Reproducing kernels, Engliš algebras and some applications (Q2804312)

Let \({\mathcal H}={\mathcal H}(\Omega)\) be a reproducing kernel Hilbert space and \(\widehat{k}_{{\mathcal H},\lambda}\) be the normalized reproducing kernel of \({\mathcal H}\). Typically, \({\mathcal H}\) is a Hardy space \(H ^2\), a Bergman space \(L_a^2\), or a model space \(K_\theta:=H^2\ominus\theta H^2\) for an inner function \(\theta\). For an operator \(T\in{\mathcal B}({\mathcal H})\), its Berezin symbol is defined by \(\widetilde{T}(\lambda)= \langle T\widehat{k}_{{\mathcal H},\lambda},\widehat{k}_{{\mathcal H},\lambda}\rangle\) for \(\lambda\in\Omega\). \textit{M. Engliš} [Linear Algebra Appl. 223--224, 171--204 (1995; Zbl 0827.47017)] proved that the sets \({\mathcal F}_{\mathcal H}:=\{T\in {\mathcal B}({\mathcal H}): \|T\widehat{k}_{{\mathcal H},\lambda}\|,\|T^*\widehat{k}_{{\mathcal H},\lambda}\|\to 0 \text{ as } \lambda\to\partial\Omega\}\), \({\mathcal A}_1:=\{T_\Phi+T:\Phi\in L^\infty({\mathbb T}),\;T\in{\mathcal F}_{H^2}\}\), \({\mathcal A}_{\mathcal H}:=\{T\in{\mathcal B}({\mathcal H}): \|T\widehat{k}_{{\mathcal H},\lambda}\|^2-|\widetilde{T}(\lambda)|^2\), \(\| T^*\widehat{k}_{{\mathcal H},\lambda}\|^2-|\widetilde{T}(\lambda)|^2\to 0 \text{ as } \lambda\to\partial\Omega\}\) are \(C^*\)-algebras and the set \({\mathcal A}_{\mathcal H}^0:=\{T\in{\mathcal B}({\mathcal H}): \|T\widehat{k}_{{\mathcal H},\lambda}\|^2-|\widetilde{T}(\lambda)|^2\to 0 \text{ as } \lambda\to\partial\Omega\}\) is an algebra. The authors prove several results on the above algebras and their applications. A criterion for a truncated Toeplitz operator to belong to the algebra \({\mathcal A}_{K_\theta}^0\) is proved. It is shown that if the operator Riccati equation \(XAX+XB-CX-D=0\) is solvable on an appropriate subset of \({\mathcal A}_{\mathcal H}\), then the solution is unique and it is represented in terms of Berezin symbols of \(A\) and \(D\). The next result says that if \(T_{\varphi_1},\dots,T_{\varphi_n}\) are Toeplitz operators with symbols \(\varphi_1,\dots,\varphi_n\in L^\infty({\mathbb T})\) and \(H_1,\dots,H_n\in{\mathcal F}_{H^2}\) are such that \((T_{\varphi_1}+H_1)\dots (T_{\varphi_n}+H_n)=0\), then \(\varphi_1\dots\varphi_n=0\). Some results for zero products of Toeplitz operators on the Bergman space \(L_a^2\) are also obtained. An Axler-Chang-Sarason-Volberg type theorem for semi-commutators of Toeplitz operators on the Bergman space \(L_a^2\) is proved. Finally, an application of Engliš algebras to the existence of a nontrivial invariant subspace in \(H^2\) in terms of reproducing kernels and Duhamel operators is given.