Strictly singular operators on 𝐿_{𝑝} spaces and interpolation

Hernández, Francisco; Semenov, Evgeny; Tradacete, Pedro

doi:10.1090/S0002-9939-09-10089-8

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Strictly singular operators on spaces and interpolation

Authors: Francisco L. Hernández, Evgeny M. Semenov and Pedro Tradacete
Journal: Proc. Amer. Math. Soc. 138 (2010), 675-686
MSC (2000): Primary 47B38; Secondary 47B07, 46B70
Posted: October 13, 2009
MathSciNet review: 2557184
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Abstract: We study the class of strictly singular non-compact operators on spaces. This allows us to obtain interpolation results for strictly singular operators on spaces. Given $1\leq p<q\leq\infty$ , it is shown that if an operator bounded on and is strictly singular on for some $p\leq r\leq q$ , then it is compact on for every .

[AA] Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. MR 1921782 (2003h:47072)
[AO] Dale Alspach and Edward Odell, 𝐿_{𝑝} spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 123–159. MR 1863691 (2003b:46013), http://dx.doi.org/10.1016/S1874-5849(01)80005-X
[B] O. J. Beucher, On interpolation of strictly (co-)singular linear operators, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 263–269. MR 1014656 (90j:46064), http://dx.doi.org/10.1017/S0308210500018734
[C] J. W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. (2) 42 (1941), 839–873. MR 0005790 (3,208c)
[CG] V. Caselles, M. González. Compactness properties of strictly singular operators in Banach lattices. Semesterbericht Funktionalanalysis. Tübingen, Sommersemester (1987), 175-189.
[CMMM] F. Cobos, A. Manzano, A. Martínez, and P. Matos, On interpolation of strictly singular operators, strictly co-singular operators and related operator ideals, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 5, 971–989. MR 1800087 (2001j:46113), http://dx.doi.org/10.1017/S0308210500000524
[D] Leonard E. Dor, On projections in 𝐿₁, Ann. of Math. (2) 102 (1975), no. 3, 463–474. MR 0420244 (54 #8258)
[GMF] I. T. Gohberg, A. S. Markus, I. A. Feldman. On normal solvable operators and related ideals. Amer. Math. Soc. Transl. (2), Vol. 61, Amer. Math. Soc., Providence, RI, 1967, 63-84.
[G] Seymour Goldberg, Unbounded linear operators, Dover Publications Inc., Mineola, NY, 2006. Theory and applications; Reprint of the 1985 corrected edition [MR0810617]. MR 2446016
[H] Stefan Heinrich, Closed operator ideals and interpolation, J. Funct. Anal. 35 (1980), no. 3, 397–411. MR 563562 (81f:47045), http://dx.doi.org/10.1016/0022-1236(80)90089-0
[KP] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces 𝐿_{𝑝}, Studia Math. 21 (1961/1962), 161–176. MR 0152879 (27 #2851)
[K] Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 0107819 (21 #6541)
[Kr] M. A. Krasnosel′skiĭ, On a theorem of M. Riesz, Soviet Math. Dokl. 1 (1960), 229–231. MR 0119086 (22 #9852)
[KZPS] M. A. Krasnosel′skiĭ, P. P. Zabreĭko, E. I. Pustyl′nik, and P. E. Sobolevskiĭ, Integral operators in spaces of summable functions, Noordhoff International Publishing, Leiden, 1976. Translated from the Russian by T. Ando; Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis. MR 0385645 (52 #6505)
[LT] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056 (58 #17766)
[LST] Mikael Lindström, Eero Saksman, and Hans-Olav Tylli, Strictly singular and cosingular multiplications, Canad. J. Math. 57 (2005), no. 6, 1249–1278. MR 2178561 (2006h:47058), http://dx.doi.org/10.4153/CJM-2005-050-7
[M] V. D. Milman. Operators of classes and . Functions theory, functional analysis and appl., Vol. 10 (1970), 15-26 (in Russian).
[P] A. Pełczyński, On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in 𝐶(𝑆)-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 31–36. MR 0177300 (31 #1563)
[PR] A. Pełczyński and H. P. Rosenthal, Localization techniques in 𝐿^{𝑝} spaces, Studia Math. 52 (1974/75), 263–289. MR 0361729 (50 #14174)
[R] C. J. Read, Strictly singular operators and the invariant subspace problem, Studia Math. 132 (1999), no. 3, 203–226. MR 1669678 (2000e:47012)
[Ru] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062 (51 #1315)
[SS] E. M. Semënov and F. A. Sukochev, The Banach-Saks index, Mat. Sb. 195 (2004), no. 2, 117–140 (Russian, with Russian summary); English transl., Sb. Math. 195 (2004), no. 1-2, 263–285. MR 2068953 (2005d:46063), http://dx.doi.org/10.1070/SM2004v195n02ABEH000802
[W1] L. Weis, On perturbations of Fredholm operators in 𝐿_{𝑝}(𝜇)-spaces, Proc. Amer. Math. Soc. 67 (1977), no. 2, 287–292. MR 0467377 (57 #7236), http://dx.doi.org/10.1090/S0002-9939-1977-0467377-X
[W2] Lutz Weis, Integral operators and changes of density, Indiana Univ. Math. J. 31 (1982), no. 1, 83–96. MR 642619 (84a:47066), http://dx.doi.org/10.1512/iumj.1982.31.31010
[Whi] R. J. Whitley, Strictly singular operators and their conjugates, Trans. Amer. Math. Soc. 113 (1964), 252–261. MR 0177302 (31 #1565), http://dx.doi.org/10.1090/S0002-9947-1964-0177302-2

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Additional Information

Francisco L. Hernández
Affiliation: Departmento de Análisis Matemático, Universidad Complutense de Madrid, 28040, Madrid, Spain
Email: pacoh@mat.ucm.es

Evgeny M. Semenov
Affiliation: Department of Mathematics, Voronezh State University, Voronezh 394006, Russia
Email: semenov@func.vsu.ru

Pedro Tradacete
Affiliation: Departmento de Análisis Matemático, Universidad Complutense de Madrid, 28040, Madrid, Spain
Email: tradacete@mat.ucm.es

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10089-8
PII: S 0002-9939(09)10089-8
Keywords: Strictly singular operator, $L_p$ space, interpolation
Received by editor(s): February 18, 2009
Received by editor(s) in revised form: June 18, 2009
Posted: October 13, 2009
Additional Notes: The first and third authors were partially supported by grants MICINN MTM2008-02652 and Santander/Complutense PR34/07-15837. The second author was partly supported by the Russian Fund. of Basic Research grants 08-01-00226-a and a Universidad Complutense grant. The third author was partially supported by grant MEC AP-2004-4841.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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