Kurina, G. A.; Martynenko, G. V. 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 \(X^+=\{(x,x):x\in X\}\). Reviewer: Yuri I. Karlovich (Cuernavaca) Cited in 7 Documents 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 Keywords:Hamiltonian operator function; Hilbert space; Riesz projection; reducibility; block-diagonal matrix PDFBibTeX XMLCite \textit{G. A. Kurina} and \textit{G. V. Martynenko}, Math. Notes 74, No. 5, 744--748 (2003; Zbl 1130.47302); translation from Mat. Zametki 74, No. 5, 789--792 (2003) Full Text: DOI