Burnside's Theorem for the Fréchet Space .
Certain invariant subspaces for operators with rich eigenvalues.
Cohyponormal operators with the single valued extension property.
Commutants of the Square of Differentiation on the Half-Line
MSC 2010: Primary: 447B37; Secondary: 47B38, 47A15
Commutators of quasinilpotents and invariant subspaces
It is proved that the set Q of quasinilpotent elements in a Banach algebra is an ideal, i.e. equal to the Jacobson radical, if (and only if) the condition [Q,Q] ⊆ Q (or a similar condition concerning anticommutators) holds. In fact, if the inner derivation defined by a quasinilpotent element p maps Q into itself then p ∈ Rad A. Higher commutator conditions of quasinilpotents are also studied. It is shown that if a Banach algebra satisfies such a condition, then every quasinilpotent element has some...
Compact operators on the weighted Bergman space A¹(ψ)
We show that a bounded linear operator S on the weighted Bergman space A¹(ψ) is compact and the predual space A₀(φ) of A¹(ψ) is invariant under S* if and only if as z → ∂D, where is the normalized reproducing kernel of A¹(ψ). As an application, we give conditions for an operator in the Toeplitz algebra to be compact.
Complétude des noyaux reproduisants dans les espaces modèles
Soit une suite de Blaschke du disque unité et une fonction intérieure. On suppose que la suite de noyaux reproduisants est complète dans l’espace modèle , . On étudie, dans un premier temps, la stabilité de cette propriété de complétude, à la fois sous l’effet de perturbations des fréquences mais également sous l’effet de perturbations de la fonction . On retrouve ainsi un certain nombre de résultats classiques sur les systèmes d’exponentielles. Puis, si on suppose de plus que la suite ...
Conditions ensuring T-1(Y) ⊂ Y.
The following theorem is the main result of the paper: Let X be a complex Banach space and T belong to L(X). Suppose that 0 lies at the unbounded component of the set of those l such that lI - T is a Fredholm operator. Let Y be a dense subspace of the dual space X' and S be a closed operator from Y to X such that T'(Y) is contained in Y and TSy = ST'y for every y belonging to Y. Then for every vector x belonging to X', T'x belongs to Y if and only if x belongs to Y.
Contractions de classe et sous-éspaces invariants
Convergence of derivations on nearest algebras.
Corrections to “Radial limits in co-invariant subspaces"
Das Lomonosov-Lemma für topologische Vektorräume.
Decomposable embeddings, complete trajectories, and invariant subspaces
We produce closed nontrivial invariant subspaces for closed (possibly unbounded) linear operators, A, on a Banach space, that may be embedded between decomposable operators on spaces with weaker and stronger topologies. We show that this can be done under many conditions on orbits, including when both A and A* have nontrivial non-quasi-analytic complete trajectories, and when both A and A* generate bounded semigroups that are not stable.
Decompositions and asymptotic limit for bicontractions
The asymptotic limit of a bicontraction T (that is, a pair of commuting contractions) on a Hilbert space is used to describe a Nagy-Foiaş-Langer type decomposition of T. This decomposition is refined in the case when the asymptotic limit of T is an orthogonal projection. The case of a bicontraction T consisting of hyponormal (even quasinormal) contractions is also considered, where we have .
Definition Theory and Systems of Simultaneous Equations in the Perdual of an Operator Algebra. II.
Der spektrale Abbildungssatz für nichtanalytische Funktionalkalküle in mehreren Veränderlichen.
Derivations of Nest Algebras.
Description of invariant subspaces of Lp(μ) by multiplication operators.
Direct sums of irreducible operators
It is known that every operator on a (separable) Hilbert space is the direct integral of irreducible operators, but not every one is the direct sum of irreducible ones. We show that an operator can have either finitely or uncountably many reducing subspaces, and the former holds if and only if the operator is the direct sum of finitely many irreducible operators no two of which are unitarily equivalent. We also characterize operators T which are direct sums of irreducible operators in terms of the...
Doubly commuting submodules of the Hardy module over polydiscs
In this note we establish a vector-valued version of Beurling’s theorem (the Lax-Halmos theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the “weak” completion problem in .