Strictly singular operators on $L_p$ spaces and interpolation
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- by Francisco L. Hernández, Evgeny M. Semenov and Pedro Tradacete PDF
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Abstract:
We study the class $V_p$ of strictly singular non-compact operators on $L_p$ spaces. This allows us to obtain interpolation results for strictly singular operators on $L_p$ spaces. Given $1\leqslant p<q\leqslant \infty$, it is shown that if an operator $T$ bounded on $L_p$ and $L_q$ is strictly singular on $L_r$ for some $p\leqslant r\leqslant q$, then it is compact on $L_s$ for every $p<s<q$.References
- 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, DOI 10.1090/gsm/050
- 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, DOI 10.1016/S1874-5849(01)80005-X
- 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, DOI 10.1017/S0308210500018734
- 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 5790, DOI 10.2307/1968771
- V. Caselles, M. González. Compactness properties of strictly singular operators in Banach lattices. Semesterbericht Funktionalanalysis. Tübingen, Sommersemester (1987), 175–189.
- 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, DOI 10.1017/S0308210500000524
- Leonard E. Dor, On projections in $L_{1}$, Ann. of Math. (2) 102 (1975), no. 3, 463–474. MR 420244, DOI 10.2307/1971039
- 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.
- Seymour Goldberg, Unbounded linear operators, Dover Publications, Inc., Mineola, NY, 2006. Theory and applications; Reprint of the 1985 corrected edition [MR0810617]. MR 2446016
- Stefan Heinrich, Closed operator ideals and interpolation, J. Functional Analysis 35 (1980), no. 3, 397–411. MR 563562, DOI 10.1016/0022-1236(80)90089-0
- M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$, Studia Math. 21 (1961/62), 161–176. MR 152879, DOI 10.4064/sm-21-2-161-176
- Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 107819, DOI 10.1007/BF02790238
- M. A. Krasnosel′skiĭ, On a theorem of M. Riesz, Soviet Math. Dokl. 1 (1960), 229–231. MR 0119086
- M. A. Krasnosel′skiĭ, P. P. Zabreĭko, E. I. Pustyl′nik, and P. E. Sobolevskiĭ, Integral operators in spaces of summable functions, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leiden, 1976. Translated from the Russian by T. Ando. MR 0385645
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
- Mikael Lindström, Eero Saksman, and Hans-Olav Tylli, Strictly singular and cosingular multiplications, Canad. J. Math. 57 (2005), no. 6, 1249–1278. MR 2178561, DOI 10.4153/CJM-2005-050-7
- V. D. Milman. Operators of classes $C_0$ and $C_0^*$. Functions theory, functional analysis and appl., Vol. 10 (1970), 15–26 (in Russian).
- 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 177300
- A. Pełczyński and H. P. Rosenthal, Localization techniques in $L^{p}$ spaces, Studia Math. 52 (1974/75), 263–289. MR 361729
- C. J. Read, Strictly singular operators and the invariant subspace problem, Studia Math. 132 (1999), no. 3, 203–226. MR 1669678, DOI 10.4064/sm-132-3-203-226
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- 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, DOI 10.1070/SM2004v195n02ABEH000802
- L. Weis, On perturbations of Fredholm operators in $L_{p}(\mu )$-spaces, Proc. Amer. Math. Soc. 67 (1977), no. 2, 287–292. MR 467377, DOI 10.1090/S0002-9939-1977-0467377-X
- Lutz Weis, Integral operators and changes of density, Indiana Univ. Math. J. 31 (1982), no. 1, 83–96. MR 642619, DOI 10.1512/iumj.1982.31.31010
- R. J. Whitley, Strictly singular operators and their conjugates, Trans. Amer. Math. Soc. 113 (1964), 252–261. MR 177302, DOI 10.1090/S0002-9947-1964-0177302-2
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
- MR Author ID: 840453
- Email: tradacete@mat.ucm.es
- Received by editor(s): February 18, 2009
- Received by editor(s) in revised form: June 18, 2009
- Published electronically: 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 675-686
- MSC (2000): Primary 47B38; Secondary 47B07, 46B70
- DOI: https://doi.org/10.1090/S0002-9939-09-10089-8
- MathSciNet review: 2557184