Berezin transform and Weyl-type unitary operators on the Bergman space (Q2845861)

Let \(Op(H)\) denote the algebra of all bounded linear operators on a complex Hilbert space \(H\) and \({\mathbb{D}}\) denote the open unit disc. In the case where \(H\) is the Bergman Hilbert space \(L^2_a({\mathbb{D}})\) or one of a large family of reproducing kernel Hilbert spaces, we have, for each \(c\) in \({\mathbb{D}}\), a reproducing kernel function \(K(\cdot, c)\) so that, for any \(f\) in \(L^2_a({\mathbb{D}})\), \(f(c) = \langle f, K(\cdot, c) \rangle\). The normalized kernel function is defined by \(k_c(\cdot) = K(\cdot, c)K(c,c)^{-1/2}\).NEWLINENEWLINEFor every bounded linear operator \(X\), the Berezin transform is defined by \(\tilde{X}(c) = \langle X k_c, k_c \rangle\). Let \(\mathrm{Ber}(X) = \tilde{X}\), let \(BC(({\mathbb{D}}))\) denote the Banach space of bounded continuous functions on \({\mathbb{D}}\), and let \(\mathrm{Aut}({\mathbb{D}})\) denote the full group, given for \(\lambda\) in \({\mathbb{C}}\) with \(|\lambda|=1\) and \(c,z\) in \({\mathbb{D}}\), \([\lambda, c](z) = \lambda \frac{z-c}{1- \bar{c}z}\). It is standard that \((V_{[\lambda, c]}f)(z) = k_c(z)f(\lambda \frac{z-c}{1-\bar{c}z})\) is a unitary transformation from \(L^2_a({\mathbb{D}})\) to itself. Then the author establishes the following theorems.NEWLINENEWLINETheorem 1: The range of \(\mathrm{Ber} : Op[L^2_a({\mathbb{D}})] \rightarrow BC(({\mathbb{D}}))\) is invariant under \(\mathrm{Aut}({\mathbb{D}})\).NEWLINENEWLINETheorem 2: \(\|\tilde{V}_{[1,c]}\|_\infty = 1- |c|^2\) and \(\|\tilde{V}_{[-1,c]}\|_\infty = 1\) for all \(c\) in \({\mathbb{D}}\).NEWLINENEWLINETheorem 3: For \(X_c = V_{[1, c]} + V_{[1, -c]}\), \(\|\tilde{X}_c\|_\infty = 2(1-|c|^2)\) and \(\|\widetilde{X_c^2}\|_\infty = 4(1+ |c|^4)/(1+ |c|^2)^2\) for all \(c\) in \({\mathbb{D}}\). Thus, \(\|\widetilde{X_c^2}\|_\infty > \|\tilde{X}_c\|_\infty^2\) for all \(c\) with \(1> |c| > 0\).NEWLINENEWLINEThe author next considers the Weyl-type unitary operators on the Segal-Bargmann space \(H^2({\mathbb{C}}^n, d\mu)\). The analogues of \(V_{[1,c]}\) and \(V_{[-1,c]}\) are the Weyl unitary operators acting on \(H^2({\mathbb{C}}^n, d\mu)\) by \((W_c f)(z) = k_c(z)f(z-c)\) and the involutive unitary operators \((U_c f)(z) = k_c(z)f(c-z)\).NEWLINENEWLINETheorem 4: \(\|\tilde{W}_c\|_\infty = \exp (-|c|^2/4)\) and \(\| \tilde{U}_c \|_\infty = 1\) for all \(c\) in \({\mathbb{C}}^n\).NEWLINENEWLINETheorem 5: For \(Y_c = W_c + W_{-c}\), \(\|\tilde{Y}_c\|_\infty = 2\exp (-|c|^2/4)\) and \(\|\widetilde{Y_c^2}\|_\infty = 2(1+ \exp (-|c|^2))\) for all \(c\) in \({\mathbb{C}}^n\). Thus, \(\|\widetilde{Y_c^2}\|_\infty > \|\tilde{Y}_c\|_\infty^2\) for all \(c\) with \(|c| \neq 0\).NEWLINENEWLINEThe author next considers extensions to a general bounded symmetric domain \(\Omega\) in \({\mathbb{C}}^n\). Let \(U_c\) denote the involutive unitary operators \((U_c f)(z)=k_c(z)f(\phi_c(z))\) on \(L^2_a(\Omega)\).NEWLINENEWLINETheorem 6: For \(a, c\) in \(\Omega\), \((U_c k_a)(z) = \lambda(c,a)k_{\phi_c(a)}(z)\) for all \(\lambda(c,a)\) in \({\mathbb{C}}\) with \(|\lambda(c,a)|=1\). Taking \(a=a(c)\) to be the fixed point of \(\phi_c\), we find that \(\|\widetilde{U_c}\|_\infty = 1\).