Let A be a bounded self-adjoint operator on a separable Hilbert space H and H0 ⊂ H a closed invariant subspace of A. Assuming that H0 is of codimension 1, we study the variation of the invariant subspace H0 under bounded self-adjoint perturbations V of A that are off-diagonal with respect to the decomposition H = H0 ⊕ H1. In particular, we prove the existence of a one-parameter family of dense ...

WAVE EQUATIONS AND ASSOCIATED DIRAC-TYPE OPERATORS FRITZ GESZTESY, JEROME A. GOLDSTEIN, HELGE HOLDEN, AND GERALD TESCHL Abstract. We discuss the unitary equivalence of generators GA,R associated with abstract damped wave equations of the type ü+Ru̇+A∗Au = 0 in some Hilbert spaceH1 and certain non-self-adjoint Dirac-type operatorsQA,R (away from the nullspace of the latter) in H1 ⊕ H2. The operat...

Let A be a bounded self-adjoint operator on a separable Hilbert space H and H0 ⊂ H a closed invariant subspace of A. Assuming that H0 is of codimension 1, we study the variation of the invariant subspace H0 under bounded self-adjoint perturbations V of A that are off-diagonal with respect to the decomposition H = H0 ⊕H1. In particular, we prove the existence of a oneparameter family of dense no...

We study the spectrum of a periodic self-adjoint operator on the axis perturbed by a small localized nonself-adjoint operator. It is shown that the continuous spectrum is independent of the perturbation, the residual spectrum is empty, and the point spectrum has no finite accumulation points. We address the existence of the embedded eigenvalues. We establish the necessary and sufficient conditi...

S U M M A R Y We carry out high-frequency analyses of Claerbout’s double-square-root equation and its (numerical) solution procedures in heterogeneous media. We show that the double-square-root equation generates the adjoint of the single-scattering modelling operator upon substituting the leading term of the generalized Bremmer series for the background Green function. This adjoint operator yi...

We give a mathematically rigorous analysis which confirms the surprising results in a recent paper [2] of Benilov, O’Brien and Sazonov about the spectrum of a highly singular non-self-adjoint operator that arises in a problem in fluid mechanics. MSC-class: 34Lxx, 76Rxx, 65F15, 65Q05 keywords: spectrum, non-self-adjoint, fluid mechanics, pseudospectra, eigenvalue, basis.

Among all linear operators on Hilbert spaces, the compact ones (defined below) are the simplest, and most imitate the more familiar linear algebra of finite-dimensional operator theory. In addition, these are of considerable practical value and importance. We prove a spectral theorem for self-adjoint operators with minimal fuss. Thus, we do not invoke broader discussions of properties of spectr...

Let à be a self-adjoint extension in K̃ of a fixed symmetric operator A in K ⊆ K̃. An analytic characterization of the eigenvalues of à is given in terms of the Q-function and the parameter function in the Krein–Naimark formula. Here K and K̃ are Krein spaces and it is assumed that à locally has the same spectral properties as a self-adjoint operator in a Pontryagin space. The general results are ...

In this paper, we investigate about the behavior of unbounded operators in Г-Hilbert Space. Here discussed adjoint, self-adjoint, symmetric and other related properties densely defined operator. We proof some theorems corollaries will show characterizations Г -Hilbert

Here we give brief account of hermitian symplectic spaces, showing that they are intimately connected to symmetric as well as self-adjoint extensions of a symmetric operator. Furthermore we find an explicit parameterisation of the Lagrange Grassmannian in terms of the unitary matrices U(n). This allows us to explicitly describe all self-adjoint boundary conditions for the Schrödinger operator o...