Isabelle Chalendar

Approximation in reflexive Banach spaces and applications to the invariant subspace problem

AbstractWe formulate a general approximation probleminvolving reflexive and smoothBanach spaces, and give its explicit solution. Two applications are presented—the first is to the Bounded Completion Problem involving approximation ofHardy class functions, while the second involves the construction of minimal vec-tors and hyperinvariant subspaces of linear operators, generalizing the Hilbertspace technique of Ansari and Enflo. 2000 Mathematics Subject Classification: 41A29, 47A15, 46B20, 46E15.Keywords: Constrained approximation, Smoothness, Invariantsubspaces, Hardyspaces,Extremal problems. 1 Introduction The construction of minimal vectors, corresponding to an operator on a Hilbert spacewas introduced by Ansari and Enflo [3], who used it to give simpler proofs of theexistence of invariant subspaces for certain classes of operator, including compact op-erators and normal operators. The minimal vectors are given as the solution of certainconstrained approximation problems, and in [9] a more general such problem was for-mulated and solved, which has other applications, notably in the theories of systemsidentification, signal processing and inverse problems (see [8] for a survey of this area).It is possible to define minimal vectors in a more general Banach space context,although the associated approximation problem is rather more difficult to resolve. An-droulakis [2] generalized the techniques of Ansari and Enflo to produce a new suffi-cient condition for the existence of hyperinvariant subspaces of certain operators on

Constrained approximation and invariant subspaces

We begin by formulating and solving a general constrained approximation problem. Two special cases are of particular interest. The first includes a problem studied in [9], which has several applications in the theories of systems identification, signal processing and inverse problems (see [5] for a survey of this area). The second special case is an application to the construction of (backward) minimal vectors. This has recently been introduced by Ansari and Enflo [2] as a more explicit technique for constructing hyperinvariant subspaces of bounded linear operators on Hilbert spaces. Their technique is particularly interesting since it provides a new unified method for showing that every compact operator and every normal operator has a hyperinvariant subspace. By investigating the further possibilities of their method, we extract a somewhat more general theorem, which we use to find hyperinvariant subspaces for bounded linear operators which are neither quasinilpotent nor polynomially compact. We illustrate the main result by applying it to weighted shift operators, which are not covered by the existing theorems. Another situation in which minimal vectors have been considered is in the case of operators T of multiplication by outer functions on the Hardy space H 2, as in the work of Spalsbury [14]. Here we use the theory of Toeplitz operators and the Fejer–Riesz theorem to provide explicit expressions for the backward minimal vectors yn for rational outer functions, and to provide a detailed analysis of the convergence of the sequence (T yn), which is the key to the techniques introduced in [2].

Approximation problems in some holomorphic spaces, with applications

We review a selection of extremal problems to do with constrained approximation in certain Banach spaces of holomorphic functions, including the classical Hardy spaces and Paley-Wiener spaces. In many cases the solutions are best interpreted in terms of linear operators. Applications of the problems under discussion to systems identification, signal processing, inverse problems for partial differential equations, and operator theory are presented.

The group of the invariants of a finite blaschke product

If b is a finite Blaschke product, we prove that the set of continuous functions such that is a cyclic group. We also study the possibility of extending analytically such functions u in ũ with where ũ is well-defined. For that purpose we localize the zeros of the derivative of a Blaschke product (not necessarily finite).

Convergence properties of minimal vectors for normal operators and weighted shifts

We study the behaviour of the sequence of minimal vectors corresponding to certain classes of operators on reflexive L p spaces, including multiplication operators and bilateral weighted shifts. The results proved are based on explicit formulae for the minimal vectors, and provide extensions of results due to Ansari and Enflo, and also Wiesner. In many cases the convergence of sequences associated with the minimal vectors leads to the construction of hyperinvariant subspaces for cyclic operators.

Determination of Inner Functions by their Value Sets on the Circle

Let Θ be an inner function with finitely many singularities on the unit circle. Then, given distinct points α and β of unit modulus, an explicit expression for Θ is given, in terms of the sets of points on the circle at which Θ takes the values α and β. If Θ has just one singularity, then it is unique to within a hyperbolic automorphism: this property need not hold in general. At the same time, a full characterization of the possible sets at which such values may be taken is obtained. The methods used involve an extension of the Hermite-Biehler Theorem to functions with finitely many essential singularities. Further applications in the theory of isometric composition operators, as well as uniqueness results for restricted shifts involving their numerical ranges are discussed.

When do finite Blaschke products commute

We study the following questions. Which finite Blaschke products are eigenvectors of the composition operators T u : f ↦ f ∘ u , what are the possible eigenvalues, and which pairs ( B , C ) of finite Blaschke products commute (that is, satisfy B ∘ C = C ∘ B ).

The Method of Minimal Vectors Applied to Weighted Composition Operators

We study the behavior of the sequence of minimal vectors corresponding to certain classes of operators on L 2 spaces, including weighted composition operators such as those induced by Mobius transformations. In conjunction with criteria for quasinilpotence, the convergence of sequences associated with the minimal vectors leads to the construction of hyperinvariant subspaces.

L1 factorizations for some perturbations of the unilateral shift

Abstract We say that T∈ L ( H ) (where H is a Hilbert space) factorizes f∈ L 1 ( T ) if there exist x,y∈ H such that f (n)=(T ∗n x,y) if n≥0 and f (−n)=(T n x,y) if n≥1. By virtue of one of Bourgains results, the unilateral shift S∈ L ( H 2 ) of multiplicity one factorizes f∈ L 1 ( T ) if and only if log |f|∈ L 1 ( T ) . We study the absolutely continuous contractions A such that the operator S⊕A factorizes all functions in L 1 ( T ) .

Binomial sums, moments and invariant subspaces

AbstractThe main result of this paper is that if a sequence of complex numbers (an)n≥0 satisfies