OPERATOR THEORY, SYSTEM THEORY AND SCATTERING THEORY: MULTIDIMENSIONAL GENERALIZATIONS

Organizers: D. Alpay and V. Vinnikov

Date: Ben-Gurion University of the Negev, Beer-Sheva
truecm June 11 - 13, 2001

truecm

ABSTRACTS

Some finite dimensional backward shift invariant subspaces of the ball and related interpolation problems

D. Alpay

We solve Gleason's problem in the reproducing kernel Hilbert space with reproducing kernel . We define and study some finite-dimensional resolvent-invariant subspaces that generalize the finite-dimensional de Branges-Rovnyak spaces to the setting of the ball.

Fréchet space structure of spaces of analytic functions

A. Aytuna

Let M be a Stein manifold and O(M) be space of analytic functions on M equipped with the the topology of uniform convergence on compact subsets of M. In the first part of talk I will look at these spaces from the nuclear Fre`chet space theory point of view and for certain M 's, identify the linear topological type of O(M). In the second part of the talk I will give some results along the theme " How much and what sort of information does O(M) carry about the complex analytic structure of M ? ". In the last part of the talk (if time permits) some applications of these considerations will be given. .

Unitary colligations, Lax-Phillips scattering and operator model theory: the ball

J. Ball

It is now classical that unitary colligation (or discrete-time, conservative, input-state-output linear systems) can be embedded in a Lax-Phillips scattering scattering system which in turn can be identified as the unitary dilation space for a contraction operator. In this way a Schur-class function simultaneously plays the role of transfer function for the unitary system, scattering function for the scattering system, and characteristic operator function for the contraction operator. Here we examine a multivariable analogue of this triptych where the nonnegative time axis for the conservative, linear system is the free emigroup on (say) d letters, the evolution operator for the Lax-Phillips scattering system is an isometric representation of the Cuntz algebra, and the contraction operator T is replaced by a row contraction . In this case, the structure of the negative time-axis is rather subtle and, unlike the classical case, there are non-equivalent scattering functions corresponding to a given characteristic function, and in general another invariant in addition to the characteristic function is required to determine a completely nonunitary row contraction up to unitary equivalence. Unitary colligations, Lax-Phillips scattering and operator model theory: the polydisk

J. Ball

We present another multivariable analogue of the triptych formed by a unitary system, Lax-Phillips scattering system and operator model,

where the system is a multidimensional system of Roesser-Agler form, the evolution for the Lax-Phillips scattering system is a representation of , and the contraction operator has block-matrix structure. In this case, there are scattering systems which do not correspond to a input-state-output system; as a corollary we arrive at a scattering-theoretic interpretation of the distinction between the Schur-Agler class and the Schur class on the unit polydisk.

Unitary colligations, Lax-Phillips scattering and operator model theory: unimodular algebraic curves.

J. Ball

We present yet another multivariable analogue of the triptych discussed in Lectures I and II. For simplicity, we consider only the case d=2. The Lax-Phillips scattering system now has a two-parameter evolution defined by a pair of commuting, unitary operators on a Hilbert space, but modeled as as multiplication operators on the L²-space over the real part of a real Riemann surface of dividing type rather than on L² on the torus as in the second lecture. The conservative, linear system now is a 2D discrete-time, linear system but with overdetermined state dynamics forcing compatibility conditions on input and output signals and with a more subtle energy conservation rule. The transfer function becomes a bundle map between the input and output bundles which is contractive with respect to certain parahermitian forms on these bundles. These bundles are kernel bundles over a unimodular curve in two-dimensional projective space; here the unimodular property refers to invariance of the curve under the Cremona transformation . The associated system theory is a discrete-time analogue of the overdetermined, continuous-time, conservative systems treated in the book of Livsic-Kravitsky-Markus-Vinnikov, Theory of Commuting Nonselfadjoint Operators. From the operator theory point of view, this structure leads to an analogue of the Sz.-Nagy-Foias model which models a pair of commuting contractions; system theory applications should be of interest as well.

Boundary interpolation in the ball

C. Dubi

We solve a boundary interpolation problem in the reproducing kernel Hilbert space of functions analytic in the unit ball of with reproducing kernel . We introduce the notion of Brune factor (or Blaschke-Potapov factor of the third kind) in this setting.

Evolutionary equations of population genetics

Y. Lyubich

The structure of the evolutionary equations will be discussed in the context of the dynamical effects arising in an infinite population under natural selection.

Hybrid differential operators and modelling of nano-electronic devices

B. Pavlov

Modern nano-electronic devices are constructed as planar networks on the surface of a semiconductor combined of two-dimensional quantum wells and one-dimensional quantum rings connected by one-dimensional quantum wires. Dynamics of electrons on the network in simplest case is described by a system of Schrödinger equations on the elements of the network which have edifferent dimensions and joined by proper boundary conditions. The resulting object is a Hybrid Schrödinger Operator (HS), which may be considered as a reasonably realistic approximation of a Schrödinger operator on physical networks. Using the described approximation we solve the classical problem of description the electrons current through the splitting of channels, see, for instance ³, where the problem was mentioned as an actual mathematical problem of quantum electronics. We reduce the problem to the scattering problem on a quantum well or quantum ring with few one-dimensional wires attached to it. The Schrödinger operators on the wires have the simplest form

- u'' := l u ,

but the potential of the Schrödinger operator on the well (ring) should be specified such that the manipulation of the current may be possible. In particular the following statement is proved for a special sort of a circular quantum well with four quantum wires attached to the boundary at the contact points and the Neumann boundary condition on the remaining part of the boundary:

Theorem The magnitude of the electric field in the Schrödinger operator on the well

may be chosen such that the transmission coefficient T_s from the incoming wire attached to a₀ to one of outgoing wires attached at a_s is non-zero and two others outgoing wires blocked , .

The redirecting of the quantum current to some other wire with remaining wires blocked may be done just by turning of the govering electric field in the plane parallel to the device : .

Our approach ⁴ is based on Krein formula ¹ and operator version of Rouchet theorem ². A large amount of calculation was done with Mathematica. Literature

¹ M.Krein, C R (Doklady) Acad. Sci. URSS (N.S.),52 (1946), 651-654.
²I.S. Gohberg and E.I. Sigal. Operator extension of the theorem about logarithmis residue and Rouchet theorem. Mat. sbornik. 84, 607 (1971).
³ P.Exner, J.Phys.A: Math.Gen. 29 (1996), 87-102.
⁴ A.Mikhaylova, B.Pavlov. Quantum domain as a triadic relay. Publ. in : Unconventional Models of Computations UMC'2K,( Proceedings of the UMC'2K Conference Brussels, Dec 2000 ) eds.I. Antoniou, C. Calude, M.J.Dinneen, Springer Verlag Series for Discrete Mathematics and Theoretical Computer Science (2001), pp 167-186.

Turing halting problem and Quantum Computing

B. Pavlov

One possible way to state the famous Merchant's problem is as follows:

A merchant learns than one of his five stacks of gram coins contains only false coins, grams heavier than normal ones. Can he find the odd stack by a single ``weighting"?

The well known solution of this problem includes actually some typical features of Quantum Computing such as paralleleizm : to solve the problem we should operate with specially selected set of coins collected from each stack, assuming that we have enough coins in each stack. This approach does not work, if we have N stacks of coins and we know that at most one stack may contain false coins. We are allowed to take just one coin from each stack and we want to see whether all coins are true or there is a stack of false coins. Can we solve this problem with just one weighting?

On this simple example we show the advantage of Quantum Computing based on superposition and Hilbert Space approach. The connections of this simplest example with Turing's halting problem will be discussed.

Literature

¹ C. Calude, B. Pavlov. Coins, Quantum Measurements , and Turing's Barrier

Standard operator models for some commuting multioperators

F. Vasilescu

Positivity plays an important role in the Hilbert space operator theory. The simplest positivity condition is, perhaps, that characterizing a Hilbert space contraction C, i.e. . We intend to present natural related versions of this positivity condition, valid for one or several (commuting) operators, and to discuss some of their consequences. We exhibit examples of multioperators satisfying such conditions, which we call standard models. Then we describe the structure of arbitrary multioperators satisfying a given positivity condition, showing that such a tuple is essentially the restriction of the corresponding standard model to an invariant subspace, modulo some addtional terms. In most interesting cases, the addtional terms can also be completely described.

Moment problems in unbounded sets via dimensional extensions

V. Vasilescu

Due to the fact that for n>1 not every nonnegative polynomial in can be written as a sum of squares of polynomials, the moment problems in n variables are more difficult than the classical one variable problems. We start from the idea of solving moment problems by a change of basis via an embedding of into a submanifold of a higher dimensional Euclidean space. We prove that certain extensions of a moment sequence can be characterized by positivity conditions and moreover, these extensions parametrize all possible solutions of the moment problems. We obtain, in particular, characterizations of the representing measures for moment sequences, whose support lie in an arbitrary (generally unbounded) semi-algebraic

Operator models and the symmetrised bidisc

N. Young (Joint work with Jim Agler, UCSD)

First Lecture: Control engineers have asked for analogues of the classical Nevanlinna-Pick theory for interpolation by analytic functions from the open unit disc into various sets in . We have investigated a very special case, in which the target set is the symmetrised bidisc, defined to be the set

Our approach is to study the family of commuting pairs of operators for which is a spectral set. Such commuting pairs are called -contractions. We obtain a characterisation of -contractions, and this leads to a necessary condition of Pick type for interpolation into and to a commutant lifting theorem.

Second Lecture: There are analogues of many of the notions of Nagy-Foias model theory for the family of -contractions. We introduce -unitaries, -isometries and we establish a functional model for a general -contraction. Roughly speaking, every -contraction is unitarily equivalent to the restriction to a common invariant subspace of of the orthogonal direct sum of a -unitary and a ``model -co-isometry'' of the form where are suitable Toeplitz operators on a vectorial Hardy space and for some opeerator A having numerical radius no greater than 1.