Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Rademacher bounded families of operators on $ L_1$

Author(s): N. J. Kalton; T. Kucherenko
Journal: Proc. Amer. Math. Soc. 136 (2008), 263-272.
MSC (2000): Primary 47D06, 46E30
Posted: October 5, 2007
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We give a characterization of R-bounded families of operators on $ L_1.$ We then use this result to study sectorial operators on $ L_1$. We show that if $ A$ is an R-sectorial operator on $ L_1$, then, for any $ \epsilon>0,$ there is an invertible operator $ U:L_1\to L_1$ with $ \Vert U-I\Vert<\epsilon$ such that for some strictly positive Borel function $ w$, $ U(\D(A))$ contains the weighted $ L_1$-space $ L_1(w).$

References:

[1]
R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. MR 0450957 (56:9247)

[2]
M. Cowling, I. Doust, A. McIntosh, and A. Yagi, Banach space operators with a bounded $ H\sp \infty $ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51-89. MR 1364554 (97d:47023)

[3]
J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, 1984. MR 737004 (85i:46020)

[4]
H. Fakhoury, Représentations d'opérateurs à valeurs dans $ L\sp {1}(X,\,\Sigma ,\,\mu )$, Math. Ann. 240 (1979), 203-212 (French). MR 526843 (80e:47027)

[5]
M. Hoffmann, N. Kalton, and T. Kucherenko, R-bounded approximating sequences and applications to semigroups, J. Math. Anal. Appl. 294 (2004), 373-386. MR 2061331 (2005e:46034)

[6]
N. J. Kalton, The endomorphisms of $ L\sb{p} (0\leq p\leq 1)$, Indiana Univ. Math. J. 27 (1978), 353-381. MR 0470670 (57:10416)

[7]
N. J. Kalton, Linear operators on $ L\sb {p}$ for $ 0<p<1$, Trans. Amer. Math. Soc. 259 (1980), 319-355. MR 0567084 (81d:47022)

[8]
N. J. Kalton and L. Weis, The $ H\sp \infty $-calculus and sums of closed operators, Math. Ann. 321 (2001), 319-345. MR 1866491 (2003a:47038)

[9]
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, II, Function spaces, Vol. 97, Springer-Verlag, 1979. MR 540367 (81c:46001)

[10]
L. Weis, On the representation of order continuous operators by random measures, Trans. Amer. Math. Soc. 285 (1984), 535-563. MR 752490 (85g:47047)

[11]
L. Weis, Operator-valued Fourier multiplier theorems and maximal $ L\sb p$-regularity, Math. Ann. 319 (2001), 735-758. MR 1825406 (2002c:42016)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47D06, 46E30

Retrieve articles in all Journals with MSC (2000): 47D06, 46E30

Additional Information:

N. J. Kalton
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Email: nigel@math.missouri.edu

T. Kucherenko
Affiliation: Department of Mathematics, University of California Los Angeles, Box 951555, Los Angeles, California 90095-1555
Email: tamara@math.ucla.edu

DOI: 10.1090/S0002-9939-07-09046-6
PII: S 0002-9939(07)09046-6
Keywords: Sectorial operators, representation of regular operators, R-boundedness
Received by editor(s): September 21, 2005
Received by editor(s) in revised form: December 6, 2006 and December 13, 2006
Posted: October 5, 2007
Additional Notes: The authors acknowledge support from NSF grants DMS-0244515 and DMS-0555670
Communicated by: N. Tomczak-Jaegermann
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google