Estimates for norms of resolvents of operators on tensor products of Hilbert spaces
Let \(H= E_{1}\otimes E_{2}\) be the tensor product of the Hilbert spaces \(E_{1}\), \(E_{2}\) and let \(M_{1}\), \(M_{2}\) be operators acting on \(E_{1}\), \(E_{2}\), respectively. Under the conditions that the domains of \(M_{j}\) and \(M_{j}^\ast\) coincide and that the difference \(M_{j} -M_{j}^\ast \in \tilde{C}_{2p_{j}}\) for some Neumann--Schatten class ideal \(\tilde{C}_{2p_{j}}\), \(p_{j}\in {\mathbb N}\), the author shows for the operator \(A= M_{1} \otimes I_{2} + I_{1}\otimes M_{2}\) the estimate \[ \| (A-\lambda)^{-1}\| E \leq \sum_{n=0}^{\infty} {b_{n}(M_{1},M_{2}) \over \varrho(A,\lambda)^{n+1}}, \] where \(\varrho(A,\lambda) = \inf_{t\in\sigma(A)}| t-\lambda| \). The general result is improved in the special cases where \(H\) is finite-dimensional or \(M_{j}-M_{j}^\ast\), \(j=1,2\), are Hilbert--Schmidt operators. As an application, estimates for the spectral variation are given in the situation \(B= A +Z\), where \(Z\) is bounded and \(A\) of the form given above. Finally, an integro-differential operator is discussed.
scientific article; zbMATH DE number 2186367