Tomas Ya. Azizov

The Schur algorithm for generalized Schur functions I: coisometric realizations

The Schur algorithm as developed by C. Chamfy [C] and J. Dufresnoy [D] (see also [BDGPS]) is related to a sequence of characteristic functions of closely outerconnected coisometric colligations in Pontryagin spaces.

The Schur algorithm for generalized Schur functions III: J -unitary matrix polynomials on the circle

The main result is that for J = ((1)(0) (0)(-1)) every J-unitary 2 x 2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2 x 2-matrix polynomials which are either of degree 1 or 2k. This is shown by means of the generalized Schur transformation introduced in [Ann. Inst. Fourier 8 (1958) 211; Ann. Acad. Sci. Fenn. Ser. A I 250 (9) (1958) 1-7] and studied in [Pisot and Salem Numbers, Birkhauser Verlag, Basel, 1992; Philips J. Res. 41 (1) (1986) 1-54], and also in the first two parts [Operator Theory: Adv. Appl. 129, Birkhauser Verlag, Basel, 2000, p. 1; Monatshefte fur Mathematik, in press] of this series. The essential tool in this paper are the reproducing kernel Pontryagin spaces associated with generalized Schur functions

On the boundedness of Hamiltonian operators

We show that a non-negative Hamiltonian operator whose domain contains a maximal uniformly positive subspace is bounded.

Standard symmetric operators in Pontryagin spaces: a generalized von Neumann formula and minimality of boundary coefficients

Certain meromorphic matrix valued functions on C\R; the so-called boundary coefficients, are characterized in terms of a standard symmetric operator S in a Pontryagin space with finite (not necessarily equal) defect numbers, a meromorphic mapping into the defect subspaces of S; and a boundary mapping for S: Under some simple assumptions the boundary coefficients also satisfy a minimality condition. It is shown that these assumptions hold if and only if for S a generalized von Neumann equality is valid. r 2002 Elsevier Science (USA). All rights reserved. MSC: primary 47B50; 47B25; 34B07; 47B32; secondary 46C20; 47A06

PT symmetric, Hermitian and P-self-adjoint operators related to potentials in PT quantum mechanics

In the recent years, a generalization H = p2 + x2(ix)e of the harmonic oscillator using a complex deformation was investigated, where e is a real parameter. Here, we will consider the most simple case: e even and x real. We will give a complete characterization of three different classes of operators associated with the differential expression H: The class of all self-adjoint (Hermitian) operators, the class of all PT symmetric operators, and the class of all P-self-adjoint operators. Surprisingly, some of the PT symmetric operators associated to this expression have no resolvent set.

A Theorem on Existence of Invariant Subspaces for J -binoncontractive Operators

Let \( \mathcal{H} \) be a J-space and let \( V = \left( {\begin{array}{*{20}c} {V_1 } & {V_{12} } \\ {V_{21} } & {V_2 } \\ \end{array} } \right) \) be the matrix representation of a J-binoncontractive operator V with respect to the canonical decomposition \( \mathcal{H} = \mathcal{H}^ + \oplus \mathcal{H}^ - \) of \( \mathcal{H} \). The main aim of this paper is to show that the assumption

On domains of {\cal P}{\cal T} symmetric operators related to −y″(x) + (− 1)nx2ny(x)

V_{12} (V_2 - V_{21} V_1^{ - 1} V_{12} ) \in \mathfrak{S}_\infty

Linear Operators in Almost Krein Spaces

(0.1) implies the existence of a V-invariant maximal nonnegative subspace. Let us note that (0.1) is a generalization of the well-known M.G. Krein condition \( V_{12} \in \mathfrak{S}_\infty \). The set of all operators satisfying (0.1) is described via Potapov-Ginsburg transform.