Vladimir Derkach

Generalized resolvents and the boundary value problems for Hermitian operators with gaps

A Hermitian operator A with gaps (αj, βj) (1 ⩽ j ⩽ m ⩽ ∞) is studied. The self-adjoint extensions which put exactly kj < ∞ eigenvalues into each gap (αj, βj), in particular (for kj = 0, 1 ⩽ j ⩽ m) the extensions preserving the gaps, are described in terms of boundary conditions. The generalized resolvents of the extensions with the indicated properties are described also. A solvability criterion and description of all the solutions of the Hamburger moment problem with supports in R/⋃j=1m(αj,βj) are obtained in terms of the Nevanlinna matrix.

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.

ON WEYL FUNCTION AND GENERALIZED RESOLVENTS OF A HERMITIAN OPERATOR IN A KREIN SPACE

AbstractA description of generalized resolvents for a densely defined Hermitian operatorA in a Krein space

On Linear Fractional Transformations Associated with Generalized J-Inner Matrix Functions

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Asymptotic Expansions of Generalized Nevanlinna Functions and their Spectral Properties

is given under explicit consideration of the number of negative squares of the inner product on the extending space and of the forms [A·,·], [A·,·],A being a selfadjoint extension ofA which corresponds to the generalized resolvent. New classesNkκ of analytic functions are introduced for this purpose. An application to a Sturm-Liouville operator with indefinite weight function is discussed.

Darboux transformations of Jacobi matrices and Padé approximation

The characteristic operator-functions W(λ) are studied of the almost solvable extensions of an Hermitian operator. The inverse problem is solved, a multiplication theorem is proved, and a formula is derived expressing W(λ) in terms of the Weyl function and the boundary operator. Characteristic functions are computed of various differential and difference operators, with the help of which are proved theorems of the completeness of the systems of proper and adjoint vectors.

Generalized Nevanlinna Functions with Polynomial Asymptotic Behaviour at Infinity and Regular Perturbations

Abstract.A class