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A note on \(J\)-positive block operator matrices. (English) Zbl 1342.47056

Block operators of the form \( {\mathcal A} = \left(\begin{smallmatrix} A & C^*\\ C & B \end{smallmatrix}\right) \) are considered in a Hilbert space \({\mathfrak H} = {\mathcal H} \times {\mathcal H}\) based on another Hilbert space \({\mathcal H}\). General conditions are assumed such that \({\mathcal A}\) is self-adjoint (i.e., the closure of an essentially self-adjoint operator, which is more precisely defined). These conditions include that \(A\) and \(B\) are self-adjoint and the form \(\langle {\mathcal A} f , f \rangle\) has at most finitely many negative squares. The fundamental symmetry \( {\mathcal J} = \left(\begin{smallmatrix} 0 & i I\\ -i I & 0 \end{smallmatrix}\right) \) turns \({\mathfrak H}\) into a Krein space and \({\mathcal L} = {\mathcal J} {\mathcal A}\) is \({\mathcal J}\)-self-adjoint. The spectra of \({\mathcal A}\) and of \({\mathcal L}\) are studied. The definitizability of \({\mathcal L}\) (i.e., here \(\rho ({\mathcal L}) \neq \emptyset\)) is characterized in terms of \(T(z) := B - (C + iz) A^{-1}(C^* - iz)\) and examples of non-definitizable operators \({\mathcal L}\) are presented.
Under the assumption \(\sigma ({\mathcal L}) \subset {\mathbb R}\), the main focus is on the similarity of \({\mathcal L}\) to a self-adjoint operator, or, in other words, on the regularity of the critical points of \({\mathcal L}\), in particular, of \(\infty\). A well-known necessary condition is the so-called linear resolvent growth condition which is here reformulated in terms of \(T(z)\). This condition gives rise to a number of examples where \(\infty\) is a singular critical point of \({\mathcal L}\). The main example is the operator \({\mathcal L}\) with \(A = - d^2/dx^2 + m^2 + V(x)\), \(C = \nu \, d/dx\), \(B = I\) and \({\mathcal H} = L^2({\mathbb R})\) induced by the nonlinear relativistic Ginzburg-Landau equation. Explicit conditions on the coefficients are obtained such that \(\infty\) is a singular critical point. This result indicates the limits of earlier research on eigenfunction expansions [A. Komech and E. Kopylova, J. Stat. Phys. 154, No. 1–2, 503–521 (2014; Zbl 1300.34195)].

MSC:

47B50 Linear operators on spaces with an indefinite metric
47A40 Scattering theory of linear operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators

Citations:

Zbl 1300.34195
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References:

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