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

Ramesh, G.

doi:10.1090/S0002-9939-2010-10557-9

The Horn-Li-Merino formula for the gap and the spherical gap of unbounded operators
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Proc. Amer. Math. Soc. 139 (2011), 1081-1090 Request permission

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 $1$ and $\sqrt {2}$ 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.

References

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
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, DOI 10.4153/CJM-2005-010-1
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, DOI 10.1017/s0022112062000440
Dragana Cvetković, On gaps between bounded operators, Publ. Inst. Math. (Beograd) (N.S.) 72(86) (2002), 49–54. MR 1997610, DOI 10.2298/PIM0272049C
Javad Faghih Habibi, The gap of the graph of a matrix, Linear Algebra Appl. 186 (1993), 55–57. MR 1217198, DOI 10.1016/0024-3795(93)90284-U
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, DOI 10.1016/0024-3795(94)90419-7
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, DOI 10.1137/1.9780898719093
Simone Gramsch and Eberhard Schock, Ill-posed equations with transformed argument, Abstr. Appl. Anal. 13 (2003), 785–791. MR 1996924, DOI 10.1155/S1085337503303021
C. W. Groetsch, Spectral methods for linear inverse problems with unbounded operators, J. Approx. Theory 70 (1992), no. 1, 16–28. MR 1168372, DOI 10.1016/0021-9045(92)90053-Q
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, DOI 10.1142/9789812798886_{0}016
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, DOI 10.1016/0024-3795(94)00185-5
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
William E. Kaufman, A stronger metric for closed operators in Hilbert space, Proc. Amer. Math. Soc. 90 (1984), no. 1, 83–87. MR 722420, DOI 10.1090/S0002-9939-1984-0722420-9
Fuad Kittaneh, On some equivalent metrics for bounded operators on Hilbert space, Proc. Amer. Math. Soc. 110 (1990), no. 3, 789–798. MR 1027097, DOI 10.1090/S0002-9939-1990-1027097-6
S. H. Kulkarni and G. Ramesh, A formula for gap between two closed operators, Linear Algebra and its Applications 432 (2010), 3012–3017.
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, DOI 10.1007/s12044-008-0047-z
A. MacIntosh, Heinz inequalities and perturbation of spectral families, Macquarie math report (1979).
Ritsuo Nakamoto, Gap formulas of operators and their applications, Math. Japon. 42 (1995), no. 2, 219–232. MR 1356379
Ritsuo Nakamoto, The spherical gap of operators, Linear Algebra Appl. 251 (1997), 89–95. MR 1421267, DOI 10.1016/0024-3795(95)00697-4
Gert K. Pedersen, Analysis now, Graduate Texts in Mathematics, vol. 118, Springer-Verlag, New York, 1989. MR 971256, DOI 10.1007/978-1-4612-1007-8
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
Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815