Operator semigroups in Banach space theory

Aiena, Pietro ; González, Manuel ; Martínez-Abejón, Antonio

Bollettino della Unione matematica italiana, Série 8, 4B (2001) no. 1, pp. 157-205.

Résumé

In questo lavoro, motivati dalla teoria di Fredholm in spazi di Banach e dalla cosiddetta teoria degli ideali di operatori nel senso di Pietsch, viene definito un nuovo concetto di semigruppo di operatori. Questa nuova definizione include quella di molte classi di operatori già studiate in letteratura, come la classe degli operatori di semi-Fredholm, quella degli operatori tauberiani ed altre ancora. Inoltre permette un nuovo ed unificante approccio ad una serie di problemi in teoria degli operatori su spazi di Banach. Ad un ideale di operatori $\mathcal{A}$ vengono associati, in modo naturale, due semigruppi di operatori $\mathcal{A}_{+}$ e $\mathcal{A}_{-}$. In particolare, se $W$ è la classe degli operatori debolmente compatti, il semigruppo di operatori associato $W_{+}$ è la classe degli operatori tauberiani. Questo lavoro contiene, oltre che una panoramica sulle proprietá, gli esempi e le applicazioni di tali semigruppi, diversi nuovi risultati. Vengono inoltre posti in evidenza una serie di nuovi problemi aperti che meritano di essere studiati

Bibliographie

[1] P. Aiena, An internal characterization of inessential operators, Proc. Amer. Math. Soc., 102 (1988), 625-626. | MR | Zbl

[2] P. Aiena, On Riesz and inessential operators, Math. Z., 201 (1989), 521-528. | MR | Zbl

[3] P. Aiena-M. González, Essentially incomparable Banach spaces and Fredholm theory, Proc. R. Ir. Acad. A, 93 (1993), 49-59. | MR | Zbl

[4] P. Aiena-M. González, On the perturbation classes of semi-Fredholm and Fredholm operators, Rendiconti Circ. Mat. Palermo Suppl., 40 (1996), 37-46. | MR | Zbl

[5] P. Aiena-M. González, On inessential and improjective operators, Studia Math., 131 (1998), 271-287. | fulltext mini-dml | MR | Zbl

[6] P. Aiena-M. González, Examples of improjective operators, Math. Z., to appear. | MR | Zbl

[7] P. Aiena-M. González-A. Martínez-Abejón, Incomparable Banach spaces and semigroups of operators, Preprint, 1999.

[8] P. Aiena-M. González-A. Martínez-Abejón, On the operators which are invertible modulo an operator ideal, Preprint, 1999. | MR | Zbl

[9] T. Alvarez-M. González, Some examples of tauberian operators, Proc. Amer. Math. Soc., 111 (1991), 1023-1027. | MR | Zbl

[10] T. Alvarez-M. González-V. M. Onieva, Totally incomparable Banach spaces and three-space ideals, Math. Nachr., 131 (1987), 83-88. | MR | Zbl

[11] T. Alvarez-M. González-V. M. Onieva, Characterizing two classes of operator ideals, in Contribuciones Matemáticas. Homenaje Prof. Antonio Plans, Univ. Zaragoza 1990, 7-21.

[12] K. Astala, On measures of noncompactness and ideal variations in Banach spaces, Ann. Acad. Sci. Fennicae Ser. A I Math., Dissertationes, 29 (1980), 42 pp. | MR | Zbl

[13] K. Astala-H.-O. Tylli, Seminorms related to weak compactness and to tauberian operators, Math. Proc. Cambridge Phil. Soc., 107 (1990), 365-375. | MR | Zbl

[14] F. Atkinson, Relatively regular operators, Acta Sci. Math. Szeged, 15 (1953), 38-56. | MR | Zbl

[15] M. Basallote, Representabilidad finita por cocientes y operadores, Doctoral Thesis, Univ. Sevilla, 1998.

[16] B. Beauzamy, Opérateurs uniformément convexifiants, Studia Math., 57 (1976), 1023-1027. | fulltext mini-dml | MR | Zbl

[17] B. Beauzamy, Introduction to Banach spaces and their Geometry, North Holland 68, 1985. | MR | Zbl

[18] S. Bellenot, The $J$-sum of Banach spaces, J. Funct. Anal., 48 (1982), 95-106. | MR | Zbl

[19] F. Bombal-C. Fierro, Compacidad débil en espacios de Orlicz de funciones vectoriales, Rev. Real Acad. Ciencias Madrid, 78 (1984), 157-163. | MR | Zbl

[20] F. Bombal-B. Hernando, A double-dual characterization of Rosenthal and semitauberian operators, Proc. R. Ir. Acad. A, 95 (1995), 69-75. | MR | Zbl

[21] J. Bourgain-H. P. Rosenthal, Applications of the theory of semi-embeddings to Banach space theory, J. Funct. Anal., 52 (1983), 149-188. | MR | Zbl

[22] J. Bonet-M. Ramanujan, Two classes of operators between Fréchet spaces, Functional analysis (Trier 1994), 53-58, de Gruyter, 1996. | MR | Zbl

[23] S. Caradus-W. Pfaffenberger-B. Yood, Calkin algebras and algebras of operators in Banach spaces, M. Dekker Lecture Notes in Pure & Appl. Math. 9, 1974. | Zbl

[24] P. G. Casazza-T. J. Shura, Tsirelson space, Springer Lectures Notes in Math. 1363, 1989. | MR | Zbl

[25] J. M. F. Castillo-M. González, Three-space problems in Banach space theory, Springer Lecture Notes in Math. 1667, 1997. | MR | Zbl

[26] R. W. Cross, Linear transformations of tauberian type in normed spaces, Note di Mat. volume dedicated to Prof. Köthe (1980), 193-203. | MR | Zbl

[27] R. W. Cross, On a theorem of Kalton and Wilansky concerning tauberian operators, J. Math. Anal. Appl., 171 (1992), 156-170. | MR | Zbl

[28] R. W. Cross, A characterisation of almost-reflexive normed spaces, Proc. R. Ir. Acad. A, 92 (1992), 225-228. | MR | Zbl

[29] R. W. Cross, $F_+$-operators are tauberian, Quaestiones Math., 18 (1995), 129-132. | MR | Zbl

[30] R. W. Cross, Multivalued linear operators, M. Dekker Pure and Appl. Math. Series 213, 1998. | MR | Zbl

[31] W. J. Davis-T. Figiel-W. B. Johnson-A. Pełcczyński, Factoring weakly compact operators, J. Funct. Anal., 19 (1974), 311-327. | MR | Zbl

[32] J. Diestel-J. Uhl, Vector measures, Amer. Math. Soc., Math. Surveys 15, 1977. | MR | Zbl

[33] N. Ghoussoub, Some remarks concerning $G_{\delta}$-embeddings and semi-quotient maps, Longhorn Notes, Univ. Texas (1982-83), 109-122. | MR

[34] N. Ghoussoub-B. Maurey, $G_{\delta}$-embeddings in Hilbert space II, J. Funct. Anal., 78 (1988), 271-305. | MR | Zbl

[35] N. Ghoussoub-H.P. Rosenthal, Martingales, $G_{\delta}$-embeddings and quotients of $L_1$, Math. Ann., 264 (1983), 321-332. | MR | Zbl

[36] S. Goldberg, Unbounded linear operators, McGraw-Hill, 1966. | MR | Zbl

[37] M. González, Properties and applications of tauberian operators, Extracta Math., 5 (1990), 91-107. | MR | Zbl

[38] M. González, Dual results of factorization for operators, Ann. Acad. Sci. Fennicae, 18 (1993), 3-11. | MR | Zbl

[39] M. González, On essentially incomparable Banach spaces, Math. Z., 215 (1994), 621-629. | MR | Zbl

[40] M. González-A. Martínez-Abejón, Supertauberian operators and perturbations, Arch. Math., 64 (1995), 423-433.

[41] M. González-A. Martínez-Abejón, Tauberian operators on $L_1(\mu)$-spaces, Studia Math., 125 (1997), 289-303.

[42] M. González-A. Martínez-Abejón, Ultrapowers and semi-Fredholm operators, Bolletino U.M.I. B, 11 (1997), 415-433.

[43] M. González-A. Martínez-Abejón, Quotients of $L_1$ by reflexive subspaces, Extracta Math., 12 (1997), 139-143.

[44] M. González-A. Martínez-Abejón, Lifting unconditionally converging series and semigroups, Bull. Austral. Math. Soc., 57 (1998), 135-146.

[45] M. González-A. Martínez-Abejón, Local reflexivity of dual Banach spaces, Pacific J. Math., 189 (1999), 263-278.

[46] M. González-A. Martínez-Abejón, Tauberian operators on $L_1(\mu)$ and ultrapowers, Rendiconti Circ. Mat. Palermo Suppl., 56 (1998), 128-138.

[47] M. González-A. Martínez-Abejón, Ultrapowers and semigroups of operators, Integral Equations Operator Theory, to appear.

[48] M. González-A. Martínez-Abejón, Ultrapowers of $L_1(\mu)$ and the subsequence splitting property, Preprint, 1998.

[49] M. González-A. Martinón, Operational quantities derived from the norm and measures of noncompactness, Proc. R. Ir. Acad. A, 91 (1991), 63-70. | MR | Zbl

[50] M. González-A. Martinón, Operational quantities derived from the norm and generalized Fredholm theory, Comment. Math. Univ. Carolinae, 32 (1991) 645-657. | fulltext mini-dml | MR | Zbl

[51] M. González-A. Martinón, Fredholm theory and space ideals, Boll. U.M.I. B, 7 (1993), 473-488. | Zbl

[52] M. González-A. Martinón, On incomparability of Banach spaces, Banach Center Publ., 30 (1994), 161-174. | Zbl

[53] M. González-A. Martinón, Operational quantities characterizing semi-Fredholm operators, Studia Math., 114 (1995), 13-27. | fulltext mini-dml | MR | Zbl

[54] M. González-V. M. Onieva, On incomparability of Banach spaces, Math. Z., 192 (1986), 581-585. | MR | Zbl

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

[56] M. González-V. M. Onieva, Semi-Fredholm operators and semigroups associated with some classical operator ideals II, Proc. R. Ir. Acad. A, 88 (1988), 119-124. | MR | Zbl

[57] M. González-V. M. Onieva, Lifting results for sequences in Banach spaces, Math. Proc. Cambridge Phil. Soc., 105 (1989), 117-121. | MR | Zbl

[58] M. González-V. M. Onieva, Characterizations of tauberian operators and other semigroups of operators, Proc. Amer. Math. Soc., 108 (1990), 399-405. | MR | Zbl

[59] M. González-E. Saksman-H.-O. Tylli, Representing non-weakly compact operators, Studia Math., 113 (1995), 265-282. | fulltext mini-dml | MR | Zbl

[60] W. T. Gowers, A solution to Banach's hyperplane problem, Bull. London Math. Soc., 26 (1994), 523-530. | MR | Zbl

[61] W. T. Gowers-B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc., 6 (1993), 851-874. | MR | Zbl

[62] A. Grothendieck, Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$, Canadian J. Math., 5 (1953), 129-173. | MR | Zbl

[63] R. Harte, Invertibility and singularity for bounded linear operators, M. Dekker, 1987. | MR | Zbl

[64] S. Heinrich, Finite representability of operators, Proc. Int. Conf. operators algebras, ideals and applications, Leipzig (1977), 33-39. | MR | Zbl

[65] S. Heinrich, Ultraproducts in Banach space Theory, J. Reine Angew. Math., 313 (1980), 72-104. | MR | Zbl

[66] S. Heinrich, Closed operator ideals and interpolation, J. Funct. Anal., 35 (1980), 397-411. | MR | Zbl

[67] R. H. Herman, Generalizations of weakly compact operators, Trans. Amer. Math. Soc., 132 (1968), 377-386. | MR | Zbl

[68] J. R. Holub, Characterizations of tauberian and related operators on Banach spaces, J. Math. Anal. Appl., 178 (1993), 280-288. | MR | Zbl

[69] N. Kalton-A. Wilansky, Tauberian operators in Banach spaces, Proc. Amer. Math. Soc., 57 (1976), 251-255. | MR | Zbl

[70] D. Kleinecke, Almost-finite, compact, and inessential operators, Proc. Amer. Math. Soc., 14 (1963), 863-868. | MR | Zbl

[71] A. Lebow-M. Schechter, Semigroups of operators and measures of noncompactness, J. Funct. Anal., 7 (1971), 1-26. | MR | Zbl

[72] J. Lindenstrauss-L. Tzafriri, Classical Banach Spaces I. Sequence spaces, Springer-Verlag, Berlin, Heidelberg, New York, 1977. | MR | Zbl

[73] R. Lohman, A note on Banach spaces containing $l_1$, Canad. Math. Bull., 19 (1976), 365-367. | MR | Zbl

[74] H. Lotz-N. Peck-H. Porta, Semi-embeddings of Banach spaces, Proc. Edinburgh Math. Soc., 12 (1979), 233-240. | MR | Zbl

[75] D. H. Martin-J. Swart, A characterisation of semi-Fredholm operators defined on an almost-reflexive normed spaces, Proc. R. Ir. Acad. A, 86 (1986), 91-93. | MR | Zbl

[76] A. Martínez-Abejón, Semigrupos de operadores y ultrapotencias, Ph. D. Thesis Univ. Cantabria, 1994.

[77] J. Martínez-Maurica-T. Pellón, Non-archimedian tauberian operators, Proc. Conf. on $p$p-adic analysis Hengelhoef (1986), 101-111. | MR | Zbl

[78] A. Martinón, Cantidades operacionales en teoria de Fredholm, Ph. D. Thesis Univ. La Laguna, 1989. | MR

[79] R. Neidinger, Properties of tauberian operators in Banach spaces, Ph. D. Thesis Univ. Texas at Austin, 1984.

[80] R. Neidinger-H. P. Rosenthal, Norm-attainment of linear functionals on subspaces and characterizations of tauberian operators, Pacific J. Math., 118 (1985), 215-228. | fulltext mini-dml | MR | Zbl

[81] A. Pietsch, Inessential operators in Banach spaces, Integral Equations and Operator Theory, 1 (1978), 589-591. | MR | Zbl

[82] A. Pietsch, Operator ideals, North-Holland, Amsterdam, New York, Oxford, 1980. | MR | Zbl

[83] H. P. Rosenthal, On totally incomparable Banach spaces, J. Funct. Anal., 4 (1969), 167-175. | MR | Zbl

[84] H. P. Rosenthal, On wide-$(s)$ sequences and their applications to certain classes of operators, Pacific J. Math., 189 (1999), 311-338. | MR | Zbl

[85] W. Schachermayer, For a Banach space isomorphic to its square the Radon-Nikodym property and the Krein-Milman property are equivalent, Studia Math., 81 (1985), 329-339. | MR | Zbl

[86] M. Schechter, Quantities related to strictly singular operators, Indiana Univ. Math. J., 21 (1972), 1061-1071. | MR | Zbl

[87] D. G. Tacon, Generalized semi-Fredholm transformations, J. Austral. Math. Soc. A, 34 (1983), 60-70. | MR | Zbl

[88] D. G. Tacon, Generalized Fredholm transformations, J. Austral. Math. Soc. A, 37 (1984), 89-97. | MR | Zbl

[89] M. Talagrand, The three-space problem for $L^1$, J. Amer. Math. Soc., 3 (1990), 9-29. | MR | Zbl

[90] E. Tarafdar, On further properties of improjective operators, J. Austral. Math. Soc., 14 (1972), 352-363. | MR | Zbl

[91] A. E. Taylor-D. C. Lay, Introduction to functional analysis, 2nd ed., Wiley, 1980. | MR | Zbl

[92] H.-O. Tylli, Two approximation conditions relative to closed operator ideals, Preprint, 1990.

[93] L. Weis, On perturbations of Fredholm operators in $L_p(\mu)$-spaces, Proc. Amer. Math. Soc., 67 (1977), 287-292. | MR | Zbl

[94] L. Weis, Perturbation classes of semi-Fredholm operators, Math. Z., 178 (1981), 429-442. | MR | Zbl

[95] R. J. Whitley, Strictly singular operators and their conjugates, Trans. Amer. Math. Soc., 113 (1964), 252-261. | MR | Zbl

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

[97] K. W. Yang, Operator invertible modulo the weakly compact operators, Pacific J. Math., 71 (1977), 559-564. | fulltext mini-dml | MR | Zbl

[98] J. Zemanek, Geometric characteristics of semi-Fredholm operators and their asymptotic behaviour, Studia Math., 80 (1984), 219-234. | fulltext mini-dml | MR | Zbl