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


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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