Factorization of Montel operators

• Autorzy: S. Dierolf , P. Domański
• Języki publikacji: EN

Abstrakty

EN

Consider the following conditions. (a) Every regular LB-space is complete; (b) if an operator T between complete LB-spaces maps bounded sets into relatively compact sets, then T factorizes through a Montel LB-space; (c) for every complete LB-space E the space C (βℕ, E) is bornological. We show that (a) ⇒ (b) ⇒ (c). Moreover, we show that if E is Montel, then (c) holds. An example of an LB-space E with a strictly increasing transfinite sequence of its Mackey derivatives is given.

Słowa kluczowe

EN

Fréchet space; Fréchet-Montel space; complete LB-space; Montel LB-space; regular LB-space; Mackey completion of an LB-space; bornologicity of C(K,E)

Kategorie tematyczne

• 46E40: Spaces of vector- and operator-valued functions
• 46A17: Bornologies and related structures; Mackey convergence, etc.
• 46A04: Locally convex Fr\'echet spaces and (DF)-spaces
• 46A45: Sequence spaces (including K\"othe sequence spaces)
• 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.)
• 46A11: Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
• 47B07: Operators defined by compactness properties
• 46A50: Compactness in topological linear spaces; angelic spaces, etc.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1993
• Tom: 107
• Numer: 1
• Strony: 15-32

Daty

wydano

1993

otrzymano

1992-08-21

Twórcy

autor

S. Dierolf

• FB IV Mathematik, Universität Trier, W-5500 Trier, Germany

autor

P. Domański

• Institute of Mathematics, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland

Bibliografia

• [1] K. D. Bierstedt, An introduction to locally convex inductive limits, in: Functional Analysis and its Applications, World Sci., Singapore, 1988, 35-133.
• [2] K. D. Bierstedt and J. Bonet, Stefan Heinrich's density condition for Fréchet spaces and the characterization of distinguished Köthe echelon spaces, Math. Nachr. 135 (1988), 149-180.
• [3] K. D. Bierstedt and J. Bonet, Dual density conditions in (DF)-spaces, Results Math. 14 (1988), 242-274.
• [4] J. Bonet, Sobre ciertos espacios de funciones continuas con valores vectoriales, Rev. Real Acad. Cienc. Madrid 75 (3) (1981), 757-767.
• [5] J. Bonet and S. Dierolf, Fréchet spaces of Moscatelli type, Rev. Mat. Univ. Complutense Madrid 2 (1990), 77-92.
• [6] J. Bonet and S. Dierolf, On (LB)-spaces of Moscatelli type, Doğa Mat. 13 (1990), 9-33.
• [7] J. Bonet and J. Schmets, Examples of bornological C(X,E) spaces, ibid. 10 (1986), 83-90.
• [8] J. Bonet and J. Schmets, Bornological spaces of type $C(X) ⊗_ε E$ and C(X,E), Funct. Approx. Comment. Math. 17 (1987), 37-44.
• [9] B. Cascales and J. Orihuela, Metrizability of precompact subsets in (LF)-spaces, Proc. Roy. Soc. Edinburgh 103A (1986), 293-299.
• [10] B. Cascales and J. Orihuela, On compactness in locally convex spaces, Math. Z. 195 (1987), 365-381.
• [11] W. Davis, T. Figiel, W. B. Johnson and A. Pełczyński, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311-327.
• [12] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1984.
• [13] S. Heinrich, Closed operator ideals and interpolation, J. Funct. Anal. 35 (1980), 397-411.
• [14] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.
• [15] H. Junek, Locally Convex Spaces and Operator Ideals, Teubner, Leipzig, 1983.
• [16] G. Köthe, Topological Vector Spaces, Springer, Berlin, 1969.
• [17] V. B. Moscatelli, Fréchet spaces without continuous norm and without a basis, Bull. London Math. Soc. 12 (1980), 63-66.
• [18] J. Mujica, A completeness criterion for inductive limits of Banach spaces, in: Functional Analysis, Holomorphy and Approximation Theory II, J. A. Barroso (ed.), North-Holland, Amsterdam, 1984, 319-329.
• [19] H. Pfister, Bemerkungen zum Satz über die Separabilität der Fréchet-Montel Räume, Arch. Math. (Basel) 27 (1976), 86-92.
• [20] J. Schmets, Spaces of Vector-Valued Continuous Functions, Lecture Notes in Math. 1003, Springer, Berlin, 1983.
• [21] M. Valdivia, Semi-Suslin and dual metric spaces, in: Functional Analysis, Holomorphy and Approximation Theory, J. A. Barroso (ed.), North-Holland, Amsterdam, 1982, 445-459.
• [22] S. Dierolf and P. Domański, Compact subsets of coechelon spaces, to appear.
• [23] J. Bonet, P. Domański and J. Mujica, Complete spaces of vector-valued holomorphic germs, to appear.