## 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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