The collection of papers contained in this book is intended to be a warm homage to Antonio Plans, on the occasion of his 65 th. birthday, which at present in Spain marks the point of retirement from academic undergraduate teaching.

We have chosen the topic of Banach Spaces since it has been the center of his Mathematical interests for the last few years. The papers included have been written by friends, or students of Professor Plans, or by a few of the mathematicians who have been in touch with him for scientific reasons.

We feel sorry to restrict ourselves to Just one topic, since there are many colleagues who, for sure, would have liked to write something for this event. As a matter of fact Professor Plans is widely known by his results on Knot Theory, a field he worked on at the beginning of his research career, and an active field of research today.

We now briefly sketch Professor Plans' personality and Mathematical work. The latter will necessarily be incomplete since at this very moment he is vigorously active and producing new results. For example, he is at present giving advice to four students for their doctoral dissertations.

Born in Madrid, he was the sixth of seven children. He was brought up in the cheerful atmosphere that characterizes such large families.

Soon after there came harder times; the social environment of his country was full of tension, the religious persecution started, and the growing insecurity and anxiety ended up in the civil war (1936–39).

Let us consider the following long standing question:

(*) Suppose that X, Y are infinite dimensional Banach spaces. Does then exist a compact non-nuclear operator T:X → Y?

The problem has positive answer under quite general assumptions. Positive answer of the finite–dimensional version of (*) was given in. Close to (*) is another problem posed by Grothendieck and solved by the example of Pisier–s space P: P is an infinite dimensional Banach space such that

1. 1) p ⊗ εp = p ⊗ πp

and

1. 2) P and P' are of cotype 2.

G. Pisier already observed that if P were reflexive or merely if P* had R.N. property then every bounded operator T:P → P' would be nuclear and thus (*) would have negative answer. Indeed, 1) implies that every T:P → P' is integral. Unfortunately no reflexive Pisier's space P is known to exist. We may also translate the problem into the category of locally convex spaces:

Suppose that X, Y are non-nuclear locally convex spaces. Does then exist a compact non-nuclear operator T:X→Y? This generalized problem has a negative answer,: There is a non-nuclear F space X and a non-nuclear DF space Z = Y' such that every continuous operator T: X→Z is (strongly) nuclear. X and Z are obtained via Pisier's observation mentioned above from an example with properties similar to 1):

There are non-nuclear F spaces X, Y such that X ⊗ ε Y = X ⊗ ε Y. Moreover X, Y are hilbertizable.

INTRODUCTION

This largely expository article describes some recent results concerning the mixed summing norm Πp, 1 as applied to operators on ℓ∞-spaces (especially the finite-dimensional spaces). The whole subject of summing operators and their norms can be said to have started with a result on Π2, 1 – the theorem of Orlicz (1933) that the identity operators in ℓ2 and ℓ1 are (2, 1)-summing. However, since that time the study of mixed summing norms has been somewhat neglected in favour of the elegant and powerful theory of the “unmixed” summing norms Πlp. A breakthrough hasnow been provided by the theorem of Pisier, which does for mixed summing norms what the fundamental theorem of Pietsch does for unmixed ones (see e.g.). One version of Pisier's theorem states that the operator can be factorised through a Lorentz function space Lp, 1 (λ). Such spaces were introduced in, and are discussed in and. However, it is not easy to find a really simple outline of the definition and basic properties of these spaces adapted to the (obviously simpler) finite-dimensional case, so the present paper includes a brief attempt to provide one.

It is of particular interest to compare the value of Π2, 1 and Π2 for operators on ℓ∞ or. It is a well-known fact, underlying the famous Grothendieck inequality, that there is a constant K, independent of n, such that for all operators T from to ℓ1 or ℓ2, we have Π2 (T)≤ K∥T∥. This equates to saying that Π2 (T) s K'Π2, 1 (T) for such T.

Abstract. We prove a representation theorem in terms of finite rank operators for operators belonging to. Some information on the tensor product of operators belonging to these ideals is also obtained.

INTRODUCTION.

The n-th approximation number an (T) of a bounded linear operator T∈ (E, F) acting between the Banach spaces E and F, is defined as

For the ideal is formed by all operators T betwen Banach space, with a finite quasi-norm

Weyl ideals are defined in a similar way, by substituting approximation numbers for Weyl numbers (xn (T)). Ideals have been studied by the authors in. Since xn (T) ≤ an (T) for al 1 n ∈ N (see), as a direct consequence of, Thm. 3, we have

Theorem 1. Let 0 < n ∞. Then there is a constant M = M such that for any complex Banach space E and any operator T ∈∞, ∞ (E, E) the following holds

Here (λn (T)) denotes the sequence of all eigenvalues of the compact operator T counted accoding to their algebraic multiplicities and ordered such that |λ1 (T)| ≥ |λ2(T)| ≥ … ≥ 0.

In this note we continue the study of -ideals. We derive a representation theorem for the elements of in terms of finite rank operators. This result is on the same lines as we established in for the case of the ideals (0 < q < ∞). We also obtain some information on the tensor pordect of operator belonging to the scale of the ideals.