Abstract
Denoting byC wu p (E) the algebra of allCp-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∈Hb(U,F), where\(U \subseteq E\) is a balanced open subset, andE,F are complex Banach spaces, lettingA:Hb(F)→Hb(U) be the homomorphism given byA(f)=f∘g for allf∈Hb(F), we prove thatA is compact if and only ifg is compact.
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