The Horn-Li-Merino formula for the gap and the spherical gap of unbounded operators

Author:
G. Ramesh

Journal:
Proc. Amer. Math. Soc. **139** (2011), 1081-1090

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 Horn-Li-Merino 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.

**1.**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****2.**Bernhelm Booss-Bavnbek, Matthias Lesch, and John Phillips,*Unbounded Fredholm operators and spectral flow*, Canad. J. Math.**57**(2005), no. 2, 225–250. MR**2124916**, 10.4153/CJM-2005-010-1**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), 693–719. MR**0162142****4.**Dragana Cvetković,*On gaps between bounded operators*, Publ. Inst. Math. (Beograd) (N.S.)**72(86)**(2002), 49–54. MR**1997610**, 10.2298/PIM0272049C**5.**Javad Faghih Habibi,*The gap of the graph of a matrix*, Linear Algebra Appl.**186**(1993), 55–57. MR**1217198**, 10.1016/0024-3795(93)90284-U**6.**Javad Faghih Habibi,*The spherical gap of the graph of a linear transformation*, Proceedings of the 3rd ILAS Conference (Pensacola, FL, 1993), 1994, pp. 501–503. MR**1306995**, 10.1016/0024-3795(94)90419-7**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****8.**Simone Gramsch and Eberhard Schock,*Ill-posed equations with transformed argument*, Abstr. Appl. Anal.**13**(2003), 785–791. MR**1996924**, 10.1155/S1085337503303021**9.**C. W. Groetsch,*Spectral methods for linear inverse problems with unbounded operators*, J. Approx. Theory**70**(1992), no. 1, 16–28. MR**1168372**, 10.1016/0021-9045(92)90053-Q**10.**C. W. Groetsch,*Inclusions for the Moore-Penrose 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. 203–211. MR**1299760**, 10.1142/9789812798886_0016**11.**Roger A. Horn, Chi-Kwong Li, and Dennis I. Merino,*Distances between the graphs of matrices*, Linear Algebra Appl.**240**(1996), 65–77. MR**1387286**, 10.1016/0024-3795(94)00185-5**12.**Tosio Kato,*Perturbation theory for linear operators*, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR**0407617****13.**William E. Kaufman,*A stronger metric for closed operators in Hilbert space*, Proc. Amer. Math. Soc.**90**(1984), no. 1, 83–87. MR**722420**, 10.1090/S0002-9939-1984-0722420-9**14.**Fuad Kittaneh,*On some equivalent metrics for bounded operators on Hilbert space*, Proc. Amer. Math. Soc.**110**(1990), no. 3, 789–798. MR**1027097**, 10.1090/S0002-9939-1990-1027097-6**15.**S. H. Kulkarni and G. Ramesh,*A formula for gap between two closed operators*, Linear Algebra and its Applications**432**(2010), 3012-3017.**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, 613–625. MR**2511129**, 10.1007/s12044-008-0047-z**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, 219–232. MR**1356379****19.**Ritsuo Nakamoto,*The spherical gap of operators*, Linear Algebra Appl.**251**(1997), 89–95. MR**1421267**, 10.1016/0024-3795(95)00697-4**20.**Gert K. Pedersen,*Analysis now*, Graduate Texts in Mathematics, vol. 118, Springer-Verlag, New York, 1989. MR**971256****21.**Michael Reed and Barry Simon,*Methods of modern mathematical physics. I*, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR**751959****22.**Frigyes Riesz and Béla Sz.-Nagy,*Functional analysis*, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR**0071727****23.**Walter Rudin,*Functional analysis*, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR**1157815**

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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:
https://doi.org/10.1090/S0002-9939-2010-10557-9

Keywords:
Closed operator,
gap metric,
spherical gap,
Horn-Li-Merino 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.