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Reducibility of a class of operator functions to block-diagonal form. (English. Russian original) Zbl 1130.47302
Math. Notes 74, No. 5, 744-748 (2003); translation from Mat. Zametki 74, No. 5, 789-792 (2003).
Let \(K\) be a compact set and \(H\) a Hilbert space. A continuous operator-valued function \(D(t):K\to L(X)\) is called conditionally \((X^-,X^+)\)-reducible if there exists a continuous operator function \(V:K\to L(X)\) such that \(V(t)\) is an isomorphism for all \(t\in K\) and \(V^{-1}(t)D(t)V(t)=\text{diag}\{D_-(t),D_+(t)\}\), where \(D_\pm(t)\in L(X^\pm)\), \(X=X^-\dot{+}X^+\), and for each \(t\in K\) the spectrum of \(D_-(t)\) (resp., \(D_+(t)\)) is located in the left (resp., right) open half-plane. The following result is obtained: If \(A,B,C:K\to L(X)\) are continuous operator functions and if the operator
\[ \begin{pmatrix} A(t)\mu B(t)\\ \mu C(t) -A^*(t)\end{pmatrix} :X\dot{+}X\to X\dot{+}X\tag{1} \] has no points of its spectrum on the imaginary axis for all \(\mu\in[0,1]\) and all \(t\in K\), then the operator function (1) is conditionally \((X^-,X^+)\)-reducible for any \(\mu\in[0,1]\), where \(X^-=\{(x,-x):x\in X\}\) and \(X^+=\{(x,x):x\in X\}\).

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
34G10 Linear differential equations in abstract spaces
47B50 Linear operators on spaces with an indefinite metric
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