Seppo Hassi

Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space h, let H be an auxiliary Hilbert space, let [GRAPHICS] and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the Kre. in space (H-2, J(H)) is called a boundary relation for the adjoint S* if ker Gamma = S. The corresponding Weyl family M(lambda) is de. ned as the family of images of the defect subspaces (n) over cap (lambda), lambda is an element of C \ R under Gamma. Here Gamma need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : ( H-2, J(H)) -> (H-2, J(H)), it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every H-valued maximal dissipative (for lambda is an element of C+) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

Boundary relations and generalized resolvents of symmetric operators

The Kreĭn-Naĭmark formula provides a parametrization of all selfadjoint exit space extensions of a (not necessarily densely defined) symmetric operator in terms of maximal dissipative (in ℂ+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct generalized resolvents from given parameter families. A general version of the coupling method is introduced and the role of the boundary relations and their Weyl families in the Kreĭn-Naĭmark formula is investigated and explained. These notions lead to several new results and new types of solutions to problems involving generalized resolvents and their applications, e.g., in boundary-value problems for (ordinary and partial) differential operators. For instance, an old problem going back to M. A. Naĭmark and concerning the analytic characterization of the so-called Naĭmark extensions is solved.

On the class of extremal extensions of a nonnegative operator

A nonnegative selfadjoint extension Aof a nonnegative operator A is called extremal if inf {(A)(ϕ) - f),ϕ - f) : ∈ dom A} = 0 for all ϕ ∈ dom A.A new construction of all extremal extensions of a nonnegative densely defined operator will be presented.It employs a fixed auxiliary Hilbert space to factorize each extremal extension.Various functional-analytic interpretations of extremal extensions are studied and some new types of characterizations are obtained.In particular,a purely analytic description of extremal extensions is established,based on a class of functions introduced by M.G.Krein and I.E.Ovearenko.

On projections in a space with an indefinite metric

Abstract We consider G-projectors (orthogonal projections) defined on an indefinite inner product space, and derive in a systematic way the indefinite counterparts of a number of useful results known to hold for ordinary projectors in Hilbert space. Some of the topological considerations encountered in the literature are avoided here, and several results are obtained using quite elementary matrix-type arguments. In particular, the relation between G-projectors and contractions in an indefinite inner product space is studied. For example, a convergence result is given for a nondecreasing sequence of G-contractive G-projectors. We also prove a result characterizing G-projectors within the class of idempotents, generalizing the corresponding result in Hilbert space.

Smooth rank one perturbations of selfadjoint operators

Let A be a selfadjoint operator in a Hilbert space aleph with inner product [.,.]. The rank one perturbations of A have the form A+tau [.,omega]omega, tau epsilon R, for some element omega epsilon aleph. In this paper we consider smooth perturbations, i.e. we consider omega epsilon dom \A\(k/2) for some k epsilon N boolean OR {0}. Function-theoretic properties of their so-called Q-functions and operator-theoretic consequences will be studied.

On rank one perturbations of selfadjoint operators

LetA be a selfadjoint operator in a Hilbert space ℌ. Its rank one perturbations A+τ(·,ω)ω, ℝ, are studied when ω belongs to the scale space ℌ−2 associated with ℌ+2=domA and (·,·) is the corresponding duality. IfA is nonnegative and ω belongs to the scale space ℌ−1, Gesztesy and Simon [4] prove that the spectral measures ofA(τ), ℝ, converge weakly to the spectral measure of the limiting perturbationA(∞). In factA(∞) can be identified as a Friedrichs extension. Further results for nonnegative operatorsA were obtained by Kiselev and Simon [14] by allowing ω∈ℌ−2, Our purpose is to show that most results of Gesztesy, Kiselev, and Simon are valid for rank one perturbations of selfadjoint operators, which are not necessarily semibounded. We use the fact that rank one perturbations constitute selfadjoint extensions of an associated symmetric operator. The use of so-calledQ-functions [6, 8] facilitates the descriptions. In the special case that ω belongs to the scale space ℌ−1 associated with ℌ+2=dom |A|1/2 the limiting perturbationA(∞) is shown to be the generalized Friedrichs extension [5].

A reproducing kernel space model for _{}-functions

A new model for generalized Nevanlinna functions Q ∈ N K will be presented. It involves Bezoutians and companion operators associated with certain polynomials determined by the generalized zeros and poles of Q. The model is obtained by coupling two operator models and expressed by means of abstract boundary mappings and the corresponding Weyl functions.

Extremal extensions for the sum of nonnegative selfadjoint relations

The sum A + B of two nonnegative selfadjoint relations (multivalued operators) A and B is a nonnegative relation. The class of all extremal extensions of the sum A + B is characterized as products of relations via an auxiliary Hilbert space associated with A and B. The so-called form sum extension of A+B is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with A and B. Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on A and B.

Sesquilinear forms corresponding to a non-semibounded Sturm-Liouville operator

Let - DpD be a differential operator on the compact interval [-b, b] whose leading coefficient is positive on (0, b] and negative on [b,0), with fixed, separated, self-adjoint boundary conditions at h and b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded sesquilinear forms by a generalization of Katos representation theorems. The theory of non-semibounded sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal sesquilinear form. Moreover, among all closed forms associated with the self-ad joint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Krein-Feller operators with a single concentrated mass.

Asymptotic Expansions of Generalized Nevanlinna Functions and their Spectral Properties

Asymptotic expansions of generalized Nevanlinna functions Q are investigated by means of a factorization model involving a part of the generalized zeros and poles of nonpositive type of the function Q. The main results in this paper arise from the explicit construction of maximal Jordan chains in the root subspace R∞(S F) of the so-called generalized Friedrichs extension. A classification of maximal Jordan chains is introduced and studied in analytical terms by establishing the connections to the appropriate asymptotic expansions. This approach results in various new analytic characterizations of the spectral properties of selfadjoint relations in Pontryagin spaces and, conversely, translates analytic and asymptotic properties of generalized Nevanlinna functions into the spectral theoretical properties of self-adjoint relations in Pontryagin spaces.