The HornLiMerino formula for the gap and the spherical gap of unbounded operators
Author:
G. Ramesh
Journal:
Proc. Amer. Math. Soc. 139 (2011), 10811090
MSC (2010):
Primary 47A55
Published electronically:
October 1, 2010
MathSciNet review:
2745658
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Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this article we obtain the HornLiMerino formula for computing the gap as well as the spherical gap between two densely defined unbounded closed operators. As a consequence we prove that the gap and the spherical gap of an unbounded closed operator are and respectively. With the help of these formulae we establish a relation between the spherical gap and the gap of unbounded closed operators. We discuss some properties of the spherical gap similar to those of the gap metric.
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 N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover Publications. Inc., New York, 1993. Translated from the Russian and with a preface by Merlynd Nestell. Reprint of the 1961 and 1963 translations, two volumes bound as one. MR 1255973 (94i:47001)
 2.
 Bernhelm BoossBavnbek, Matthias Lesch, and John Phillips, Unbounded Fredholm operators and spectral flow, Canad. J. Math. 57 (2005), no. 2, 225250. MR 2124916 (2006a:58029)
 3.
 H. O. Cordes and J. P. Labrousse, The invariance of the index in the metric space of closed operators, J. Math. Mech. 12 (1963), 693719. MR 0162142 (28:5341)
 4.
 Dragana Cvetković, On gaps between bounded operators, Publ. Inst. Math. (Beograd) (N.S.) 72(86) (2002), 4954. MR 1997610 (2004d:47003)
 5.
 Javad Faghih Habibi, The gap of the graph of a matrix, Linear Algebra Appl. 186 (1993), 5557. MR 1217198 (94c:15039)
 6.
 , The spherical gap of the graph of a linear transformation, Proceedings of the 3rd ILAS Conference (Pensacola, FL, 1993), vol. 212/213, 1994, pp. 501503. MR 1306995
 7.
 Israel Gohberg, Peter Lancaster, and Leiba Rodman, Invariant subspaces of matrices with applications, Classics in Applied Mathematics, vol. 51, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006, reprint of the 1986 original. MR 2228089 (2007k:15001)
 8.
 Simone Gramsch and Eberhard Schock, Illposed equations with transformed argument, Abstr. Appl. Anal. (2003), no. 13, 785791. MR 1996924 (2004f:47019)
 9.
 C. W. Groetsch, Spectral methods for linear inverse problems with unbounded operators, J. Approx. Theory 70 (1992), no. 1, 1628. MR 1168372 (93g:47011)
 10.
 , Inclusions for the MoorePenrose inverse with applications to computational methods, Contributions in numerical mathematics, World Sci. Ser. Appl. Anal., vol. 2, World Sci. Publ., River Edge, NJ, 1993, pp. 203211. MR 1299760 (95h:65041)
 11.
 Roger A. Horn, ChiKwong Li, and Dennis I. Merino, Distances between the graphs of matrices, Linear Algebra Appl. 240 (1996), 6577. MR 1387286 (97e:15004)
 12.
 Tosio Kato, Perturbation theory for linear operators, second ed., SpringerVerlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617 (53:11389)
 13.
 William E. Kaufman, A stronger metric for closed operators in Hilbert space, Proc. Amer. Math. Soc. 90 (1984), no. 1, 8387. MR 722420 (85a:47010)
 14.
 Fuad Kittaneh, On some equivalent metrics for bounded operators on Hilbert space, Proc. Amer. Math. Soc. 110 (1990), no. 3, 789798. MR 1027097 (91b:47017)
 15.
 S. H. Kulkarni and G. Ramesh, A formula for gap between two closed operators, Linear Algebra and its Applications 432 (2010), 30123017.
 16.
 S. H. Kulkarni, M. T. Nair and G. Ramesh, Some properties of unbounded operators with closed range, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), no. 4, 613625. MR 2511129 (2010e:47006)
 17.
 A. MacIntosh, Heinz inequalities and perturbation of spectral families, Macquarie math report (1979).
 18.
 Ritsuo Nakamoto, Gap formulas of operators and their applications, Math. Japon. 42 (1995), no. 2, 219232. MR 1356379 (96k:47033)
 19.
 , The spherical gap of operators, Linear Algebra Appl. 251 (1997), 8995. MR 1421267 (98e:47034)
 20.
 Gert K. Pedersen, Analysis now, Graduate Texts in Mathematics, vol. 118, SpringerVerlag, New York, 1989. MR 971256 (90f:46001)
 21.
 Michael Reed and Barry Simon, Methods of modern mathematical physics. I, Functional Analysis, second ed., Academic Press. Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. MR 751959 (85e:46002)
 22.
 Frigyes Riesz and Béla Sz.Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. MR 0071727 (17:175i)
 23.
 Walter Rudin, Functional analysis, second ed., International Series in Pure and Applied Mathematics, McGrawHill Inc., New York, 1991. MR 1157815 (92k:46001)
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Additional Information
G. Ramesh
Affiliation:
Statistics and Mathematics Unit, Indian Statistical Institute Bangalore, Bangalore, India 560 059
Email:
ramesh@isibang.ac.in
DOI:
http://dx.doi.org/10.1090/S000299392010105579
PII:
S 00029939(2010)105579
Keywords:
Closed operator,
gap metric,
spherical gap,
HornLiMerino formula
Received by editor(s):
October 14, 2009
Received by editor(s) in revised form:
April 2, 2010
Published electronically:
October 1, 2010
Additional Notes:
The author is thankful to the NBHM for financial support and ISI Bangalore for providing necessary facilities to carry out this work.
Communicated by:
Marius Junge
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
