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• W2073560255 abstract "We consider operator matrices documentclass{article}pagestyle{empty}begin{document}${rm{H = }}left( {{_{B_{10} }^{A{_0}}} {_{A_{1} }^{B{_{01}}}} } right)$end{document} with self - adjoint entries Ai, i = 0,1, and bounded B10 = B10 acting in the orthogonal sum μ=μ0 ⊕ μ1 of Hilbert spaces μ0 and μ1. We are especially interested in the case where the spectrum of, say, A1 is partly or totally embedded into the continuous spectrum of A0 and the transfer function V1(z) = B10 where (χ - A0)−1 B01, admits analytic continuation (as an operator - valued function) through the cuts along branches of the continuous spectrum of the entry A0 into the unphysical sheet(s) of the spectral parameter plane. The values of χ in the unphysical sheets where M1−1(z) and consequently the resolvent (H - z)−1 have poles are usually called resonances. A main goal of the present work is to find non - selfadjoint operators whose spectra include the resonances as well as to study the completeness and basis properties of the resonance eigenvectors of M1(z) in μ1. To this end we first construct an operator - valued function V1(Y) on the space of operators in μ1 possessing the property: V1(Y)ϕ1 = V1(z)ϕ1 for any eigenvector ϕ1 of Y corresponding to an eigenvalue z and then study the equation H1 = A1 + V1(H1). We prove the solvability of this equation even in the case where the spectra of A0 and A1 overlap. Using the fact that the root vectors of the solutions H1 are at the same time such vectors for M1(z), we prove completeness and even basis properties for the root vectors (including those for the resonances)." @default.
• W2073560255 created "2016-06-24" @default.
• W2073560255 creator A5021072645 @default.
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• W2073560255 date "1999-01-01" @default.
• W2073560255 modified "2023-09-27" @default.
• W2073560255 title "Operator Interpretation of Resonances Arising in Spectral Problems for 2 × 2 Operator Matrices" @default.
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• W2073560255 doi "https://doi.org/10.1002/mana.19992010107" @default.
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