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Composition operators between algebras of differentiable functions

Llavona, José G. and Gutiérrez, Joaquín M. (1993) Composition operators between algebras of differentiable functions. Transactions of the American Mathematical Society , 338 (2). pp. 769-782. ISSN 0002-9947

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Abstract

Let E, F be real Banach spaces, U subset-or-equal-to E and V subset-equal-to F non-void open subsets and C(k)(U) the algebra of real-valued k-times continuously Frechet differentiable functions on U, endowed with the compact open topology of order k. It is proved that, for m greater-than-or-equal-to p, the nonzero continuous algebra homomorphisms A: C(m)(U) --> C(p)(V) are exactly those induced by the mappings g: V --> U satisfying phi . g is-an-element-of C(p)(V) for each phi is-an-element-of E*, in the sense that A(f) = fog for every f is-an-element-of C(m)(U). Other homomorphisms are described too. It is proved that a mapping g: V --> E** belongs to C(k)(V, (E**, w*)) if and only if phi . g is-an-element-of C(k)(V) for each phi is-an-element-of E*. It is also shown that if a mapping g: V --> E verifies phi . g is-an-element-of C(k)(V) for each phi is-an-element-of E*, then g is-an-element-of C(k-1)(V, E).


Item Type:Article
Uncontrolled Keywords:Differentiable mappings between banach spaces; Algebras of differentiable functions; Homomorphisms; Composition operators
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:16383
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