Showing papers in "Mathematische Annalen in 1974"

Spaces of Compact Operators

Abstract: In this paper we study the structure of the Banach space K(E, F) of all compact linear operators between two Banach spaces E and F. We study three distinct problems: weak compactness in K(E, F), subspaces isomorphic to l~ and complementation of K(E, F) in L(E, F), the space of bounded linear operators. In § 2 we derive a simple characterization of the weakly compact subsets of K(E, F) using a criterion of Grothendieck. This enables us to study reflexivity and weak sequential convergence. In § 3 a rather different problem is investigated from the same angle. Recent results of Tong [20] indicate that we should consider when K(E, F) may have a subspace isomorphic to l~. Although L(E, F) often has this property (e.g. take E = F =/2) it turns out that K(E, F) can only contain a copy of l~o if it inherits one from either E* or F. In § 4 these results are applied to improve the results obtained by Tong and also to approach the problem investigated by Tong and Wilken [21] of whether K(E, F) can be non-trivially complemented in L(E,F) (see also Thorp [19] and Arterburn and Whitley [2]). It should be pointed out that the general trend of this paper is to indicate that K(E, F) accurately reflects the structure of E and F, in the sense that it has few properties which are not directly inherited from E and F. It is also worth stressing that in general the theorems of the paper do not depend on the approximation property, which is now known to fail in some Banach spaces; the paper is constructed independently of the theory of tensor products. These results were presented at the Gregynog Colloquium in May

Holomorphic mappings from the ball and polydisc

Abstract: Introduction. The holomorphic self-homeomorphisms ("automorphisms") of the open unit ball B, in ~L TM have long been known [1] - they are given by certain rational functions which are holomorphic on a neighborhood of Bn and induce a homeomorphism of the boundary, bB,, of the ball. Our first result can be viewed as a local characterization of these automorphisms: For n > 1, a nonconstant holomorphic mapping into ~" which is defined in a neighborhood of a point of bB, and which maps bB, into itself is necessarily an automorphism, or, more precisely, extends to be an automorphism. We apply this to obtain some information on the as yet unsettled question as to whether every proper holomorphic self-mapping of Bn is an automorphism. In particular, we recover (Cor. 1.1) a result of Pelles ([3, 5]). In the second part, we consider holomorphic mappings from polydiscs. According to a classical theorem of Poincar6, there exists no biholomorphism from the polydisc U z in C 2 with the ball B2. We obtain some integral formulas which yield a quantitative explanation of this phenomenon, Finally I wish to acknowledge that the above characterization of automorphisms may have been known to the late Professor L6wner, at least for two complex variables. I want to thank Professors L. Bers and C. Titus for this information on their oral communication with L6wner.

An example of unirational surfaces in characteristic p

Abstract: § 1. Let p be an odd prime number, and n a natural number such that n @ 0 (modp). Let X, denote the Fermat surface of degree n x~ + x~ + x~ + x,~ = 0 0 ) defined over a field k of characteristic p. Then Xn is a non-singular irreducible surface, which is rational only for n N 3, K 3 for n = 4, and of general type for n ~ 5. In this note, a surface X will be called unirational if its function field k(X) has a finite extension, which is a purely transcendental extension of dimension two over the constant field k. For the sake of simplicity, we assume that the constant field k is algebraically closed. Proposition 1. Suppose that there is a power q = pV o f p such that q----1 (modn). Then X , is unirational. Proof. (i) First let us consider the case n = q + 1. By an obvious change of coordinates, we can rewrite (t) as

On the construction of Galois extensions of function fields and number fields

Abstract: This paper consists of two parts and an appendix. In Part 1, we investigate Galois converings and consider the problem of reducing their fields of definition. We restrict ourselves to PSL 2 (Z/pZ)-coverings in Part 2. The results of Part t are applied to obtain Galois extensions with P S L 2 (Z/p Z) as Galois group. We show that if p is an odd prime such that 2, 3 or 7 is a quadratic non-residue modulo p, then PSL2(Z/pZ ) occurs as Galois groups over the rationals. To prove this, Shimura's theory of canonical system of models is used to reduce the fields of definition of certain Galois coverings. Previously, our result is only known for p = 3, 5 and 7. In the appendix, we discuss the classification of Galois coverings, which is necessary in verifying Weil's criterion in certain cases. We also indicate how to use the theory developed in Part t to show Hilbert's result that alternating groups can be realized as Galois groups over Q. This paper is based on the author's doctoral dissertation. He would like to thank Professor Goro Shimura for several valuable suggestion during the course of the research. Notation. For an associative ring S with an identity element, we denote by S x the group of all invertible elements of S.

Non-existence of Continuous Convex Functions on Certain Riemannian Manifolds

Abstract: The present note is written in response to a question of Wu. In [51 Wu proved that every complete non-compact convex hypersurface in euclidean space has infinite volume. He then asked whether this is true for all complete non-compact non-negatively curved manifold or not. The structure theorem of Cheeger and Gromoll, I-2], says that such a manifold has a continuous convex exhaustion. On the other hand, a theorem of Bishop and O'Neill, [1], says that there is no non-trivial smooth convex function on a complete manifold with finite volume. However, as was remarked by Greene and Wu [3], the passage from continuity to smoothness is highly non-trivial. The main theorem here is that there is no non-trivial continuous convex function on a complete manifold with finite volume. I would like to thank Professor Wu for his constant interest in this problem. Recently he informed us that he and Greene also proved that a complete non-compact manifold with non-negative curvature has infinite volume.