Eduard Tsekanovskii

Conservative realizations of Herglotz-Nevanlinna functions

Preface.- 1 Extensions of Symmetric Operators.- 2 Rigged Hilbert Spaces.- 3 Bi-extensions of Closed Symmetric Operators.-.4 Quasi-self-adjoint Extensions.- 5 The Livsic Canonical Systems with Bounded Operators.- 6 Herglotz-Nevanlinna functions and Rigged Canonical Systems.- 7 Classes of realizable Herglotz-Nevanlinna functions.- 8 Normalized Canonical Systems.- 9 Canonical L-systems with Contractive and Accretive Operators.- 10 Systems with Schrodinger operator.- 11 Non-self-adjoint Jacobi Matrices and System Interpolation.- 12 Non-canonical Systems.- Notes and Comments.- References.- Index.

On von Neumann's problem in extension theory of nonnegative operators

The solution of von Neumanns problem about parametrization of all nonegative selfadjoint extensions of a nonnegative densely defined operator in terms of his formulas is obtained.

On Classes of Realizable Operator-valued R-functions

In this paper we consider realization problems (see [5]–[7]) for operator-valued R-functions acting on a Hilbert space E (dim E < ∞) as linear-fractional transformations of the transfer operator-valued functions (characteristic functions) of linear stationary conservative dynamic systems (Brodskii-Livsic rigged operator colligations). We specialize three subclasses of the class of all realizable operator-valued R-functions [7]. We give complete proofs of direct and inverse realization theorems for each subclass announced in [5], [6].

Interpolation theory in Sectorial Stieltjes Classes and Explicit System Solutions

Abstract We introduce sectorial classes of matrix-valued Stieltjes functions in which we solve the bitangential interpolation problem of Nudelman and Ball–Gohberg–Rodman. We consider also a new type of solutions of Nevanlinna–Pick interpolation problems, so-called explicit system solutions generated by Brodskii–Livsic colligations, and find conditions on interpolation data of their existence and uniqueness. We point out the connections between sectorial Stieltjes classes and sectorial operators, and find out new properties of the classical Nevanlinna–Pick interpolation matrices (in the scalar case). We present in terms of interpolation data the exact formula for the angle of sectoriality of the main operator in the explicit system solution as well as the criterion for this operator to be extremal.The interpolation model for nonselfadjoint matrices is established.

Inverse Stieltjes-like Functions and Inverse Problems for Systems with Schrodinger Operator

A class of scalar inverse Stieltjes-like functions is realized as linear-fractional transformations of transfer functions of conservative systems based on a Schrodinger operator T h in L 2au][a,+∞) with a non-selfadjoint boundary condition. In particular it is shown that any inverse Stieltjes function of this class can be realized in the unique way so that the main operator \( \mathbb{A} \) possesses a special semi-boundedness property. We derive formulas that restore the system uniquely and allow to find the exact value of a non-real boundary parameter h of the operator T h as well as a real parameter μ that appears in the construction of the elements of the realizing system. An elaborate investigation of these formulas shows the dynamics of the restored parameters h and μ in terms of the changing free term α from the integral representation of the realizable function.

Commutative and Noncommutative Representations of Matrix-Valued Herglotz-Nevanlinna Functions

Operator realizations of matrix-valued Herglotz-Nevanlinna functions play an important and essential role in system theory, in the spectral theory of bounded nonselfadjoint operators, and in interpolation problems. Here, a generalization for realization results of the Brodskiǐ-Livsic type is given for Herglotz-Nevanlinna functions whose spectral measures are compactly supported.

An Addendum to the Multiplication and Factorization Theorems of Brodskiĭ-Livšic-Potapov

In this paper the classical Brodskiǐ—Livšic— operator colligation is generalized to certain pairs of bounded linear operators and the corresponding characteristic operator—valued (transfer) function is introduced. The fundamental results due to M.S. Brodskiǐ, M.S. Livšic, and V.P. Potapov are then extended to such colligations. These new types of colligations can be used to obtain, for instance, realization results for general Herglotz—Nevanlinna functions.

Extensions of Symmetric Operators

In this chapter we deal with extensions of densely and non-densely defined symmetric operators. The parametrization of the domains of all self-adjoint extensions in both dense and non-dense cases is given in terms of von Neumann’s and \({\rm Krasnoselski\breve{i}^\prime s}\) formulas, respectively. The so-called admissible unitary operators serve as parameters in \({\rm Krasnoselski\breve{i}^\prime s}\) formulas.

Geometry of Rigged Hilbert Spaces

In this chapter we study extensions of symmetric non-densely defined operators in the triplets H+⊂ H ⊂H- of rigged Hilbert spaces. The Krasnoselskiĭi formulas discussed in Section 1.7 are based upon the indirect decomposition (1.33), where deficiency subspaces and the domain of symmetric operator may be linearly dependent. Introduction of the rigged Hilbert spaces allows us to obtain the direct decomposition and parameterization for the domain of the adjoint operator. This direct decomposition is written in terms of the semi-deficiency subspaces and is an analogue of the von Neumann formulas (1.7) and (1.13) for the case of the symmetric operator ?A whose domain is not dense in H.

The Livšsic Canonical Systems with Bounded Operators

In this chapter we present the foundations of the theory of the Livssic canonical open systems with bounded state-space (main) operators. We provide an analysis of such a type of systems in terms of transfer functions and their linear-fractional transformations. We also consider couplings of these systems and present multiplication and factorization theorems of the transfer functions.