Title:
|
Interpolation theory and measures related to operator ideals (English) |
Author:
|
Cobos, Fernando |
Language:
|
English |
Journal:
|
Nonlinear Analysis, Function Spaces and Applications |
Volume:
|
Vol. 6 |
Issue:
|
1998 |
Year:
|
|
Pages:
|
93-118 |
. |
Category:
|
math |
. |
Summary:
|
Given any operator ideal $\mathcal{I}$, there are two natural functionals $\gamma_{\mathcal{I}}(T)$, $\beta_{\mathcal{I}}(T)$ that one can use to show the deviation of the operator $T$ to the closed surjective hull of $\mathcal{I}$ and to the closed injective hull of $\mathcal{I}$, respectively. We describe the behaviour under interpolation of $\gamma_{\mathcal{I}}$ and $\beta_{\mathcal{I}}$. The results are part of joint works with A. Martínez, A. Manzano and P. Fernández-Martínez. (English) |
MSC:
|
46B70 |
MSC:
|
46M35 |
MSC:
|
47B10 |
MSC:
|
47L20 |
idZBL:
|
Zbl 0964.47010 |
idMR:
|
MR1777713 |
. |
Date available:
|
2009-10-08T09:47:13Z |
Last updated:
|
2012-08-03 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/702473 |
. |
Reference:
|
[1] Aksoy A. G., Maligranda L. : Real interpolation and measure of weak noncompactness.Math. Nachr. 175 (1995), 5–12. Zbl 0843.46013, MR 1355009 |
Reference:
|
[2] 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. MR 0575533 |
Reference:
|
[3] Astala K., Tylli H.-O. : Seminorms related to weak compactness and to Tauberian operators.Math. Proc. Cambridge Philos. Soc. 107 (1990), 367–375. Zbl 0709.47009, MR 1027789 |
Reference:
|
[4] Beauzamy B. : Espaces d’Interpolation Réels: Topologie et Géométrie.Springer-Verlag, Lecture Notes in Math. 666, Berlin 1978. Zbl 0382.46021, MR 0513228 |
Reference:
|
[5] Bergh J., Löfström J. : Interpolation spaces. An Introduction.Springer-Verlag, Berlin 1976. Zbl 0344.46071, MR 0482275 |
Reference:
|
[6] Beucher O. J. : On interpolation of strictly (co-)singular linear operators.Proc. Royal Soc. Edinburgh 112 A (1989), 263–269. Zbl 0691.46048, MR 1014656 |
Reference:
|
[7] Cobos F., Edmunds D. E., Potter A. J. B. : Real interpolation and compact linear operators.J. Funct. Anal. 88 (1990), 351–365. Zbl 0704.46049, MR 1038446 |
Reference:
|
[8] Cobos F., Fernandez D. L. : On interpolation of compact operators.Ark. Mat. 27 (1989), 211–217. Zbl 0691.46047, MR 1022277 |
Reference:
|
[9] Cobos F., Martínez P. Fernández,- Martínez A. : Interpolation of the measure of non-compactness by the real method., preprint. |
Reference:
|
[10] Cobos F., Kühn T., Schonbek T. : One-sided compactness results for Aronszajn-Gagliardo functors.J. Funct. Anal. 106 (1992), 274–313. Zbl 0787.46061, MR 1165856 |
Reference:
|
[11] Cobos F., Manzano A., Martínez A. : Interpolation theory and measures related to operator ideals., preprint. Zbl 0945.46009 |
Reference:
|
[12] Cobos F., Martínez A. : Remarks on interpolation properties of the measure of weak non-compactness and ideal variations.To appear in Math. Nachr. Zbl 0944.46012, MR 1719795 |
Reference:
|
[13] Cobos F., Martínez A. : Extreme estimates for interpolated operators by the real method.To appear in J. London Math. Soc. Zbl 0940.46011, MR 1753819 |
Reference:
|
[14] Cobos F., Peetre J. : Interpolation of compactness using Aronszajn-Gagliardo functors.Israel J. Math. 68 (1989), 220–240. Zbl 0716.46054, MR 1035891 |
Reference:
|
[15] Cwikel M. : Real and complex interpolation and extrapolation of compact operators.Duke Math. J. 65 (1992), 333–343. Zbl 0787.46062, MR 1150590 |
Reference:
|
[16] Davis W. J., Figiel T., Johnson W. B., Pelczyński A. : Factoring weakly compact operators.J. Funct. Anal. 17 (1974), 311–327. MR 0355536 |
Reference:
|
[17] 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. Zbl 0365.46015, MR 0482402 |
Reference:
|
[18] González M., Saksman E., Tylli H.-O. : Representing non-weakly compact operators.Studia Math. 113 (1995), 265–282. Zbl 0832.47039, MR 1330211 |
Reference:
|
[19] Heinrich S. : Closed operator ideals and interpolation.J. Funct. Anal. 35 (1980), 397–411. Zbl 0439.47029, MR 0563562 |
Reference:
|
[20] Jarchow H., Matter U. : Interpolative constructions for operator ideals.Note Mat. 8 (1988), 45–56. Zbl 0712.47039, MR 1050508 |
Reference:
|
[21] Kantorovich L. V., Akilov G. P. : Functional analysis.Pergamon, Oxford 1982. Zbl 0484.46003, MR 0664597 |
Reference:
|
[22] skii M. A. Krasnosel,’ : On a theorem of M. Riesz.Soviet Math. Dokl. 1 (1960), 229–231. MR 0119086 |
Reference:
|
[23] 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. MR 0385645 |
Reference:
|
[24] Lebow A., Schechter M. : Semigroups of operators and measures of non-compactness.J. Funct. Anal. 7 (1971), 1–26. MR 0273422 |
Reference:
|
[25]
: .,, |
Reference:
|
[26] Lions J. L., Peetre J. : Sur une classe d’espaces d’interpolation.Inst. Hautes Etudes Sci. Publ. Math. 19 (1964), 5–68. Zbl 0148.11403, MR 0165343 |
Reference:
|
[27] Maligranda L., Quevedo A. : Interpolation of weakly compact operators.Arch. Math. 55 (1990), 280–284. Zbl 0696.46049, MR 1075053 |
Reference:
|
[28] Mastylo M. : On interpolation of weakly compact operators.Hokkaido Math. J. 22 (1993), 105–114. Zbl 0789.46061, MR 1226586 |
Reference:
|
[29] Neidinger R. D. : Factoring operators through hereditarily-$\ell _p$ spaces.In: Banach Spaces, Proc. Conf. Columbia /Mo. 1984, Lecture Notes in Math. 1166, Springer-Verlag, Berlin 1985, 116–128. MR 0827767 |
Reference:
|
[30] 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. MR 0967087 |
Reference:
|
[31] 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. MR 0177300 |
Reference:
|
[32] Pietsch A. : Operator Ideals.North-Holland, Amsterdam 1980. Zbl 0455.47032, MR 0582655 |
Reference:
|
[33] Rosenthal H. P. : A characterization of Banach spaces containing $\ell _1$.Proc. Nat. Acad. Sci. USA 71 (1974), 2411–2413. MR 0358307 |
Reference:
|
[34] Teixeira M. F., Edmunds D. E. : Interpolation theory and measure of non-compactness.Math. Nachr. 104 (1981), 129–135. MR 0657887 |
Reference:
|
[35] Triebel H. : Interpolation theory, function spaces, differential operators.North-Holland, Amsterdam 1978. Zbl 0387.46033, MR 0503903 |
Reference:
|
[36] Tylli H.-O. : The essential norm of an operator is not self-dual.Israel J. Math. 91 (1995), 93–110. Zbl 0838.47010, MR 1348307 |
Reference:
|
[37] Weis L. : On the surjective (injective) envelope of strictly (co-)singular operators.Studia Math. 54 (1976), 285–290. Zbl 0323.47016, MR 0399908 |
. |