Coincidence of topologies on tensor products of Köthe echelon spaces

• Autorzy: J. Bonet , A. Defant , A. Peris , M. S. Ramanujan
• Języki publikacji: EN

Abstrakty

EN

We investigate conditions under which the projective and the injective topologies coincide on the tensor product of two Köthe echelon or coechelon spaces. A major tool in the proof is the characterization of the επ-continuity of the tensor product of two diagonal operators from $l_p$ to $l_q$. Several sharp forms of this result are also included.

Słowa kluczowe

EN

Köthe echelon spaces   topological tensor products   injective and projective topologies   tensor products of diagonal operators  

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46M05: Tensor products
• 46A32: Spaces of linear operators; topological tensor products; approximation properties
• 46A45: Sequence spaces (including K\"othe sequence spaces)
• 46B45: Banach sequence spaces
• 46B28: Spaces of operators; tensor products; approximation properties

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994
• Tom: 111
• Numer: 3
• Strony: 263-281

Daty

wydano

1994

otrzymano

1993-11-08

poprawiono

1994-01-31

Twórcy

autor

J. Bonet

• Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, E-46071 Valencia, Spain ,

autor

A. Defant

• FB Mathematik, Universität Oldenburg, D-26121 Oldenburg, Germany ,

autor

A. Peris

• Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, E-46071 Valencia, Spain ,

autor

M. S. Ramanujan

• Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, U.S.A. ,

Bibliografia

• [1] V. Bartík, K. John and J. Korbaš, Tensor products of operators and Horn's inequality, in: Oper. Theory: Adv. Appl. 14, Birkhäuser, Basel, 1984, 39-46.
• [2] K. D. Bierstedt und R. Meise, Induktive Limiten gewichteter Räume stetiger und holomorpher Funktionen, J. Reine Angew. Math. 282 (1976), 186-220.
• [3] K. D. Bierstedt, R. Meise and W. H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), 107-160.
• [4] K. D. Bierstedt, R. Meise and W. H. Summers, Köthe sets and Köthe sequence spaces, in: Functional Analysis, Holomorphy and Approximation Theory, North-Holland Math. Stud. 71, North-Holland, 1982, 27-91.
• [5] B. Carl, B. Maurey und J. Puhl, Grenzordnungen von absolut-(r, p)-summierenden Operatoren, Math. Nachr. 82 (1978), 205-218.
• [6] A. Defant, Barrelledness and nuclearity, Arch. Math. (Basel) 44 (1985), 168-170.
• [7] A. Defant and K. Floret, Topological tensor products and the approximation property of locally convex spaces, Bull Soc. Roy. Sci. Liège 58 (1989), 29-51.
• [8] A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, 1993.
• [9] A. Defant and V. Mascioni, Limit orders of ε-π continuous operators, Math. Nachr. 145 (1990), 337-344.
• [10] A. Defant and L. A. Moraes, On unconditional bases in spaces of holomorphic functions in infinite dimensions, Arch. Math. (Basel) 56 (1991), 163-173.
• [11] S. Dineen and R. M. Timoney, Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), 227-234.
• [12] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
• [13] H. Jarchow, Locally Convex Spaces, Teubner, 1981.
• [14] H. Jarchow and K. John, On the equality of injective and projective tensor products, Czechoslovak Math. J. 38 (1988), 464-472.
• [15] H. Jarchow and K. John, Bilinear forms and nuclearity, preprint, 1992.
• [16] K. John, Counterexample to a conjecture of Grothendieck, Math. Ann. 265 (1983), 169-179.
• [17] K. John, On the compact non-nuclear operator problem, ibid. 287 (1990), 509-514.
• [18] G. Köthe, Topological Vector Spaces I, II, Springer, 1969, 1979.
• [19] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Math. Stud. 131, North-Holland, 1987.
• [20] A. Pietsch, Nuclear Locally Convex Spaces, Springer, 1972.
• [21] A. Pietsch, Operator Ideals, Deutscher Verlag Wiss., 1978.
• [22] A. Pietsch, Eigenvalues and s-Numbers, Cambridge University Press, 1987.
• [23] G. Pisier, Counterexamples to a conjecture of Grothendieck, Acta Math. 151 (1983), 180-208.
• [24] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.