We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces $H^p$ 0 < p < ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,Lh-order bounded (we write (X,Lh)-ob) with $X=H^p$ or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into $L_h$. We give a complete characterization...

For every symmetric operator acting in a Hilbert space, we introduce the families of p-analytic and p-quasi-analytic vectors (p>0), indexed by positive numbers. We prove various properties of these families. We make use of these families to show that certain results guaranteeing essential selfadjointness of an operator with sufficiently large sets of quasi-analytic and Stieltjes vectors are optimal.

E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $\left|\right|Tⁿ\left|\right|\sim n^{1/4}$). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $li{m}_{n\to \infty }\left|\right|Tⁿ\left|\right|/n^{1-\gamma }=\infty .IntheclassofpositiveoperatorstheseexamplesarethebestpossibleinthesensethatforeverysuchoperatorTthereexistsa\gamma ₀>0suchthat$lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0$.$A class of numerical sequences αₙ, intimately related to the...

A characterization of the structure of positive maps is presented. This sheds some more light on the old open problem studied both in Quantum Information and Operator Algebras. Our arguments are based on the concept of exposed points, links between tensor products and mapping spaces and convex analysis.

By applying the results of the first part of the paper, we establish some Korovkin-type theorems for continuous positive linear operators in the setting of regular vector lattices of continuous functions. Moreover, we present simple methods to construct Korovkin subspaces for finitely defined operators and for the identity operator and we determine those classes of operators which admit finite-dimensional Korovkin subspaces. Finally, we give a Korovkin-type theorem for continuous positive projections....

In this paper we get some relations between the boundary point spectrum of the generator A of a C0-semigroup and the generator A* of the dual semigroup. This relations combined with the results due to Lyubich-Phong and Arendt-Batty, yield stability results on positive C0-semigroups.

Let $L$, $M$ be Archimedean Riesz spaces and ${ℒ}_b(L,M)$ be the ordered vector space of all order bounded operators from $L$ into $M$. We define a Lamperti Riesz subspace of ${ℒ}_b(L,M)$ to be an ordered vector subspace $ℒ$ of ${ℒ}_b(L,M)$ such that the elements of $ℒ$ preserve disjointness and any pair of operators in $ℒ$ has a supremum in ${ℒ}_b(L,M)$ that belongs to $ℒ$. It turns out that the lattice operations in any Lamperti Riesz subspace $ℒ$ of ${ℒ}_b(L,M)$ are given pointwise, which leads to a generalization of the classic Radon-Nikod’ym theorem for Riesz homomorphisms....

Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X{\widehat{\otimes }}_{\tau }Y$ is their complete tensor product with respect to some tensor product topology $\tau$. A uniformly bounded $X$-valued function need not be integrable in $X{\widehat{\otimes }}_{\tau }Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau$ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index $1\le p<\infty$ and suppose that $X$ and $Y$ are $L^p$-spaces with ${\tau }_p$ the associated $L^p$-tensor product...

We introduce the notion of order weakly sequentially continuous lattice operations of a Banach lattice, use it to generalize a result regarding the characterization of order weakly compact operators, and establish its converse. Also, we derive some interesting consequences.