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Estimates for resolvents and functions of operator pencils on tensor products of Hilbert spaces. (English) Zbl 1227.47008

Let \(X, Y\) be separable Hilbert spaces. The author establishes some norm estimates for the resolvent and operator-valued functions of the operator \(A =\sum_{k=0}^m B_k\otimes S_k\), where \(B_k\,\,(k =0, \dots , m)\) are bounded operators acting on \(Y\) and \(S\) is an invertible selfadjoint operator acting in \(X\). By these estimates, he investigates spectrum perturbations of the operator pencil \(A\). Some improvements are given in the case where \(Y=\mathbb{C}^n\). The author then uses his results to study the non-selfadjoint differential operators with matrix and operator coefficients in Hilbert spaces. The reader may also want to consult L. A. Shuster [J. Math. Anal. Appl. 229, No. 2, 363–375 (1999; Zbl 0919.34070)], where the problem of localization of the spectrum for a class of semiregular boundary value problems is investigated.

MSC:

47A80 Tensor products of linear operators
47E05 General theory of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators

Citations:

Zbl 0919.34070
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Full Text: DOI

References:

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