Publication: Composition operators between algebras of differentiable functions
Loading...
Full text at PDC
Publication Date
1993-08
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Mathematical Society
Citations
Exportar
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).
Description
UCM subjects
Unesco subjects
Keywords
Citation
R. M. Aron, Compact polynomials and compact differentiable mappings between Banach spaces, Séminaire P. Lelong (Analyse), Lecture Notes in Math., vol. 524, Springer-Verlag, Berlin, 1976, pp. 213-222.
R. M. Aron, J. Gómez, and J. G. Llavona, Homomorphisms between algebras of differentiable functions in infinite dimensions, Michigan Math. J. 35 (1988), 163-178.
R. M. Aron and J. G. Llavona, Composition of weakly uniformly continuous functions, Proc. Roy. Irish Acad. Sect. A 88(1) (1988), 29-33.
R. M. Aron and J. B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195-216.
E. Beckenstein, L. Narici, and C Suffel, Topological algebras, Math. Stud., vol. 24, North- Holland, Amsterdam, 1977.
F. Bombai and J. G. Llavona, La propiedad de aproximación en espacios de funciones diferenciables, Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid 70(4) (1976), 727-741.
N. Bourbaki, Éléments de mathématique. Fase. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats, Actualités Sei. Indust., no. 1333, Hermann, Paris, 1971.
S. B. Chae, Holomorphy and calculus in normed spaces, Monographs Textbooks Pure Appl. Math., vol. 92, Dekker, New York, 1985.
J. F. Colombeau and R. Meise, Cx'-functions on locally convex and on bornological vector spaces, Functional Analysis, Holomorphy and Approximation Theory, Lecture Notes in Math., vol. 843, Springer-VerlagB, erlin, 1981, pp. 195-216.
J. Gómez, Espectro e ideales primarios del álgebra C^b{E) de funciones débilmente diferenciables sobre un espacio de Banach, Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid 75(2)(1981),pp. 514-519.
J. A. Jaramillo, An example on composite differentiable functions in infinite dimensions, Bull. Austral. Math. Soc. 40(1) (1989), 91-95.
-, Multiplicative functionals on algebras of differentiable functions, Arch. Math. 58 ( 1992), 384-387.
J. A. Jaramillo and J. G. Llavona, Homomorphisms between algebras of continuous functions, Canad. J. Math. 41(1) (1989), 132-162.
H.H. Keller, Differential calculus in locally convex spaces, Lecture Notes in Math., vol. 417, Springer-Verlag, Berlin, 1974.
J. G. Llavona, Approximations of differentiable functions, Adv. Math. Suppl. Stud. 4 (1979), 197-221.
-, Approximation of continuously differentiable functions, Math. Stud., vol. 130, North- Holland, Amsterdam, 1986.
J. Mujica, Complex analysis in Banach spaces. Math. Stud., vol. 120, North-Holland, Amsterdam, 1986.
L. Nachbin, Topology on spaces of holomorphic mappings, Ergeb. Math. Grenzgeb., vol. 47, Springer-Verlag, Berlin, 1969.
J. B. Prolla, On polynomial algebras of continuously differentiable functions, Atti Accad. Naz. Lincei Rend. Cl. Sei. Fis. Mat. Natur. (8) 57 (1974), 481-486.
L. Schwartz, Espaces de fonctions différentiables à valeurs vectorielles, J. Analyse Math. 4 (1954), 88-148.
M. Sova, Conditions of differentiability in linear topological spaces, Czechoslovak Math. J. 16(91) (1966), 339-362. (Russian)
S. Yamamuro, Differential calculus in topological linear spaces, Lecture Notes in Math., vol. 374, Springer-Verlag, Berlin, 1974.