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)

     

Restrictions of bounded linear operators: Closed range

Author(s): Bruce A. Barnes
Journal: Proc. Amer. Math. Soc. 135 (2007), 1735-1740.
MSC (2000): Primary 47A05
Posted: November 15, 2006
MathSciNet review: 2286083
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Let $ T$ be a bounded linear operator on a Banach space $ X,$ and let $ W$ be a subspace of $ X$ which is a Banach space and $ T-$ invariant. Denote by $ T_{W}$ the restriction of $ T$ to $ W. $ This paper explores the questions:

$ \circ $ If the range of $ T$ is closed, under what conditions is the range of $ T_{W}$ closed?

$ \circ $ If the range of $ T_{W}$ is closed, under what conditions is the range of $ T$ closed?

References:

[B1]
B. Barnes, The spectral and Fredholm theory of extensions of bounded linear operators, Proc. Amer. Math. Soc. 105 (1989), 941-949. MR 0955454 (89i:47008)

[B2]
B. Barnes, Common operator properties of the linear operators RS and SR, Proc. Amer. Math. Soc. 126 (1998), 1055-1061. MR 1443814 (98f:47003)

[B3]
B. Barnes, Generalized inverses of operators in some subalgebras of B(X), Acta Sci. Math.. (Szeged) 69 (2003), 349-357. MR 1991672 (2004c:47001)

[H]
P. Halmos, Introduction to Hilbert space, 2nd Edition, Chelsea Pub. Co., New York, 1957.MR 1653399 (99g:47001)

[H:R]
E. Hewitt and K. Ross, Abstract Harmonic Analysis, Vol. I, Springer-Verlag, Berlin, 1963.MR 0156915 (28:158)

[J]
K. Jorgens, Linear Integral Operators, Pitman, Boston, London, Melbourne, 1982.MR 0647629 (83j:45001)

[L:T]
D. Lay and A. Taylor, Introduction to Functional Analysis, 2nd Ed., Wiley, New York, 1980.MR 0564653 (81b:46001)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A05

Retrieve articles in all Journals with MSC (2000): 47A05

Additional Information:

Bruce A. Barnes
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: barnes@uoregon.edu

DOI: 10.1090/S0002-9939-06-08624-2
PII: S 0002-9939(06)08624-2
Keywords: Restriction of a bounded linear operator, closed range.
Received by editor(s): September 9, 2005
Received by editor(s) in revised form: December 28, 2005
Posted: November 15, 2006
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2006, 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