Operator Colligations Research Articles

Sturm - Liouville systems and nonselfadjoint operators, presented as couplings of dissipative and antidissipative operators with real absolutely continuous spectra

This paper is a continuation of the considerations of the paper [1] and it presents the connection between Sturm-Liouville systems and Livšic operator colligations theory. An usefull representation of solutions of Sturm - Liouville systems is obtained using the resolvent of operators from a large class of nonselfadjoint nondissipative operators, presented as couplings of dissipative and antidissipative operators with real spectra. A connection between Sturm-Liouville systems and the inner state of the corresponding open system of operators from the considered class is presented.

Generalized Schur\u2013Nevanlinna functions and their realizations

Pontryagin space operator valued generalized Schur functions and generalized Nevanlinna functions are investigated by using discrete-time systems, or operator colligations, and state space realizations. It is shown that generalized Schur functions have strong radial limit values almost everywhere on the unit circle. These limit values are contractive with respect to the indefinite inner product, which allows one to generalize the notion of an inner function to Pontryagin space operator valued setting. Transfer functions of self-adjoint systems such that their state spaces are Pontryagin spaces, are generalized Nevanlinna functions, and symmetric generalized Schur functions can be realized as transfer functions of self-adjoint systems with Kreĭn spaces as state spaces. A criterion when a symmetric generalized Schur function is also a generalized Nevanlinna function is given. The criterion involves the negative index of the weak similarity mapping between an optimal minimal realization and its dual. In the special case corresponding to the generalization of an inner function, a concrete model for the weak similarity mapping can be obtained by using the canonical realizations.

Minimal Passive Realizations of Generalized Schur Functions in Pontryagin Spaces

Passive discrete-time systems in Pontryagin space setting are investigated. In this case the transfer functions of passive systems, or characteristic functions of contractive operator colligations, are generalized Schur functions. The existence of optimal and ^*-optimal minimal realizations for generalized Schur functions are proved. By using those realizations, a new definition, which covers the case of generalized Schur functions, is given for defects functions. A criterion due to D.Z. Arov and M.A. Nudelman, when all minimal passive realizations of the same Schur function are unitarily similar, is generalized to the class of generalized Schur functions. The approach used here is new; it relies completely on the theory of passive systems.

Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument

We extend the classical construction of operator colligations and characteristic functions. Consider the group $G$ of finite block unitary matrices of size $\alpha+\infty+...+\infty$ ($k$ times). Consider the subgroup $K=U(\infty)$, which consists of block diagonal unitary matrices (with a block 1 of size $\alpha$ and a matrix $u\in U(\infty)$ repeated $k$ times). It appears that there is a natural multiplication on the conjugacy classes $G//K$. We construct 'spectral data' of conjugacy classes, which visualize the multiplication and are sufficient for a reconstruction of a conjugacy class.

Infinite-dimensional -adic groups, semigroups of double cosets, and inner functions on Bruhat–Tits buildings

We construct -adic analogues of operator colligations and their characteristic functions. Consider a -adic group , a subgroup of and a subgroup which is diagonally embedded in . We show that the space of double cosets admits the structure of a semigroup and acts naturally on the space of -fixed vectors of any unitary representation of . With each double coset we associate a `characteristic function' that sends a certain Bruhat– Tits building to another building (the buildings are finite- dimensional) in such a way that the image of the distinguished boundary lies in the distinguished boundary. The second building admits the structure of a (Nazarov) semigroup, and the product in corresponds to the pointwise product of characteristic functions.

Symmetries of Gaussian measures and operator colligations

Consider an infinite-dimensional linear space equipped with a Gaussian measure and the group GLO(∞) of linear transformations that send the measure to equivalent one. Limit points of GLO(∞) can be regarded as ‘spreading’ maps (polymorphisms). We show that the closure of GLO(∞) in the semigroup of polymorphisms contains a certain semigroup of operator colligations and write explicit formulas for action of operator colligations by polymorphisms of the space with Gaussian measure.

Sphericity and multiplication of double cosets for infinite-dimensional classical groups

We construct spherical subgroups in infinite-dimensional classical groups $G$ (usually they are not symmetric and their finite-dimensional analogs are not spherical). We present a structure of a semigroup on double cosets $L\setminus G/L$ for various subgroups $L$ in $G$, moreover these semigroups act in spaces of $L$-fixed vectors in unitary representations of $G$. We also obtain semigroup envelops of groups $G$ generalizing constructions of operator colligations.

Multi-operator colligations and multivariate characteristic functions

In the spectral theory of non-self-adjoint operators there is a well-known operation of product of operator colligations. Many similar operations appear in the theory of infinite-dimensional classical groups as multiplications of double cosets. We construct characteristic functions for such double cosets and get semigroups of inner matrix-valued functions in matrix balls.

Dirac structures and their composition on Hilbert spaces

Dirac structures appear naturally in the study of certain classes of physical models described by partial differential equations and they can be regarded as the underlying power conserving structures. We study these structures and their properties from an operator-theoretic point of view. In particular, we find necessary and sufficient conditions for the composition of two Dirac structures to be a Dirac structure and we show that they can be seen as Lagrangian (hyper-maximal neutral) subspaces of Kreĭn spaces. Moreover, special emphasis is laid on Dirac structures associated with operator colligations. It turns out that this class of Dirac structures is linked to boundary triplets and that this class is closed under composition.

Weyl–Titchmarsh Function of an Abstract Boundary Value Problem, Operator Colligations, and Linear Systems with Boundary Control

The paper defines the Weyl–Titchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator colligations, and the linear systems theory that suggests an approach to the study of boundary value problems based on the open systems theory founded in works of M. S. Livsic. Examples of boundary value problems for partial differential equations and calculations of their Weyl–Titchmarsh functions are offered as illustration. In particular, we give an independent derivation of the Weyl–Titchmarsh function for the three dimensional Schrodinger operator introduced by W.O. Amrein and D.B. Pearson. Relationships to the Schrodinger operator with singular potential supported by the unit sphere are clarified and other possible applications of the developed approach in mathematical physics are noted.

A general realization theorem for matrix-valued Herglotz–Nevanlinna functions

New special types of stationary conservative impedance and scattering systems, the so-called non-canonical systems, involving triplets of Hilbert spaces and projection operators, are considered. It is established that every matrix-valued Herglotz–Nevanlinna function of the form V ( z ) = Q + Lz + ∫ R 1 t - z - t 1 + t 2 d Σ ( t ) can be realized as a transfer function of such a new type of conservative impedance system. In this case it is shown that the realization can be chosen such that the main and the projection operators of the realizing system satisfy a certain commutativity condition if and only if L = 0. It is also shown that V( z) with an additional condition (namely, L is invertible or L = 0), can be realized as a linear fractional transformation of the transfer function of a non-canonical scattering F +-system. In particular, this means that every scalar Herglotz–Nevanlinna function can be realized in the above sense. Moreover, the classical Livšic systems (Brodskii˘–Livšic operator colligations) can be derived from F +-systems as a special case when F + = I and the spectral measure d Σ( t) is compactly supported. The realization theorems proved in this paper are strongly connected with, and complement the recent results by Ball and Staffans.

An application of the nonselfadjoint operators theory in the study of stochastic processes

The theory of operator colligations in Hilbert spaces gives rise to certain models for nonselfadjoint operators, called triangular models. These models generalize the spectral decomposition of selfadjoint operators. In this paper, we use the triangular model to obtain the correlation function (CF) of a nonstationary linearly representable stochastic process for which the corresponding operator is simple, dissipative, nonselfadjoint of rank 1, and has real spectrum. As a generalization, we represent the infinitesimal correlation function (ICF) of a nonhomogeneous linearly representable stochastic field in which at least one of the operators has real spectrum.

Products of Nevanlinna-Pick kernels and operator colligations

The Schur classSd of analytic functionsf ofd complex variables for whichf(T1,T2,...,Td) has norm at most one for everyd-tuple of commuting strict contractions is characterized by J. Agler [1] in terms of a Szego kernel factorization property and in terms of a transfer function of a certain type ofd-variable unitary colligation. Replacing the Szego kernel by positive kernels whose reciprocal has one positive square we can define a new Schur class in terms of a kernel factorization property. By using, in part, the approach found in Ball and Trent [3], we characterize this Schur class in terms of a transfer function of a certain type of unitary colligation. Further, with these results we establish a Nevanlinna-Pick interpolation theorem for our Schur class. Such interpolation theorems already exist in the literature for two special cases where [1] all kernels are taken to be the Szego kernel, and [2] d=1.

Linearization of boundary eigenvalue problems

It is shown that certain eigenvalue problems for ordinary differential operators with boundary conditions depending holomorphically on the eigenvalue parameter γ can be linearized by making use of the theory of operator colligations. As examples, first order systems with boundary conditions depending polynomially on γ and Sturm-Liouville problems with γ-holomorphic boundary conditions are considered.

Automorphic-invariant operator colligations and the factorization of their characteristic operator functions

Automorphic-invariant operator colligations and the factorization of their characteristic operator functions

Connection colligations on Hilbert bundles

The theory of operator colligations in a Hilbert space is extended to connection colligations on Hilbert bundles. Any invariant subbundle of a Hilbert bundle leads to a decomposition of connection colligation into simpler components. The problem of finding all invariant subbundles of a Hilbert bundle is reduced to a search of subspaces of a fixed Hilbert space which are invariant with respect to the holonomy group of the linear connection.

Regular colligations for several commuting operators in Banach space

Several important results of the theory of operator colligations, stated initially for strict commutative colligations in Hilbert space [6] [7], are extended to the more general setting of regular commutative colligations in Banach space, which are defined in the text.

Book Review: Operator colligations in Hilbert spaces

Book Review: Operator colligations in Hilbert spaces

Book Review: Operator Colligations in Hilbert Spaces

Book Review: Operator Colligations in Hilbert Spaces

OPERATOR COLLIGATIONS IN HILBERT SPACES (Translation from the original Russian edited by R. G. Douglas)

By M. S. Livshits and A. A. Yantsevich: pp. 212. £12.50. (V. H. Winston/John Wiley & Sons, 1979).