59 research outputs found
Approximation in reflexive Banach spaces and applications to the invariant subspace problem
We formulate a general approximation problem involving reflexive and smooth Banach spaces, and give its explicit solution. Two applications are presented— the first is to the Bounded Completion Problem involving approximation of Hardy class functions, while the second involves the construction of minimal vec- tors and hyperinvariant subspaces of linear operators, generalizing the Hilbert space technique of Ansari and Enflo
Approximation in reflexive Banach spaces and applications to the invariant subspace problem
We formulate a general approximation problem involving reflexive and smooth Banach spaces, and give its explicit solution. Two applications are presented— the first is to the Bounded Completion Problem involving approximation of Hardy class functions, while the second involves the construction of minimal vec- tors and hyperinvariant subspaces of linear operators, generalizing the Hilbert space technique of Ansari and Enflo
L1-Factorization forC00-Contractions with Isometric Functional Calculus
AbstractLetTbe an absolutely continuous contraction acting on a Hilbert space H. Forx, y∈H, definex·Ty∈L1(T) by its Fourier coefficients:x·Ty∧(n)=(T*nx, y) ifn<0. The main technical result of the paper is that the vanishing condition limn→∞(‖xn·Tw‖L1/H10+‖w·Txn‖L1/H10)=0,w∈H implies that limn→∞‖xn·Tw‖L1=0,w∈H. Using known factorization techniques, we exhibit a Borel setσTsuch that for anyf∈L1(σT), there existx, y∈H such thatf=(x·Ty)|σT. In the case whereT∈A∩C00, this leads to a simple proof of the fact that for everyf∈L1(T) there existsx, y∈H such thatf=x·Ty. In this case we also show, using dilation theory in the unit disk, that every strictly positive lower semicontinuous functionϕ∈L1(T) can be written in the formϕ=x·Tx. Examples show that this is the best possible result for the class A∩C00
Compactness, differentiability and similarity to isometry of composition semigroups
This paper provides sufficient conditions for eventual compactness and differentiability of C0-semigroups on the Hardy and Dirichlet spaces on the unit disc with a prescribed generator of the form Af = Gf'. Moreover, the isometric semigroups (or isometric up to a similarity) of composition operators on the Hardy space are characterized in terms of G
Clark measures and a theorem of Ritt
We determine when a finite Blaschke product B can be written, in a non-trivial way, as a composition of two finite Blaschke products (Ritt's problem) in terms of the Clark measure for B. Our tools involve the numerical range of compressed shift operators and the geometry of certain polygons circumscribing the numerical range of the relevant operator. As a consequence of our results, we can determine, in terms of Clark measures, when two finite Blaschke products commute
A class of quasicontractive semigroups acting on Hardy and Dirichlet space
This paper provides a complete characterization of quasicontractive C0-semigroups on Hardy and Dirichlet space with a prescribed generator of the form Af = Gf ′. We show that such semigroups are semigroups of composition operators, and we give simple sufficient and necessary condition on G. Our techniques are based on ideas from semigroup theory, such as the use of numerical ranges
Inner functions and operator theory
This tutorial paper presents a survey of results, both classical and new, linking inner functions and operator theory. Topics discussed include invariant subspaces, universal operators, Hankel and Toeplitz operators, model spaces, truncated Toeplitz operators, restricted shifts, numerical ranges, and interpolation
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