu)\) with \(\mu\) a Radon measure on a locally compact space~\(X\). Further, if there is a Lie group \(G\) acting on \(X\) and \(\mathcal H\) carries a unitary representation of \(G\) naturally defined by this action, then \(B\) is \(G\)-invariant. NEWLINENEWLINENEWLINEThe present paper surveys a number of results by the author on this topic, some of which have already appeared. Namely, the situations when \(\mathcal H\) are the \(K\)-irreducible subspaces of the space of polynomials on a vector space~\(V\), in the cases of \(K=U(n)\) acting on \(V=\mathbf C^n\) [\textit{E.~Fujita} and \textit{T.~Nomura}, J. Math. Kyoto Univ. 36, 877-888 (1996; Zbl 0910.43006)], \(K=U(2)\times U(2)\) acting on \(V=M(2,\mathbf C)\) [same authors, Integral Equations Oper. Theory 32, 152-179 (1998; Zbl 0910.43008)], and \(K=SO(n,\mathbf R)\) acting on \(V=\mathbf R^n\). Finally, the case of symmetric Siegel domains \(X\) is discussed, with \(G\) the identity component of the group of holomorphic automorphisms of~\(X\), and an alternative proof of the Berezin-Unterberger-Upmeier formula for the eigenvalues of \(B\) is indicated.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00041].