We establish the (polynomially) logarithmic decay of ergodic means of Cesàro bounded operators of any fractional order, under convergence of the one-sided ergodic Hilbert transform. This extends the theorem of Gomilko, Haase and Tomilov for power bounded operators. We also improve the polynomial decay of means involved in the fractional Poisson equation. The theorems are obtained as an application of a general result, also proved here, about rates of decay of means for Cesàro bounded operators.

We study spectral synthesis properties in convolution Sobolev algebras on the real line. Mainly, a description of primary closed ideals in such algebras is given. Then we address an approximation problem involving bounded representations of such Sobolev algebras, which arises naturally in relation with the asymptotic behavior of integrated semigroups

We provide the spectral picture of groups of weighted composition operators, induced by the hyperbolic group of automorphisms of the unit disc, acting on holomorphic functions. Some questions about the spectrum of single weighted hyperbolic composition operators are discussed, and results related with them in the literature are completed or partly extended. Also, our results on the weighted hyperbolic group are applied to the spectral study of two families of multiparameter weighted averaging operators, which generalize both Siskakis' operator and the reduced Hilbert matrix operator. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).

We characterize the solutions of the Poisson equation and the domain of its associated one-sided Hilbert transform for ( C , α )-bounded operators, α > 0. This extends known results for power bounded operators to the setting of Cesàro bounded operators of fractional order, thus generalizing the results substantially. In passing, we obtain a generalization of the mean ergodic theorem in our framework. Examples are given to illustrate the theory.

We characterize twisted convolutions associated with the Pedersen transform for unitary irreducible representations of nilpotent Lie groups. For $$1\le p<\infty $$ , we also prove the $$L^p$$ -boundedness for the Pedersen $$L^p$$ -multipliers in the case of unitary irreducible representations that are square-integrable modulo the center of the group under consideration, thus, generalizing an earlier result on Weyl multipliers associated to the pseudo-differential Weyl calculus.

We study reproducing kernel Hilbert spaces introduced as ranges of generalized Cesàro–Hardy operators, in one real variable and in one complex variable. Such spaces can be seen as formed by absolutely continuous functions on the positive half-line (or paths of infinite length) of fractional order, in the real case. A theorem of Paley–Wiener type is given which connects the real setting with the complex one. These spaces are related with fractional operations in the context of integrated Brownian processes. We give estimates of the norms of the corresponding reproducing kernels.

We discuss the behaviour at infinity of n-times integrated semi-groups with nonquasianalytic growth. The results obtained provide in this setting extensions of the Arendt-Batty-Lyubich-Vu theorem on stability of C-0-semigroups and of a theorem of El Mennaoui on stability of bounded once integrated semigroups.

In this paper we deal with a scale of reproducing kernel Hilbert spaces H2(n), n≥0, which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane C+. They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree. An explicit integral formula is given for the reproducing kernel Kz,n of H2(n), from which we can find the estimate ‖Kz,n‖∼|z|−1/2 for z∈C+. Then composition operators Cφ:H2(n)→H2(n), Cφf=f∘φ, on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps φ:C+→C+ to induce bounded composition operators.

We establish a general CCR (liminarity) property for uniformly bounded irreducible representations of nilpotent Lie groups on reflexive Banach spaces, extending the well-known property of unitary irreducible representations of these groups on Hilbert spaces. We also prove that this conclusion fails for many representations on non-reflexive Banach spaces. Our approach to these results blends the method of transference from abstract harmonic analysis and a systematic use of spaces of smooth vectors with respect to Lie group representations.

We discuss the behaviour at infinity of n-times integrated semigroups with nonquasianalytic growth and invertible generator. The results obtained extend in this setting a theorem of O. El Mennaoui on stability of bounded once integrated semigroups, and (partially) a theorem of Q. P. Vũ on stability of C_0-semigroups.

We investigate infinitesimal properties of sets of ordered $n$-uples of idempotents in a symmetric Banach $*$-algebra. These sets are called flag manifolds and carry several interesting bundles that hold an important role in some areas of operator theory. In this direction, we introduce and study Stiefel bundles on flag manifolds, which are extensions of the well known Stiefel bundles on Grassmannians. The main ingredient of our investigation is the notion of connection on an infinite-dimensional bundle, and we survey some equivalent ocurrences of such a notion in the literature.

We investigate the interaction between the existence of reproducing kernels on infinite-dimensional Hermitian vector bundles and the positivity properties of the corresponding bundles. The positivity refers to the curvature form of certain covariant derivatives associated to reproducing kernels on the vector bundles under consideration. The values of the curvature form are Hilbert space operators, and its positivity is thus understood in the usual sense from operator theory.

For vector bundles having an involution on the base space, Hermitian-like structures are defined in terms of such an involution. We prove a universality theorem for suitable self-involutive reproducing kernels on Hermitian-like vector bundles. This result relies on pullback operations involving the tautological bundle on the Grassmann manifold of a Hilbert space and exhibits the aforementioned reproducing kernels as pullbacks of universal reproducing kernels that live on the Hermitian-like tautological bundle. To this end we use a certain type of classifying morphisms, which are geometric versions of the coherent state maps from quantum theory. As a consequence of that theorem, we obtain some differential geometric properties of these reproducing kernels in this setting.

It is shown how reproducing kernels, in a wide class, define in a very natural manner differential geometric objects like linear connections, covariant derivatives, and curvatures. The correspondence from kernels to connections is achieved through a pullback operation from the tautological universal bundle, using a suitable classifying morphism for the given kernel. The theory is illustrated by several examples including classical kernels in function spaces, kernels occurring in dilation theory for completely positive maps, and kernels on homogeneous vector bundles.

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism of the input kernel. The aforementioned correspondence turns out to be a canonical functor between categories of kernels and linear connections. A number of examples of linear connections including the ones associated to classical kernels, homogeneous reproducing kernels and kernels occurring in the dilation theory for completely positive maps are given, together with their covariant derivatives.

Y. Katznelson and L.Tzafriri proved that if T is a power-bounded operator and f is an analytic function, in the Wiener algebra, of spectral synthesis with respect to its peripheral spectrum then lim (n ->infinity) parallel to T-n f(T)parallel to = 0. Here f(T) is given by the usual functional calculus associated with T. The analogous version for bounded C-0-semigroups of operators was obtained by J. Esterle, E. Strouse and F. Zouakia and independently by Q.P. Vu. We extend this result to alpha-times integrated semigroups. The proof is based on an analysis of homomorphisms from convolution Banach algebras of Sobolev type.

We extend results of Caffarelli–Silvestre and Stinga–Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in particular to infinitesimal generators of bounded C 0 semigroups and operators with purely imaginary symbol. We give integral representations to the extension problem in terms of solutions to the heat equation and the wave equation.

We give examples of convolution semigroups on the positive half-line and on the real line. Such semigroups are expressed in terms of special functions which arise in classical differential equations.

We study the range of the Laplace transform on convolution Banach algebras T(α)(tα), α>0, defined by fractional derivation. We introduce Banach algebras A0(α)(C+) of holomorphic functions in the right hand half-plane which are defined using complex fractional derivation along rays leaving the origin. We prove that the range of the Laplace transform on T(α)(tα) is densely contained in A0(α)(C+). The proof makes use of the so-called Kummer functions.