Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
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)

     

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
MathSciNet review: 2350412
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 and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia