We consider a large class of self-adjoint elliptic problem associated with the second derivative acting on a space of vector-valued functions. We present two different approaches to the study of the associated eigenvalues problems. The first, more general one allows to replace a secular equation (which is well-known in some special cases) by an abstract rank condition. The latter seems to apply particularly well to a specific boundary condition, sometimes dubbed "anti-Kirchhoff" in the literature, that arise in the theory of differential operators on graphs; it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities. In either case our results yield, among other, some results on the symmetry of the spectrum.
We prove an analogue to the Cayley identity for an arbitrary self-adjoint operator in a Hilbert space. We also provide two new ways to characterize vectors belonging to the singular spectral subspace in terms of the analytic properties of the resolvent of the operator, computed on these vectors. The latter are analogous to those used routinely in the scattering theory for the absolutely continuous subspace.
A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $λ_{min}=λ_1\leqslant\ldots\leqslantλ_m<σ_{ess}$. The problem of approximation of these eigenvalues by eigenvalues of some linear operator in a finite-dimensional space of the dimension $s$ is considered and solved. The accuracy of the approximation obtained becomes unlimitedly high as $s\to\infty$.
Let $H_0$ and $H$ be self-adjoint operators in a Hilbert space. In the scattering theory framework, we describe the essential spectrum of the difference $φ(H)-φ(H_0)$ for piecewise continuous functions $φ$. This description involves the scattering matrix for the pair $H$, $H_0$.
A bounded linear Hilbert space operator $S$ is said to be a $2$-isometry if the operator $S$ and its adjoint $S^*$ satisfy the relation $S^{*2}S^{2} - 2 S^{*}S + I = 0$. In this paper, we study Hilbert space operators having liftings or dilations to $2$-isometries. The adjoint of an operator which admits such liftings is characterized as the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators (i.e., operators $S$ such that $S^{*2}S^{2} - 2 S^{*}S + I \le 0$) and to operators similar to contractions or isometries…
In this note we define and study a Hilbert space-valued stochastic integral of operator-valued functions with respect to Hilbert space-valued measures. We show that this integral generalizes the classical Ito stochastic integral of adapted processes with respect to normal martingales and the Ito integral in a Fock space
We introduce a natural framework for dealing with Mourre theory in an abstract two-Hilbert spaces setting. In particular a Mourre estimate for a pair of self-adjoint operators (H,A) is deduced from a similar estimate for a pair of self-adjoint operators (H_0,A_0) acting in an auxiliary Hilbert space. A new criterion for the completeness of the wave operators in a two-Hilbert spaces setting is also presented.
We define a left quotient as well as a right quotient of two bounded operators between Hilbert spaces, and we parametrize these two concepts using the Moore-Penrose inverse. In particular, we show that the adjoint of a left quotient is a right quotient and conversely. An explicit formulae for computing left (resp. right) quotient which correspond to adjoint, sum, and product of given left (resp. right) quotient of two bounded operators are also shown.
This paper focuses on the stability of self-adjointness of linear relations under perturbations in Hilbert spaces. It is shown that a self-adjoint relation is still self-adjoint under bounded and relatively bounded perturbations. The results obtained in the present paper generalize the corresponding results for linear operators to linear relations, and some weaken the conditions of the related existing results.
In the terms of triples $D^+\to H\to D^-$ of Hilbert spaces we construct an analogue of Friedrichs's extension for operator matrices. Also we establish some general approach to construction of variational principles for such matrices.
In this paper we construct the spectral expansion for the differential operator generated in all real line by ordinary differential expression of arbitrary order with periodic complex-valued coefficients by introducing new concepts as essential spectral singularities and singular quasimomenta and using the series with parenthesis. Moreover, we find a criteria for which the spectral expansion coincides with the Gelfand expansion for the self-adjoint case.
In this paper, we give Lieb-Thirring type inequalities for isolated eigenvalues of $d$-dimensional non-selfadjoint Schrödinger operators with complex-valued and dilation analytic potentials. In order to derive them, we prove that isolated eigenvalues and their multiplicities are invariant under complex dilation.
Given a complex, separable Hilbert space $\cH$, we consider differential expressions of the type $τ= - (d^2/dx^2) + V(x)$, with $x \in (a,\infty)$ or $x \in \bbR$. Here $V$ denotes a bounded operator-valued potential $V(\cdot) \in \cB(\cH)$ such that $V(\cdot)$ is weakly measurable and the operator norm $\|V(\cdot)\|_{\cB(\cH)}$ is locally integrable. We consider self-adjoint half-line $L^2$-realizations $H_α$ in $L^2((a,\infty); dx; \cH)$ associated with $τ$, assuming $a$ to be a regular endpoint necessitating a boundary condition of the type $\sin(α)u'(a) + \cos(α)u(a)=0$, indexed by the sel…
