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.

We establish necessary and sufficient conditions under which each operator between Banach lattices is weakly compact and we give some consequences.

We establish some properties of the class of order weakly compact operators on Banach lattices. As consequences, we obtain some characterizations of Banach lattices with order continuous norms or whose topological duals have order continuous norms.

We introduce and study the class of unbounded Dunford--Pettis operators. As consequences, we give basic properties and derive interesting results about the duality, domination problem and relationship with other known classes of operators.

We establish several inequalities for the spectral radius of a positive commutator of positive operators in a Banach space ordered by a normal and generating cone. The main purpose of this paper is to show that in order to prove the quasi-nilpotency of the commutator we do not have to impose any compactness condition on the operators under consideration. In this way we give a partial answer to the open problem posed in the paper by J. Bračič, R. Drnovšek, Y. B. Farforovskaya, E. L. Rabkin, J. Zemánek...

An integral Markov operator $P$ appearing in biomathematics is investigated. This operator acts on the space of probabilistic Borel measures. Let $\mu$ and $\nu$ be probabilistic Borel measures. Sufficient conditions for weak and strong convergence of the sequence $(P^n\mu -P^n\nu )$ to $0$ are given.