In this paper we establish operator quasilinearity properties of some functionals associated with Davis–Choi–Jensen’s inequality for positive maps and operator convex or concave functions. Applications for the power function and the logarithm are provided.

We introduce some new refinements of numerical radius inequalities for Hilbert space invertible operators. More precisely, we prove that if $T\in {\mathcal{B}}({\mathcal{H}})$ is an invertible operator, then $\Vert T\Vert \leq \sqrt{2}\unicode[STIX]{x1D714}(T)$ .

Let $\unicode[STIX]{x1D6FA}$ be a domain in $\mathbb{R}^{m}$ with nonempty boundary. In Ward [‘On essential self-adjointness, confining potentials and the $L_{p}$ -Hardy inequality’, PhD Thesis, NZIAS Massey University, New Zealand, 2014] and [‘The essential self-adjointness of Schrödinger operators on domains with non-empty boundary’, Manuscripta Math.150(3) (2016), 357–370] it was shown that the Schrödinger operator $H=-\unicode[STIX]{x1D6E5}+V$ , with domain of definition $D(H)=C_{0}^{\infty }(\unicode[STIX]{x1D6FA})$ and $V\in L_{\infty }^{\text{loc}}(\unicode[STIX]{x1D6FA})$ , is essentially self-adjoint provided that $V(x)\geq (1-\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA}))/d(x)^{2}$ . Here $d(x)$ is the Euclidean distance to the boundary and $\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$ is the nonnegative constant associated to the $L_{2}$ -Hardy inequality. The conditions required for a domain to admit an $L_{2}$ -Hardy inequality are well known and depend intimately on the Hausdorff or Aikawa/Assouad dimension of the boundary. However, there are only a handful of domains where the value of $\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$ is known explicitly. By obtaining upper and lower bounds on the number of cubes appearing in the $k\text{th}$ generation of the Whitney decomposition of $\unicode[STIX]{x1D6FA}$ , we derive an upper bound on $\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x1D6FA})$ , for $p>1$ , in terms of the inner Minkowski dimension of the boundary.

Let ${\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ be the set of Cowen–Douglas operators of index $n$ on a nonempty bounded connected open subset $\unicode[STIX]{x1D6FA}$ of $\mathbb{C}$ . We consider the strong irreducibility of a class of Cowen–Douglas operators ${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ on Banach spaces. We show ${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})\subseteq {\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ and give some conditions under which an operator $T\in {\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ is strongly irreducible. All these results generalise similar results on Hilbert spaces.

We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.

Kuznetsov and co-authors in 2011‒14 introduced the family of hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. In 2014 it was shown that the definition of a hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally specified. In this short article, we push the parameter range even further.

The classical spaces ℓp+, 1 ≤ p < ∞, and L p−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, L p loc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

We prove that the classical universal Taylor series in the complex plane are never frequently universal. On the other hand, we prove the 1-upper frequent universality of all these universal Taylor series.

Let $H^{2}$ be the Hardy space over the bidisk. It is known that Hilbert–Schmidt invariant subspaces of $H^{2}$ have nice properties. An invariant subspace which is unitarily equivalent to some invariant subspace whose continuous spectrum does not coincide with $\overline{\mathbb{D}}$ is Hilbert–Schmidt. We shall introduce the concept of splittingness for invariant subspaces and prove that they are Hilbert–Schmidt.

In this paper we introduce the two possible formulations of the -functional calculus that are based on the Fueter–Sce mapping theorem in integral form and we introduce the pseudo--resolvent equation. In the case of dimension 3 we prove the -resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the -functional calculus.

We improve a previous result about the local energy decay for the damped wave equation on $\mathbb{R}^{d}$ . The problem is governed by a Laplacian associated with a long-range perturbation of the flat metric and a short-range absorption index. Our purpose is to recover the decay ${\mathcal{O}}(t^{-d+\unicode[STIX]{x1D700}})$ in the weighted energy spaces. The proof is based on uniform resolvent estimates, given by an improved version of the dissipative Mourre theory. In particular, we have to prove the limiting absorption principle for the powers of the resolvent with inserted weights.

We consider both a problem over ℝn and a boundary problem over an exterior subregion for a Douglis–Nirenberg system of differential operators under limited smoothness assumptions as well as under the assumption of parameter ellipticity in a closed sector Ꮭ in the complex plane with vertex at the origin. We pose each problem on an Lp Sobolev space setting, 1 < p < ∞, and denote by the operator induced in this setting by the problem over ℝn and by Ap the operator induced in this setting by the boundary problem under null boundary conditions. We then derive results pertaining to the Fredholm theory for each of these operators for values of the spectral parameter λ lying in Ꮭ as well as results for these values of λ pertaining to the invariance of their Fredholm domains with p.

We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if $\unicode[STIX]{x1D6E4}$ is a nonamenable group and $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.

We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T: X → X on some Banach space X with the following property: T has a bounded H∞-functional calculus with respect to the unit disc 𝔻(that is, T is polynomially bounded) but T does not have any bounded H∞-functional calculus with respect to a Stolz domain of 𝔻 with vertex at 1. Also we show that for an R-Ritt operator the unconditional Ritt condition of Kalton and Portal is equivalent to the existence of a bounded H∞-functional calculus with respect to such a Stolz domain.

We introduce efficient approaches to construct high order finite difference discretizations for solving partial differential equations, based on a composite grid hierarchy. We introduce a modification of the traditional point clustering algorithm, obtained by adding restrictive parameters that control the minimal patch length and the size of the buffer zone. As a result, a reduction in the number of interfacial cells is observed. Based on a reasonable geometric grid setting, we discuss a general approach for the construction of stencils in a composite grid environment. The straightforward approach leads to an ill-posed problem. In our approach we regularize this problem, and transform it into solving a symmetric system of linear of equations. Finally, a stencil repository has been designed to further reduce computational overhead. The effectiveness of the discretizations is illustrated by numerical experiments on second order elliptic differential equations.

An investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator $\mathsf{C}$ when acting on the weighted Banach sequence spaces $\ell _{p}(w)$, $1<p<\infty$, for a positive decreasing weight $w$, thereby extending known results for $\mathsf{C}$ when acting on the classical spaces $\ell _{p}$. New features arise in the weighted setting (for example, existence of eigenvalues, compactness) which are not present in $\ell _{p}$.

We use the best constants in the Khintchine inequality to generalise a theorem of Kato [‘Similarity for sequences of projections’, Bull. Amer. Math. Soc.73(6) (1967), 904–905] on similarity for sequences of projections in Hilbert spaces to the case of unconditional Schauder decompositions in $\ell _{p}$ spaces. We also sharpen a stability theorem of Vizitei [‘On the stability of bases of subspaces in a Banach space’, in: Studies on Algebra and Mathematical Analysis, Moldova Academy of Sciences (Kartja Moldovenjaska, Chişinău, 1965), 32–44; (in Russian)] in the case of unconditional Schauder decompositions in any Banach space.

We establish interior and trace embedding results for Sobolev functions on a class of bounded non-smooth domains. Also, we define the corresponding generalized Maz'ya spaces of variable exponent, and obtain embedding results similar as in the constant case. Some relations between the variable exponent Maz'ya spaces and the variable exponent Sobolev spaces are also achieved. At the end, we give an application of the previous results for the well-posedness of a class of quasi-linear equations with variable exponent.

We offer a Lebesgue-type decomposition of a representable functional on a *-algebra into absolutely continuous and singular parts with respect to another. Such a result was proved by Zs. Szűcs due to a general Lebesgue decomposition theorem of S. Hassi, H.S.V. de Snoo, and Z. Sebestyén concerning non-negative Hermitian forms. In this paper, we provide a self-contained proof of Szűcs' result and in addition we prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.

We continue the study of the boundedness of the operator

on the set of decreasing functions in Lp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimate was completely solved, the characterization of the weights w such that is bounded is still open for the case in which p > 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.