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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 $$X^+=\{(x,x):x\in X\}$$.

##### MSC:
 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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