Interpolation theory and measures related to operator ideals

Cobos, Fernando

Nonlinear Analysis, Function Spaces and Applications, GDML_Books, (1999), p. 93-118 / Harvested from

Résumé

Given any operator ideal $ℐ$ , there are two natural functionals $γ_{ℐ} (T)$ , $β_{ℐ} (T)$ that one can use to show the deviation of the operator $T$ to the closed surjective hull of $ℐ$ and to the closed injective hull of $ℐ$ , respectively. We describe the behaviour under interpolation of $γ_{ℐ}$ and $β_{ℐ}$ . The results are part of joint works with A. Martínez, A. Manzano and P. Fernández-Martínez.

MR MR1777713 | Zbl 0964.47010

EUDML-ID : urn:eudml:doc:220269
Mots clés:

@article{702473,
     title = {Interpolation theory and measures related to operator ideals},
     booktitle = {Nonlinear Analysis, Function Spaces and Applications},
     series = {GDML\_Books},
     publisher = {Czech Academy of Sciences, Mathematical Institute},
     address = {Praha},
     year = {1999},
     pages = {93-118},
     mrnumber = {MR1777713},
     zbl = {0964.47010},
     url = {http://dml.mathdoc.fr/item/702473}
}

Cobos, Fernando. Interpolation theory and measures related to operator ideals, dans Nonlinear Analysis, Function Spaces and Applications, GDML_Books,  (1999), pp. 93-118. http://gdmltest.u-ga.fr/item/702473/

Bibliographie

Cobos F.; Martínez P. Fernández,- Martínez A. Interpolation of the measure of non-compactness by the real method, , preprint.

Cobos F.; Manzano A.; Martínez A. Interpolation theory and measures related to operator ideals, , preprint. | Zbl 0945.46009

Cobos F.; Martínez A. Remarks on interpolation properties of the measure of weak non-compactness and ideal variations, To appear in Math. Nachr. | MR 1719795 | Zbl 0944.46012

Cobos F.; Martínez A. Extreme estimates for interpolated operators by the real method, To appear in J. London Math. Soc. | MR 1753819 | Zbl 0940.46011

, ,,

skii M. A. Krasnosel,’ On a theorem of M. Riesz, Soviet Math. Dokl. 1 (1960), 229–231. (1960) | MR 0119086

Lions J. L.; Peetre J. Sur une classe d’espaces d’interpolation, Inst. Hautes Etudes Sci. Publ. Math. 19 (1964), 5–68. (1964) | MR 0165343 | Zbl 0148.11403

Pelczyński A. On strictly singular and strictly cosingular operator. I: Strictly singular and strictly cosingular operators in $C (S)$ -spaces, Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 13 (1965), 31–41. (1965) | MR 0177300

Lebow A.; Schechter M. Semigroups of operators and measures of non-compactness, J. Funct. Anal. 7 (1971), 1–26. (1971) | MR 0273422

Davis W. J.; Figiel T.; Johnson W. B.; Pelczyński A. Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311–327. (1974) | MR 0355536

Rosenthal H. P. A characterization of Banach spaces containing $ℓ_{1}$ , Proc. Nat. Acad. Sci. USA 71 (1974), 2411–2413. (1974) | MR 0358307

Bergh J.; Löfström J. Interpolation spaces. An Introduction, Springer-Verlag, Berlin 1976. (1976) | MR 0482275 | Zbl 0344.46071

skii M. A. Krasnosel,’ Zabreiko P. P.; Pustylnik E. I.; Sobolevskii P. E. Integral operators in spaces of summable functions, Noordhoff Inter. Publ., Leyden 1976. (1976) | MR 0385645

Weis L. On the surjective (injective) envelope of strictly (co-)singular operators, Studia Math. 54 (1976), 285–290. (1976) | MR 0399908 | Zbl 0323.47016

Blasi F. S. De On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 21 (1977), 259–262. (1977) | MR 0482402 | Zbl 0365.46015

Beauzamy B. Espaces d’Interpolation Réels: Topologie et Géométrie, Springer-Verlag, Lecture Notes in Math. 666, Berlin 1978. (1978) | MR 0513228 | Zbl 0382.46021

Triebel H. Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam 1978. (1978) | MR 0503903 | Zbl 0387.46033

Astala K. On measures of non-compactness and ideal variations in Banach spaces, Ann. Acad. Sci. Fenn. Ser. A. I Math. Dissertationes 29 (1980), 1–42. (1980) | MR 0575533

Heinrich S. Closed operator ideals and interpolation, J. Funct. Anal. 35 (1980), 397–411. (1980) | MR 0563562 | Zbl 0439.47029

Pietsch A. Operator Ideals, North-Holland, Amsterdam 1980. (1980) | MR 0582655 | Zbl 0455.47032

Teixeira M. F.; Edmunds D. E. Interpolation theory and measure of non-compactness, Math. Nachr. 104 (1981), 129–135. (1981) | MR 0657887

Kantorovich L. V.; Akilov G. P. Functional analysis, Pergamon, Oxford 1982. (1982) | MR 0664597 | Zbl 0484.46003

Neidinger R. D. Factoring operators through hereditarily- $ℓ_{p}$ spaces, In: Banach Spaces, Proc. Conf. Columbia /Mo. 1984, Lecture Notes in Math. 1166, Springer-Verlag, Berlin 1985, 116–128. (1984) | MR 0827767

Neidinger R. D. Concepts in the real interpolation of Banach spaces, Longhorn Notes of the Univ. of Texas at Austin, Funct. Anal. Seminar 1986–1987, 1–15. (1986) | MR 0967087

Jarchow H.; Matter U. Interpolative constructions for operator ideals, Note Mat. 8 (1988), 45–56. (1988) | MR 1050508 | Zbl 0712.47039

Beucher O. J. On interpolation of strictly (co-)singular linear operators, Proc. Royal Soc. Edinburgh 112 A (1989), 263–269. (1989) | MR 1014656 | Zbl 0691.46048

Cobos F.; Fernandez D. L. On interpolation of compact operators, Ark. Mat. 27 (1989), 211–217. (1989) | MR 1022277 | Zbl 0691.46047

Cobos F.; Peetre J. Interpolation of compactness using Aronszajn-Gagliardo functors, Israel J. Math. 68 (1989), 220–240. (1989) | MR 1035891 | Zbl 0716.46054

Astala K.; Tylli H.-O. Seminorms related to weak compactness and to Tauberian operators, Math. Proc. Cambridge Philos. Soc. 107 (1990), 367–375. (1990) | MR 1027789 | Zbl 0709.47009

Cobos F.; Edmunds D. E.; Potter A. J. B. Real interpolation and compact linear operators, J. Funct. Anal. 88 (1990), 351–365. (1990) | MR 1038446 | Zbl 0704.46049

Maligranda L.; Quevedo A. Interpolation of weakly compact operators, Arch. Math. 55 (1990), 280–284. (1990) | MR 1075053 | Zbl 0696.46049

Cobos F.; Kühn T.; Schonbek T. One-sided compactness results for Aronszajn-Gagliardo functors, J. Funct. Anal. 106 (1992), 274–313. (1992) | MR 1165856 | Zbl 0787.46061

Cwikel M. Real and complex interpolation and extrapolation of compact operators, Duke Math. J. 65 (1992), 333–343. (1992) | MR 1150590 | Zbl 0787.46062

Mastylo M. On interpolation of weakly compact operators, Hokkaido Math. J. 22 (1993), 105–114. (1993) | MR 1226586 | Zbl 0789.46061

Aksoy A. G.; Maligranda L. Real interpolation and measure of weak noncompactness, Math. Nachr. 175 (1995), 5–12. (1995) | MR 1355009 | Zbl 0843.46013

González M.; Saksman E.; Tylli H.-O. Representing non-weakly compact operators, Studia Math. 113 (1995), 265–282. (1995) | MR 1330211 | Zbl 0832.47039

Tylli H.-O. The essential norm of an operator is not self-dual, Israel J. Math. 91 (1995), 93–110. (1995) | MR 1348307 | Zbl 0838.47010