Florian-Horia Vasilescu

Standard models for some commuting multioperators

An analogue to the Sz.-Nagy-Foia5 dilation theory is presented for several commuting operators on a Hilbert space. The dilation theory of Hilbert space contractions, a reference of which is the excellent monograph [9], has not yet found a complete counterpart valid for several (commuting) Hilbert space operators. Apart from the results of this type already contained in [9] and other results of general character (i.e., Arvesons or Stinesprings extension theorem), there are not too many contributions specialized to the multiparameter dilation theory. Among these contributions, we quote the works [7, 3-5, 10], which are closer to our topics. The aim of this paper is to analyse some positivity conditions for commuting multioperators, which ensure the unitary equivalence of these objects to some standard models consisting of backwards multishifts. We emphasize, in particular, a situation that seems to satisfy the requirements of an appropriate extension for the dilation theory of Hilbert space contractions and that exploits the results already obtained in [4, 10]. Our finer methods also imply better statements of some assertions from [10] and provide an answer to Question 4.5 from [5]. Let n > 1 be an integer, and let Zn be the set of all n-tuples of nonnegative integers (i.e., the multi-indices of length n). If a = (a, ...I, an) E Zn+, we set, as usual, lal =a,+ + -+a, and a! =a1! .. !. For a,ficZ+ wewrite a + f = (a1 + f1j ... , an + fin), and a < fi whenever aci < /i (i = 1, ... , n). Let H be a Hilbert space, and let 2(H) be the algebra of all bounded linear operators on H. An element T = (T,, ... , T,) E 2(H)n such that T,, ... , T, mutually commute will be designated as a commuting multioperator (briefly, a c.m.). Let T E Y(H)n be a c.m. We define the operator MT: Y(H) -* J(H) by the formula

An operator-valued Poisson kernel

Abstract Versions of the Cauchy and Poisson formulas in the unit ball of C n are defined, by means of appropriate operator-valued kernels. Connections with some problems of dilation theory are then made.

Hamburger and Stieltjes moment problems in several variables

In this paper we give solutions to the Hamburger and Stieltjes moment problems in several variables, in algebraic terms, via extended sequences. Some characterizations of the uniqueness of the solutions are also presented.

Banach space complexes

Introduction. I: Preliminaries. 1. Algebraic prerequisites. 2. Algebraic Fredholm pairs. 3. Paraclosed linear transformations. 4. Homogeneous operators. 5. Linear and homogeneous projections and liftings. 6. The gap between two closed subspaces. 7. Linear operators with closed range, and finite extensions. 8. Metric relations and duality. 9. Operators in quotient Banach spaces. 10. References and comments. II: Semi-Fredholm complexes. 1. Semi-Fredholm operators. 2. Semi-Fredholm complexes. 3. Essential complexes. 4. Fredholm pairs. 5. Other continuous invariants. 6. References and comments. III: Related topics. 1. Joint spectra and perturbations. 2. Spectral interpolation and perturbations. 3. Versions of Poincares and Grothendiecks lemmas. 4. Differentiable families of partial differential operators. 5. References and comments. Subject index. Notations. Bibliography.

Stability of the index of a semi-Fredholm complex of Banach spaces

Abstract We prove the stability of the index and the semicontinuity of the dimensions of the cohomology groups of semi-Fredholm complexes of Banach spaces and closed linear operators with respect to perturbations of the operators and of the underlying spaces which are small with respect to the gap topologies. It seems that, even for single semi-Fredholm operators, some of the statements are more general than the current ones. The results are applied to obtain semicontinuity for joint spectra of finite systems of commuting bounded linear operators.

Operator-Theoretic Positivstellensätze

We study the structure of positive polynomials with coefficients in an operator algebra as a non-commutative infinite-dimensional analogue of Hilbert’s 17-th problem.

Positive polynomials on semi-algebraic sets

Abstract Structure theorems, in the spirit of E. Artins Positivstellensatz, are obtained for certain positive polynomials on semi-algebraic subsets of the Euclidean space. The proofs use simple geometric constructions and a separation theorem of convex cones in topological vector spaces.

Operator moment problems in unbounded sets

Results concerning some moment problems in unbounded sets from the scalar context to the case of linear operator data are extended. The present methods also lead to a theoretic characterization of a class of subnormal multioperators.

Some properties of the commutator of two operators

1. Consider a Banach space X and let B(X) be the algebra of linear- bounded operators on 3Z. If T, S, X E B(3E) we may define on B(X) the opera- tor C( T, S) X = TX - XS, which is often called the commutator of T and S. We denote px(T, S) = !E /I C(T, S)n X jjlln. This number can be considered as a “spectral distance” from S to T with regard to X, being intimately connected with spectral properties of T and S at least in the case when these have the single-valued extension property [l] (in particular if they are decomposable [2]). If X = I then (see [3]) C(T, S)” I = (T - S)[%l = f (- l)k (;) T”-kSk

Unbounded Normal Algebras and Spaces of Fractions

We consider arbitrary families of unbounded normal operators, commuting in a strong sense, in particular algebras consisting of unbounded normal operators, and investigate their connections with some algebras of fractions of continuous functions on compact spaces. New examples and properties of spaces of fractions are also given.