During the last decades it turned out to be fruitful to apply certain Banach algebra techniques in the theory of approximation of operators by matrix sequences. Here we discuss the case of operator sequences (acting on infinite dimensional Banach spaces and which do not necessarily converge strongly) and we derive analogous results concerning the stability and Fredholm properties of such sequen...

We create a new family of Banach spaces, the James–Schreier spaces, by amalgamating two important classical Banach spaces: James’ quasi-reflexive Banach space on the one hand and Schreier’s Banach space giving a counterexample to the Banach–Saks property on the other. We then investigate the properties of these James–Schreier spaces, paying particular attention to how key properties of their ‘a...

We prove complex interpolation theorems for analytic families of multilinear operators defined on quasi-Banach spaces, with explicit constants on the intermediate spaces. We obtain analogous results for analytic families of operators defined on spaces generated by the Calderón method applied to couples of quasi-Banach lattices with nontrivial lattice convexity. As an application we derive a mul...

The embedding functions of an intermediate space A into a Banach couple (A0, A1) are defined as its embedding constants into the couples ( 1 α A0, 1 β A1), ∀α, β > 0. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak-type interpolation to any Banach couple to...

In this paper we show the weak Banach-Saks property of the Banach vector space (L p µ) m generated by m L p µ-spaces for 1 ≤ p < +∞, where m is any given natural number. When m = 1, this is the famous Banach-Saks-Szlenk theorem. By use of this property, we also present inequalities for integrals of functions that are the composition of nonnegative continuous convex functions on a convex set of ...

In this paper we present inequalities for integrals of functions that are the composition of nonnegative convex functions on an open convex set of a vector space R and vectorvalued functions in a weakly compact subset of a Banach vector space generated by m Lμ-spaces for 1 ≤ p < +∞ and inequalities when these vector-valued functions are in a weakly* compact subset of a Banach vector space gener...

In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space R and vector-valued functions in a weakly compact subset of a Banach vector space generated by m Lμ-spaces for 1 ≤ p < +∞. Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach...