Spectrum perturbations of operators on tensor products of Hilbert spaces (Q1890383)

In the paper, bounds for the resolvent and for the spectrum of a class of linear operators on tensor products of separable Hilbert spaces are established. Applications to partial integral operators and to integro-differential operators are also given. In Section 2, some estimates of \(\| (W_1+W_2)^n\| _H\) are derived, where \(W_1\) and \(W_2\) denote quasinilpotent and commuting operators. In particular, the author proves that \(W_1+W_2\) is quasinilpotent. This can be obtained from the well-known inequality \[ r_s(W_1+W_2)\leq r_s(W_1)+r_s(W_2) \] which holds for the spectral radii of commuting operators (see, for example, [\textit{F. Riesz, B. Sz.-Nagy}, ``Functional Analysis'' (1990; Zbl 0732.47001)]).