ISSN:
1436-5081
Keywords:
46E25
;
46G05
;
46G20
;
47B38
;
Algebras of differentiable functions
;
algebras of analytic functions of bounded type
;
holomorphic functions
;
composition operators
;
compact homomorphisms
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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≤∞, 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 $$U \subseteq E$$ 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01326032
Permalink
