We introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari and Fefferman are proved.

In this note semi-bounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on the parameters in the boundary space to induce self-adjoint realizations, and to relate the decay of the Weyl function to estimates on the lower bound of the spectrum. The abstract results are illustrated with uniformly elliptic second-order partial differential equations on domains with non-compact boundaries.

Let ${\mathcal{A}}$ be a unital torsion-free algebra over a unital commutative ring ${\mathcal{R}}$ . To characterise Lie $n$ -higher derivations on ${\mathcal{A}}$ , we give an identity which enables us to transfer problems related to Lie $n$ -higher derivations into the same problems concerning Lie $n$ -derivations. We prove that: (1) if every Lie $n$ -derivation on ${\mathcal{A}}$ is standard, then so is every Lie $n$ -higher derivation on ${\mathcal{A}}$ ; (2) if every linear mapping Lie $n$ -derivable at several points is a Lie $n$ -derivation, then so is every sequence $\{d_{m}\}$ of linear mappings Lie $n$ -higher derivable at these points; (3) if every linear mapping Lie $n$ -derivable at several points is a sum of a derivation and a linear mapping vanishing on all $(n-1)$ th commutators of these points, then every sequence $\{d_{m}\}$ of linear mappings Lie $n$ -higher derivable at these points is a sum of a higher derivation and a sequence of linear mappings vanishing on all $(n-1)$ th commutators of these points. We also give several applications of these results.

The goal of this paper is to characterize the operating functions on modulation spaces $M^{p,1}(\mathbb{R})$ and Wiener amalgam spaces $W^{p,1}(\mathbb{R})$ . This characterization gives an affirmative answer to the open problem proposed by Bhimani (Composition Operators on Wiener amalgam Spaces, arXiv: 1503.01606) and Bhimani and Ratnakumar (J. Funct. Anal. 270 (2016), pp. 621–648).

We carry out an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand, we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron–Frobenius type spectral theorems. We furthermore prove a Kreĭn–Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.

In this paper, we study spectral properties and local spectral properties of ∞-complex symmetric operators T. In particular, we prove that if T is an ∞-complex symmetric operator, then T has the decomposition property (δ) if and only if T is decomposable. Moreover, we show that if T and S are ∞-complex symmetric operators, then so is T ⊗ S.

For a von Neumann subalgebra $A \, \subseteq \, {\cal B}({\cal H})$ and any two elements a, b ∈ A with a normal, such that the corresponding derivations d a and d b satisfy the condition ‖d b (x)‖ ≤ ‖d a (x)‖ for all x ∈ A, there exist completely bounded (a)ʹ-bimodule map $\varphi : {\cal B}({\cal H}) \rightarrow {\cal B}({\cal H})$ such that d b |A = φ d a |A=d a φ|A. (In particular d b (A) ⊆ d a (A).) Moreover, if A is a factor, then φ can be taken to be normal and these equalities hold on ${\cal B}({\cal H})$ instead of just on A. This result is not true for general (even primitive) C*-algebras ${\cal A}$ .

Let $(X,d,\unicode[STIX]{x1D707})$ be a metric measure space endowed with a distance $d$ and a nonnegative, Borel, doubling measure $\unicode[STIX]{x1D707}$ . Let $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$ . Assume that the (heat) kernel associated to the semigroup $e^{-tL}$ satisfies a Gaussian upper bound. In this paper, we prove that for any $p\in (0,\infty )$ and $w\in A_{\infty }$ , the weighted Hardy space $H_{L,S,w}^{p}(X)$ associated with $L$ in terms of the Lusin (area) function and the weighted Hardy space $H_{L,G,w}^{p}(X)$ associated with $L$ in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duong et al. [‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’, J. Geom. Anal.26 (2016), 1617–1646], who proved that $H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$ for $p\in (0,1]$ and $w\in A_{\infty }$ by imposing an extra assumption of a Moser-type boundedness condition on $L$ . Our result is new even in the unweighted setting, that is, when $w\equiv 1$ .

We analyze domination properties and factorization of operators in Banach spaces through subspaces of $L^{1}$ -spaces. Using vector measure integration and extending classical arguments based on scalar integral bounds, we provide characterizations of operators factoring through subspaces of $L^{1}$ -spaces of finite measures. Some special cases involving positivity and compactness of the operators are considered.

We use a generalised Nevanlinna counting function to compute the Hilbert–Schmidt norm of a composition operator on the Bergman space $L_{a}^{2}(\mathbb{D})$ and weighted Bergman spaces $L_{a}^{1}(\text{d}A_{\unicode[STIX]{x1D6FC}})$ when $\unicode[STIX]{x1D6FC}$ is a nonnegative integer.

We consider the boundary-value problem

where H: [0,+∞) → ℝ and f : [0, 1] × ℝ → ℝ are continuous and λ > 0 is a parameter. We show that if H satisfies a boundedness condition on a specified compact set, then this, together with an assumption that H is either affine or superlinear at +∞, implies existence of at least one positive solution to the problem, even in the case where we impose no growth conditions on f. Finally, since it can hold that f(t, y) < 0 for all (t, y) ∈ [0, 1]×ℝ, the semipositone problem is included as a special case of the existence result.

Wu [‘An order characterization of commutativity for $C^{\ast }$ -algebras’, Proc. Amer. Math. Soc.129 (2001), 983–987] proved that if the exponential function on the set of all positive elements of a $C^{\ast }$ -algebra is monotone in the usual partial order, then the algebra in question is necessarily commutative. In this note, we present a local version of that result and obtain a characterisation of central elements in $C^{\ast }$ -algebras in terms of the order.

This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$ -space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.

We analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius r ess(P |𝓁²(𝜋)) of P |𝓁²(𝜋) derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG2) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG2) is shown to hold under the condition α0≔∑m=−N N lim supi→+∞(P(i,i+m)P *(i+m,i)1∕2<1. Moreover, r ess(P |𝓁²(𝜋)≤α0. Effective bounds on the convergence rate can be provided from a truncation procedure.

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.

In this paper we discuss the continuity of the Hausdorff dimension of the invariant set of generalised graph-directed systems given by contractive infinitesimal similitudes on bounded complete metric spaces. We use the theory of positive linear operators to show that the Hausdorff dimension varies continuously with the functions defining the generalised graph-directed system under suitable assumptions.

We study two-weight norm inequalities for a vector-valued operator from a weighted $L^{p}(\unicode[STIX]{x1D70E})$ -space to mixed norm $L_{l^{s}}^{q}(\unicode[STIX]{x1D707})$ spaces, $1<p<\infty$ , $0<q<p$ . We apply these results to the boundedness of Wolff’s potentials.

This note furnishes an example showing that the main result (Theorem 4) in Toumi [‘When lattice homomorphisms of Archimedean vector lattices are Riesz homomorphisms’, J. Aust. Math. Soc. 87 (2009), 263–273] is false.

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.

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.