Schauder type theorems for differentiable and holomorphic mappings |
| |
Authors: | Manuel González Joaquín M Gutiérrez |
| |
Institution: | (1) Departamento de Matemáticas Facultad de Ciencias, Universidad de Cantabria, E-39071 Santander, Spain;(2) Departamento de Matemática Aplicada ETS de Ingenieros Industriales, Universidad Politécnica de Madrid, C.-José Gutiérrez Abascal 2, E-28006 Madrid, Spain |
| |
Abstract: | Denoting byC
wu
p
(E) the algebra of allC
p-real-valued functions on the real Banach spaceE such that the functions and the derivatives are weakly uniformly continuous on bounded subsets, it is known that the algebra homomorphismsA:C
wu
q
(F) C
wu
p
(E) are induced by differentiable mappingsg:E F
**. We prove that, for 1 p+1 q![le](/content/x60877p2x6gux174/xxlarge8804.gif) , the following are equivalent: (a)A is compact; (b)g and its derivatives are compact; (c)g C
wu
p
(E,F
**) (the authors had proved that, forp=q< ,A is weakly] compact if and only ifg is a constant mapping, and it is known that ifq<p, thenA is always induced by a constant mapping and is therefore compact). Also, for an entire function of bounded typeg H
b
(U,F), where
is a balanced open subset, andE,F are complex Banach spaces, lettingA:H
b
(F) H
b
(U) be the homomorphism given byA(f)=f g for allf H
b
(F), we prove thatA is compact if and only ifg is compact.Supported in part by DGICYT Grant PB 94-1052 (Spain).Supported in part by DGICYT Grant PB 93-0452 (Spain). |
| |
Keywords: | 46E25 46G05 46G20 47B38 |
本文献已被 SpringerLink 等数据库收录! |
|