Compact factorization of differentiable mappings

Abstract:

Results on factorization (through linear operators) of polynomials and holomorphic mappings between Banach spaces have been obtained in recent years by several authors. In the present paper, we obtain a factorization result for differentiable mappings through compact operators. Namely, we prove that a mapping $f:X\to Y$ between real Banach spaces is differentiable and its derivative $f’$ is a compact mapping with values in the space ${\mathcal K}(X,Y)$ of compact operators from $X$ into $Y$ if and only if $f$ may be written in the form $f=g\circ S$, where the intermediate space is normed, $S$ is a precompact operator, and $g$ is a Gâteaux differentiable mapping with some additional properties. We also show that if $f’$ is uniformly continuous on bounded sets and takes values in ${\mathcal K}(X,Y)$, then $f’$ is compact if and only if $f$ is weakly uniformly continuous on bounded sets.