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Semistability of the dimension for the linear span of eigen- and associated vectors of a holomorphic operator-function. (Russian) Zbl 0658.47021
Let Y be a complex Banach space, let $$G\subset {\mathbb{C}}$$ be a bounded domain and let W($$\lambda)$$ be a holomorphic operator function in G as well as a continuous operator function in $$\bar G$$ with values in L(Y). The paper study the change of the dimension d(G,W($$\lambda)$$) of the linear span of all eigen- and associated vectors corresponding to the eigenvalues in G. Under small perturbations $$W_ 1(\lambda)$$ of W($$\lambda)$$ this dimension turns out to be nondecreasing as well as $$d(G,W_ 1(\lambda))$$ can reach every value d with d(G,W($$\lambda)$$)$$\leq d\leq m(G,W(\lambda))$$, $$d\leq \dim Y$$, where m(G,W($$\lambda)$$) is the sum of the multiplicities of all eigenvalues in the sense of M. V. Keldish.
Reviewer: N.Bozhinov
##### MSC:
 47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) 47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
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