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A note on Riesz bases of eigenvectors of certain holomorphic operator-functions. (English) Zbl 0972.47011

The operator-function \[ {\mathcal A}(\lambda)=A-\lambda I+Q(\lambda) \] is considered acting in a separable Hilbert space, where \(A\) is an operator with compact resolvent, \(Q(\lambda)\) is an A-compact linear operator for each \(\lambda\in\Omega\) (\(\Omega\) is an open connected subset of \(C\)). It is assumed that all the eigenvalues \(\{\lambda_j\}_1^{\infty}\) of \(A\) beyond a certain index are all simple, lie in \(\Omega\), and the corresponding eigenvectors form a Riesz basis of finite defect. The main theorem says that if there are bounds for \(||Q(\lambda)(A-\lambda I)^{-1}||\) on certain circles around the \(\lambda_j\) which decrease rapidly enough with \(j\), then the spectrum of \({\mathcal A}\) in each circle consists only of a single eigenvalue and the corresponding eigenvectors also form a Riesz basis of finite defect.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46A35 Summability and bases in topological vector spaces
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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