Congruence of selfadjoint operators

Given a bounded selfadjoint operator a in a Hilbert space , the aim of this paper is to study the orbit of a, i.e., the set of operators which are congruent to a. We establish some necessary and sufficient conditions for an operator to be in the orbit of a. Also, the orbit of a selfadjoint operator with closed range is provided with a structure of differential manifold.      

Nonnegative perturbations of selfadjoint operators

Let 4 be a selfadjoint operator on a Hilbert space H. The results in this paper provide necessary and sufficient conditions on A in order that there exist a nontrivial nonnegative operator D and a unitary operator U with UA = (A − D)U. In one case considered, it is required that the least subspace reducing A, U and containing the range of D is the full Hilbert space. In this case the operators U, D exist if and only if the operator A is not a scalar multiple of the identity and the maximum and minimum of the spectrum of A are not eigenvalues of finite multip icity. This result is used to complete a characterization of the absolute value of a completely nonnormal hyponormal operator.     

A characterization of semibounded selfadjoint operators

For a class of closed symmetric operators with defect numbers it is possible to define a generalization of the Friedrichs extension, which coincides with the usual Friedrichs extension when is semibounded. In this paper we provide an operator-theoretic interpretation of this class of symmetric operators. Moreover, we prove that a selfadjoint operator is semibounded if and only if each one-dimensional restriction of has a generalized Friedrichs extension.

     

Form sums of nonnegative selfadjoint operators

Summary The sum of two unbounded nonnegative selfadjoint operators is a nonnegative operator which is not necessarily densely defined. In general its selfadjoint extensions exist in the sense of linear relations (multivalued operators). One of its nonnegative selfadjoint extensions is constructed via the form sum associated with A and B. Its relations to the Friedrichs and Krein--von Neumann extensions of A+Bare investigated. For this purpose, the one-to-one correspondence between densely defined closed semibounded forms and semibounded selfadjoint operators is extended to the case of nondensely defined semibounded forms by replacing semibounded selfadjoint operators by semibounded selfadjoint relations. In particular, the inequality between two closed nonnegative forms is shown to be equivalent to a similar inequality between the corresponding nonnegative selfadjoint relations.</o:p>     

Expansions in generalized eigenfunctions of selfadjoint operators

Mathematische Zeitschrift -     

The Carleman property of functions of selfadjoint operators

Siberian Mathematical Journal -     

Smooth rank one perturbations of selfadjoint operators

Let be a selfadjoint operator in a Hilbert space with inner product . The rank one perturbations of have the form , , for some element . In this paper we consider smooth perturbations, i.e. we consider for some . Function-theoretic properties of their so-called -functions and operator-theoretic consequences will be studied.

     

Differential structure of the Thompson components of selfadjoint operators

Different equivalence relations are defined in the set of selfadjoint operators of a Hilbert space in order to extend a very well known relation in the cone of positive operators. As in the positive case, for the equivalence class admits a differential structure, which is compatible with a complete metric defined on . This metric coincides with the Thompson metric when is positive.

     

On lifting of operators to Hilbert spaces induced by positive selfadjoint operators

We introduce the notion of induced Hilbert spaces for positive unbounded operators and show that the energy spaces associated to several classical boundary value problems for partial differential operators are relevant examples of this type. The main result is a generalization of the Krein-Reid lifting theorem to this unbounded case and we indicate how it provides estimates of the spectra of operators with respect to energy spaces.     

Compressions of kth-order slant Toeplitz operators to model spaces

Lithuanian Mathematical Journal - In this paper, we consider compressions of kth-order slant Toeplitz operators to the backward shift-invariant subspaces of the classical Hardy space H2. In...     

Dynamical systems method (DSM) for selfadjoint operators

Let A be a selfadjoint linear operator in a Hilbert space H. The DSM (dynamical systems method) for solving equation Av=f consists of solving the Cauchy problem , u(0)=u₀, where Φ is a suitable operator, and proving that (i) ∃u(t)∀t>0, (ii) ∃u(∞), and (iii) A(u(∞))=f. It is proved that if equation Av=f is solvable and u solves the problem , u(0)=u₀, where a>0 is a parameter and u₀ is arbitrary, then lima→0limt→∞u(t,a)=y, where y is the unique minimal-norm solution of the equation Av=f. Stable solution of the equation Av=f is constructed when the data are noisy, i.e., fδ is given in place of f, ‖fδ−f‖?δ. The case when a=a(t)>0, , a(t)↘0 as t→∞ is considered. It is proved that in this case limt→∞u(t)=y and if fδ is given in place of f, then limt→∞u(tδ)=y, where tδ is properly chosen.     

Pairs of quaternionic selfadjoint operators with numerical range in a halfplane

The property is studied that two selfadjoint operators on a quaternionic Hilbert space have the joint numerical range in a halfplane bounded by a line passing through the origin. This property is expressed in various ways, in particular, in terms of compressions to two dimensional subpaces, and in terms of linear dependence over the reals. The canonical form for two selfadjoint quaternionic operators in finite dimensional spaces is the main technical tool.