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$AC$-operators are a generalization in the context of well-bounded\-ness of normal operators on Hilbert space. It was shown by Doust and Walden that compact $AC$-operators have a representation as a conditionally convergent sum reminiscent of the spectral representations for compact normal operators. In this representation, the eigenvalues must be taken in a particular order to ensure convergence of the sum. Here we show that one cannot replace the ordering given by Doust and Walden by the more natural one suggested in their paper. [ABSTRACT FROM AUTHOR]

Suppose that $(\Omega ,\mathcal{M},\mu )$ is a $\sigma $-finite measure space, $1

In this paper we completely solve the open problem of finding the fundamental solution of the semidiscrete fractional-spatial damped wave equation. We combine operator theory and Laplace transform methods with properties of Bessel functions to show an explicit representation of the solution when initial conditions are given. Our findings extend known results from the literature and also provide new insights into the qualitative behavior of the solutions for the studied model. As an example, we show the existence of almost periodic solutions as well as their profile in the homogeneous case. [ABSTRACT FROM AUTHOR]

The discrete counterpart of the problem related to the convergence of the Fourier-Jacobi series is studied. To this end, given a sequence, we consider the analogue of the partial sum operator related to Jacobi polynomials and characterize its convergence in the ℓp(N)-norm. [ABSTRACT FROM AUTHOR]

We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy-Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on stochastic representation of the Hilbert transform in the continuous setting. [ABSTRACT FROM AUTHOR]

We prove various extensions of the Coifman-Rubio de Francia- Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call ℓr(ℓs )-boundedness, which implies R-boundedness in many cases. The proofs are based on new Littlewood-Paley-Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors. [ABSTRACT FROM AUTHOR]

We show that any bounded analytic semigroup on Lp (with 1 < p < ∞) whose negative generator admits a bounded H∞(Σθ) functional calculus for some θ ∊ (0, π/2 ) can be dilated into a bounded analytic semigroup (Rt)t≥ 0 on a bigger Lp-space in such a way that Rt is a positive contraction for any t≥0. We also establish a discrete analogue for Ritt operators and consider the case when Lp-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations of amenable groups on Banach spaces and establish various generalizations of Dixmier's unitarization theorem. [ABSTRACT FROM AUTHOR]

Suppose E is a subset of the unit circle T and H∞ ⊂ L∞ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to znH∞. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings. [ABSTRACT FROM AUTHOR]

Dominated and Pointwise Ergodic Theorems for bounded representations of second countable locally compact amenable groups by Lamperti operators in Lp(O,F,m), p > 1 fixed, are proved; we restrict ourselves to Cesàro averages in this paper. These theorems generalize or are closely related to well-known theorems for powers of power bounded Lamperti operators. [ABSTRACT FROM AUTHOR]

We obtain restriction results of K. De Leeuw's type for maximal operators defined through Fourier multipliers of either strong or weak type for weighted Lp spaces with 1 ≤ p > ∞. Applications to the case of Höormander-Mihlin multipliers, singular integral operators and Bochner-Riesz sums are given. [ABSTRACT FROM AUTHOR]