Coincidence of topologies on tensor products of Köthe echelon spaces

Bonet, J. ; Defant, A. ; Peris, A. ; Ramanujan, M.

Studia Mathematica, Tome 108 (1994), p. 263-281 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1994-01-01

Zbl 0805.46007

EUDML-ID : urn:eudml:doc:216132

Bibliographie

[00000] [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. | Zbl 0553.47018

[00001] [2] K. D. Bierstedt und R. Meise, Induktive Limiten gewichteter Räume stetiger und holomorpher Funktionen, J. Reine Angew. Math. 282 (1976), 186-220. | Zbl 0318.46034

[00002] [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. | Zbl 0599.46026

[00003] [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. | Zbl 0504.46007

[00004] [5] B. Carl, B. Maurey und J. Puhl, Grenzordnungen von absolut-(r, p)-summierenden Operatoren, Math. Nachr. 82 (1978), 205-218. | Zbl 0407.47010

[00005] [6] A. Defant, Barrelledness and nuclearity, Arch. Math. (Basel) 44 (1985), 168-170. | Zbl 0536.46001

[00006] [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. | Zbl 0677.46057

[00007] [8] A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, 1993. | Zbl 0774.46018

[00008] [9] A. Defant and V. Mascioni, Limit orders of ε-π continuous operators, Math. Nachr. 145 (1990), 337-344. | Zbl 0726.47011

[00009] [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. | Zbl 0706.46028

[00010] [11] S. Dineen and R. M. Timoney, Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), 227-234. | Zbl 0713.46005

[00011] [12] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).

[00012] [13] H. Jarchow, Locally Convex Spaces, Teubner, 1981. | Zbl 0466.46001

[00013] [14] H. Jarchow and K. John, On the equality of injective and projective tensor products, Czechoslovak Math. J. 38 (1988), 464-472. | Zbl 0674.46044

[00014] [15] H. Jarchow and K. John, Bilinear forms and nuclearity, preprint, 1992. | Zbl 0854.46002

[00015] [16] K. John, Counterexample to a conjecture of Grothendieck, Math. Ann. 265 (1983), 169-179. | Zbl 0506.46053

[00016] [17] K. John, On the compact non-nuclear operator problem, ibid. 287 (1990), 509-514. | Zbl 0687.47012

[00017] [18] G. Köthe, Topological Vector Spaces I, II, Springer, 1969, 1979. | Zbl 0179.17001

[00018] [19] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Math. Stud. 131, North-Holland, 1987. | Zbl 0614.46001

[00019] [20] A. Pietsch, Nuclear Locally Convex Spaces, Springer, 1972.

[00020] [21] A. Pietsch, Operator Ideals, Deutscher Verlag Wiss., 1978.

[00021] [22] A. Pietsch, Eigenvalues and s-Numbers, Cambridge University Press, 1987.

[00022] [23] G. Pisier, Counterexamples to a conjecture of Grothendieck, Acta Math. 151 (1983), 180-208.

[00023] [24] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.