Mathematische Zeitschrift

Math Z 129, 1-19(1972) © by Springer-Verldg 1972

Danach Lattices of Compact Maps

John Chancy

Introduction

This is a study of some of the properties of a subspace of the space of compact linear maps where the domain or range is a Banach lattice. It is known that the space of compact maps from a Banach lattice into a C(X) is a lattice, and the nuclear maps from a Banach lattice into the space Ü for a finite measure space is a lattice. These two examples of subspaces of the space of compact linear maps which are lattices will be shown to be special cases of a general subspace of compact maps which is always a Banach lattice when the domain and range are Banach lattices. This space of compact maps, denoted M^{E\F% is shown to represent a completed normed tensor product E®mF defined when E is a Banach space and F is a Banach lattice.

Throughout this paper, E denotes a Banach space and F a Banach lattice, unless otherwise stated. Define the |/z|-norm on E® F by

l " llH = inf

II^JLv .

: u=f^x^®y^ in E®F, >',^oL

In the first section, we introduce the M-tensor product norm on E®F and show that this norm is actually an alternate description of the |/i|-norm. The completion Е®мР ^^ the M-tensor product E®^F is characterized as a space M^{E\F) of compact maps from E' into F in (1.3). In (1.7) it is shown that if both E and F are Banach lattices then M^{E\ F) is also a Banach lattice of compact maps.

In the second section, we show that M^{E'\F) is canonically morphic to the space M(£, F) of all maps T:E—>F that can be factored as:

E —^~>F

С

where T^ is a compact map into an (^M)-space С and T2 is a positive map from С into F. We also show that M^(E\F') is canonically morphic to the space Л (F, E) of all maps T\ F -^ E that can be factored as:

1 Math Z,Bd 129