Real interpolation with weighted rearrangement invariant Banach function spaces

This chapter concludes the analysis of the tracial setting by showing that real interpolation extends to the context of tracial Lp and Orlicz spaces in a manner which closely mimics the classical theory. After a brief consideration of a noncommutative Marcinkiewicz interpolation theorem, we then go on to enlarge our pool of noncommutative spaces by using the K-method of interpolation to introduce a very general class of noncommutative rearrangement invariant Banach function spaces – the so-called monotone Riesz-Fischer spaces. Our approach to this by now well-known theory is slightly non-standard in that we introduce the theory by exploiting the known theory of modular interpolation spaces. As a result we are able to get to the cut of the chase a lot quicker.

Throughout this paper we denote by L p the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on L p-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on L p-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of L p-spaces. We will discuss results related to the boundedness of positive linear operators on L p-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from L p to L q. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on L p-spaces and discuss as an example the weighted composition operators on L p-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikisin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.

New John–Nirenberg inequalities for martingales

We show that if a rearrangement invariant Banach function space E on the positive semi-axis satisfies a non-trivial lower q-estimate with constant 1 then the corresponding space E(M) of τ-measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra M equipped with a distinguished faithful, normal, semi-finite trace τ, has the uniform Kadec-Klee property for the topology of local convergence in measure. In particular, the Lorentz function spaces Lq, p and the Lorentz-Schatten classes Cg, p have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse, we show that if E has the UKK property with respect to local convergence in measure then E must satisfy some non-trivial lower q-estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower q-estimate.

Stability of iterative procedures with errors for approximating common fixed points of a couple of q-contractive-like mappings in Banach spaces

Iterative algorithm for generalized quasi-variational-like inclusions with fuzzy mappings in Banach spaces

Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces

Characterization of BMO in terms of rearrangement-invariant Banach function spaces

In this paper, we first introduce a class of nonlinear mappings called generalized nonspreading which contains the class of nonspreading mappings in a Banach space and then prove a fixed point theorem, a nonlinear mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for such nonlinear mappings in a Banach space. Using these theorems, we obtain some fixed point theorems, nonlinear mean convergence theorems and weak convergence theorems in a Banach space.

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications

Rearrangement invariance of Rademacher multiplicator spaces

In this paper, we prove strong convergence theorems of modified Halpern’s iteration for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping in a Banach space by using two hybrid methods. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and relatively nonexpansive mappings in Banach spaces.

We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holomorphic functional calculus techniques. Using the concept of \(\mathcal {R}_{s}\)-sectorial operators, which in turn is based on the notion of \(\mathcal {R}_{s}\)-bounded sets of operators introduced by Lutz Weis, we obtain a neat theory including equivalence of various norms and a precise description of real and complex interpolation spaces. Another main result of this article is that an \(\mathcal {R}_{s}\)-sectorial operator always has a bounded H∞-functional calculus in its associated generalized Triebel-Lizorkin spaces.

Modified block iterative algorithm for Quasi- ϕ-asymptotically nonexpansive mappings and equilibrium problem in banach spaces

A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications