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

C₀-semigroups of 2-isometries and Dirichlet spaces

In the context of a theorem of Richter, we establish a similarity between C₀-semigroups of analytic 2-isometries {T(t)}t≥0 acting on a Hilbert space H and the multiplication operator semigroup {Mϕt}t≥0 induced by ϕt(s)=exp(−st) for s in the right-half plane C₊ acting boundedly on weighted Dirichlet spaces on C₊. As a consequence, we derive a connection with the right shift semigroup {St}t≥0 given by Stf(x)={0 if 0≤x≤t,f(x−t) if x>t, acting on a weighted Lebesgue space on the half line R+ and address some applications regarding the study of the invariant subspaces\linebreak of C₀-semigroups of analytic 2-isometries

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

Toeplitz kernels and model spaces

We review some classical and more recent results concerning kernels of Toeplitz operators and their relations with model spaces, which are themselves Toeplitz kernels of a special kind. We highlight the fundamental role played by the existence of maximal vectors for every nontrivial Toeplitz kernel

Norms of Truncated Toeplitz Operators and Numerical Radii of Restricted Shifts

This paper gives a new approach to the calculation of the numerical radius of a restricted shift operator by linking it to the norm of a truncated Toeplitz operator (TTO), which can be calculated by various methods. Further results on the norm of a TTO are derived, and a conjecture on the existence of continuous symbols for compact TTO is resolved

Admissibility of Diagonal State-Delayed Systems with a One-Dimensional Input Space

In this paper we investigate admissibility of the control operator B in a Hilbert space state-delayed dynamical system setting of the form z˙(t)=Az(t−τ)+Bu(t) , where A generates a diagonal semigroup and u is a scalar input function. Our approach is based on the Laplace embedding between L2 and the Hardy space. The sufficient conditions for infinite-time admissibility are stated in terms of eigenvalues of the generator and in terms of the control operator itself

L₂ and BIBO stability of systems with variable delays

This paper considers L₂ and BIBO stability and stabilization issues for systems with time-varying delays which can be of retarded or neutral type. An important role is played by a nominal system with fixed delays which are close to the time-varying ones. Under stability or stabilizability conditions of this nominal system, sufficient conditions are given in order to ensure similar properties for the system with time-varying delays

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

Estimates near the origin for functional calculus on analytic semigroups

This paper provides sharp lower estimates near the origin for the functional calculus F (−uA) of a generator A of an operator semigroup defined on a sector; here F is given as the Fourier–Borel transform of an analytic functional. The results are linked to the existence of an identity element in the Banach algebra generated by the semigroup. Both the quasinilpotent and non-quasinilpotent cases are considered, and sharp results are proved extending many in the literature

On the Wandering Property in Dirichlet spaces

We show that in a scale of weighted Dirichlet spaces Dα, including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in Dα such that B satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell et al. (Indiana Univ Math J 51(4):931–961, 2002). As a particular instance, when B(z)=zk and |α|≤log(2)log(k+1), the chosen norm is the usual one in Dα