On Lorentz and Orlicz–Lorentz subspaces of bounded families and approximation type operators

Abstract

For an arbitrary Banach space X and an arbitrary index set I,we denote by $l_{\infty ,I}(X)$, the Banach space of all bounded families $\{x_i\}_{i \in I}$ in X, equipped with the sup norm; and by $l_{p,q,M,I}(X)$ and $l_{p,q,r,I}(X)$, subspaces of $l_{\infty ,I}(X)$, where p,q,r are positive reals and M is an Orlicz function. In case, X is a real Banach space which is also a $\sigma$-Dedekind complete Banach lattice, it is shown that $l_{p,q,M,I}(X)$ is $\sigma$-Dedekind complete Banach lattice containing a subspace order isometric to $l_\infty$ when $1/p-1/q <-1$. In this paper, we study their structural properties and characterize their elements. For $~X=\mathbb K$, the symbols $~l_{p,q,M}(I)$ and $~l_{p,q,r}(I)$ are being used for the subspaces $l_{p,q,M,I}(X)$ and $l_{p,q,r,I}(X)$ respectively. Besides investigating relationships amongst the spaces $l_{p,q,r}(I)$ for different positive indices p,q and r, we consider their product. Using generalized approximation numbers of bounded linear operators and these spaces, we consider operators of generalized approximation type $l_{p,q,r}$ and represent them as an infinite series of finite rank operators. We also establish the quasi-Banach ideal structure of the class of all such operators. Finally we prove results preserving various set theoretic inclusion relations amongst these operator ideals. These results generalize some of the earlier results proved for Lorentz spaces by A. Pietsch.