Ultraweakly compact operators and dual spaces

Alvarez, Teresa

Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 3, pp. 697-711.

Abstract (VO)

In this paper, the class of all bounded ultraweakly compact operators in Banach spaces is introduced and characterised in terms of their first and second conjugates. We analize the relationship between an ultraweakly compact operator and its conjugate. Examples of operators belonging to this class are exhibited. We also investigate the connection between ultraweak compactness of $T\in L(X, Y)$ and minimal subspaces of $Y'$ and we present a result of factorisation for ultraweakly compact operators.

Riassunto

In questo articolo si introduce e si caratterizza la classe di tutti gli operatori ultradebolmente compatti, definiti negli spazi di Banach per mezzo dei loro operatori coniugati. Si analizza la relazione esistente fra un operatore ultradebolmente compatti e il suo coniugato. Si presentano esempi di operatori appartenenti a questa classe. Inoltre, si studia la connessione fra la compattezza ultradebole di $T\in L(X, Y)$ e i sottospazi minimali di $Y'$ e si presenta un risultato relativo alla fattorizzazione degli operatori ultradebolmente compatti.

Bibliographie

[1] T. Álvarez-V. M. Onieva, On operators factorizable through quasi-reflexive Banach spaces, Arch. Math. Vol., 48 (1987), 85-87. | MR | Zbl

[2] J. M. F. Castillo-M. González, Three-space Problems in Banach Space Theory (Springer Lecture Notes in Math. 1667, 1997). | MR | Zbl

[3] J. R. Clark, Coreflexive and somewhat reflexive Banach spaces, Proc. Amer. Math. Soc., 36 (1972), 421-427. | MR | Zbl

[4] R. W. Cross, Multivalued linear Operators (Marcel Dekker, New York, 1998.) | MR | Zbl

[5] J. Davis-T. Figiel-W. Johnson-A. Pelczynski, Factoring weakly compact operators, J. Funct. Anal., 17 (1976), 311-327. | MR | Zbl

[6] J. Dixmier, Sur un théorème de Banach, Duke Math. J. 15 (1948), 1057-1071. | fulltext mini-dml | MR | Zbl

[7] N. Dunford-J. T. Schwartz, Linear Operators Part I (Interscience, New York, 1958). | Zbl

[8] D. Van Dulst, Ultra weak topologies on Banach spaces, Proc. of the seminar on random series, convex sets and geometry of Banach spaces, Various Publ. Series, 24 (1975), 57-66. | MR | Zbl

[9] D. Vandulst, Reflexive and superreflexiveBanach spaces (Mathematisch Centrum, Amsterdam, 1978). | MR | Zbl

[10] G. Godofrey, Espaces de Banach: Existence et unicité de certains préduax, Ann. Inst. Fourier, Grenoble, no. 3 (1978). | MR | Zbl

[11] S. Goldberg, Unbounded Linear Operators (McGraw-Hill, New York, 1966). | MR | Zbl

[12] M. González-V. M. Onieva, Semi-Fredholm operators and semigroups associated with some classical operator ideals, Proc. Royal Irish Acad., Ser. A, 88A (1988), 35-38. | MR | Zbl

[13] R. C. James, Some self-dual properties of normed linear spaces, Ann. of Math. Studies, 69 (1972), 159-175. | MR | Zbl

[14] N. I. Kalton-A. Pelczynski, Kernels of surjections from $L_1$-spaces with an applications to Sidon sets, Math. Ann. 309, no. 1 (1997), 135-158. | MR | Zbl

[15] J. Lindenstrauss-L. Tzafriri, Classical Banach spaces I, sequence spaces (Springer-Verlag, New York, 1997). | MR | Zbl

[16] R. D. Neidinger, Properties of Tauberian Operators on Banach Spaces (Doctoral dissertation, University of Texas at Austin, 1984).

[17] M. I. Ostrovski, Total subspaces in dual Banach spaces which are not norming over any infinite dimensional subspace, Studia Math., 105 (1993), 37-49. | fulltext mini-dml | MR | Zbl

[18] H. P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from $L^p(\mu)$ to $L^r(\mu)$, J. Funct. Anal., 4 (1969), 176-214. | MR | Zbl

[19] K. W. Yang, The generalized Fredholm operators, Trans. Amer. Math. Soc., 216 (1976), 313-326. | MR | Zbl