Open Problems In Theory Research Articles

Some Problems in Operator Theory on Bounded Symmetric Domains

We discuss some problems concerning the operators Km(Z,Z*), where Km is the reproducing kernel of the Peter–Weyl space with signature m and Z stands for the commuting tuple of operators of multiplication by the coordinate variables on a bounded symmetric domain. These problems have applications for construction of operator models, as well as for a possible generalization of the recent dilation theory for row contractions due to Arveson.

On the Rigidity of Bergman Submodules

Introduction. The search for and the classification of invariant subspaces of various operators acting on function spaces have proved to be two very rewarding research problems in operator theory and harmonic analysis. Not only that the questions themselves have turned out to be important, but even more so did they applications and the techniques used to solve them. For instance the classical theorem of Beurling [3] on the structure of analytically invariant subspaces of the Hardy space on the torus has been influential in many areas of modern mathematics, ranging from the dilation theory of a contraction, to interpolation problems in function theory or to the probabilistic analysis of time series. In Hilbert spaces of analytic functions depending on several complex vari? ables, such a simple classification as that of Beurling's theorem is no longer to be expected. Even the existence of inner functions can be a difficult question. The case of a Hardy space supported by a polydisk in Cn has been, however, successfully studied during the last two decades, see [12], [1]. More recently, R.G. Douglas and his collaborators have developed a general framework for investigating invariant subspaces from the point of view of Hilbert modules over function algebras, [5], [6], [7]. The importance ofthe algebraic language of mod? ules and their operations lies in its being deeply rooted into the structure theory of algebras of analytic functions. Thus the comparison between Hilbert modules and, say, sheaves of ideals of analytic functions becomes transparent and benefic. In this way Douglas and Paulsen [5] have isolated from a series a previous works a phenomenon which is specific only to multidimensional Hardy submodules: once two such submodules are analytically isomorphic they are equal, see [1], [5], [6] and [7] for the^precise statements. This rigidity behaviour of Hardy submodules contrasts with Beurling's theorem-the latter implying that all invari? ant subspaces of the Hardy space on the torus are unitarily equivalent. When compared with the purely algebraic case of ideals of a regular, Noetherian local ring, this anomaly is no longer surprising. The transition from a Hardy submodule to an ideal of a local ring, which is necessary in any proof of a rigidity result as before, is made by a localization functor, see [6].

Operators in finite distributive subspace lattices, I

AbstractThe purpose of this paper is to settle in the negative an open problem in operator theory, which asks whether in a finite distributive subspace lattice ℒ on a Hilbert space, every finite rank operator of Alg ℒ can be written as a finite sum of rank one operators from Alg ℒ. The counter-example constructed is on a specific Hilbert space realization of the free distributive lattice on three generators and the operator which fails the above property has rank two.

ON THE K-THEORY OF QUARTER-PLANE TOEPLITZ ALGEBRAS

Given a C*-algebra A which is filtered by a collection of closed ideals Ai, there is a spectral sequence which relates the K-theory of A to the K-theory of the various quotient algebras Ai/Ai−1. The d1 differentials in this spectral sequence are familiar index invariants, but the higher differentials are not well-understood. Considering the case of Toeplitz C*-algebras associated with certain cones in Z2, it is shown that a d2 differential in the spectral sequence is non-trivial. This differential turns out to be an obstruction to a classical lifting problem in operator theory. Analysis of this obstruction leads to necessary and sufficient conditions for the lifting problem. It is hoped that this example will illuminate the role of higher differentials in the K-theory spectral sequence.

Boundary value problems and best approximation by holomorphic functions

A classical problem in complex analysis consists in finding the distance of a functionfe L” on the unit circle U to H”, the space of functions which extend to a bounded holomorphic function in the unit disk D. It is closely related to some other questions, such as the Pick-Nevanlinna problem of minimizing the supremum norm over the set of bounded holomorphic functions in D, subject to a finite or infinite set of interpolation conditions [S-lo] or the problem of seeking the largest circular domain of a positive harmonic function whose first Taylor coefficients are given [2]. The problem has a remarkable variety of applications, especially in systems engineering. Recent heightening of theoretical interest was brought about by results of Adamyan, Arov, and Krejn [l] on an equivalent problem in operator theory. Nowadays a series of related interpolation and approximation problems can be handled by several alternative mathemati- cal approaches in a unified treatment (problems with matrix-valued functions included, cf. [7, 141, introduction, for instance). Much less is known about a far-reaching generalization of the above problem, which was brought into discussion in the recent paper [6] by J. W. Helton and R. E. Howe (Unfortunately, this paper is not available to me at present, therefore I refer to [S]). The authors study the following optimization problem: Given a function F T x CN + Iw, find inf sup F( t, w(t)), WEE tt~ (1.1) where E= (H” n C)” denotes the space of all continuous cCN-valued func- tions on U with holomorphic continuation into D. Assuming the existence

On almost commuting Hermitian operators

It is an old problem in operator theory whether a pair of norm one compact Hermitian operators with “small” (in norm) commutator can be “well” approximated by a commuting pair of Hermitian operators. We show that, for operators of rank not exceeding n, such approximants exist provided ||[A, B]||/n1/2 is small. This improves a result of Pearcy and Shields and sheds some new light on the original question and its relationship to a few related ones. The following is an old question in the “local” operator theory (cf. [8]): If two norm one compact Hermitian operators have small commutators, are they close to a commuting pair? More precisely, Given e > 0, does there exist δ > 0 such that, whenever A,B are norm one, compact Hermitian operators on a Hilbert space with ||[A,B]|| ≤ δ, then one can find (compact Hermitian operators) A1, B1 satisfying ||A1 − A|| ≤ e, ||B1 − B|| ≤ e and [A1, B1] = 0? (1) We are going to refer to (1) as the Main Problem. An equivalent version follows: If T is a norm one compact operator with “small” selfcommutator [T ∗, T ], is T “close” to a normal operator? This one is clearly related to the work of Brown, Douglas, Filmore [3] on essentially normal operators. By approximation, questions of the above type reduce to the case of operators acting on finite dimensional spaces (i.e., to matrices) with dimension-free dependence of δ on e. Two positive results in the direction of the Main Problem are certainly worth mentioning. First, it was proved by Pearcy and Shields [7] that, if just one of the operators is assumed to be Hermitian and they The final part of this research was done while the author was visiting Institut des Hautes Etudes Scientifiques. Supported in part by the NSF grant # DMS-8702058 and the Sloan Research Fellowship. Received by the editors on January 20, 1988. Copyright c ©1990 Rocky Mountain Mathematics Consortium 581

Padé-approximants in operator theory for the solution of nonlinear differential and integral equations

The Padé approximation problem in operator theory and its solution will be repeated briefly together with some properties of the Padé approximants. Effective methods for the solution of differential and integral equations are of great practical importance and there is a vast literature devoted to this subject. Here a few methods, resulting from the use of Padé approximants in operator theory, are introduced and illustrated by means of some typical examples (initial value problems, boundary value problems, partial differential equations, nonlinear integral equations). Among the new methods is an iterative scheme which we will call Halley's method and which proves to be very useful in the neighbourhood of singularities. Well known methods such as Newton's and Chebyshev's method prove to be special cases of the class of iterative procedures.

States in quantum mechanics and a problem in operator theory

States in quantum mechanics and a problem in operator theory

Open problems in theory of metric linear spaces

Open problems in theory of metric linear spaces

Three test problems in operator theory

Three test problems in operator theory