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CONTINUITY OF A CONDITION SPECTRUM AND ITS LEVEL SETS

  • D. SUKUMAR (a1) and S. VEERAMANI (a2)

Abstract

Let ${\mathcal{A}}$ be a complex unital Banach algebra, let $a$ be an element in it and let $0<\unicode[STIX]{x1D716}<1$ . In this article, we study the upper and lower hemicontinuity and joint continuity of the condition spectrum and its level set maps in appropriate settings. We emphasize that the empty interior of the $\unicode[STIX]{x1D716}$ -level set of a condition spectrum at a given $(\unicode[STIX]{x1D716},a)$ plays a pivotal role in the continuity of the required maps at that point. Further, uniform continuity of the condition spectrum map is obtained in the domain of normal matrices.

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Copyright

Corresponding author

Footnotes

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The second author thanks the University Grants Commission (UGC), India for the financial support (Ref. no. 23/12/2012(ii)EU-V) provided as a form of Research Fellowship to carry out this research work at IIT Hyderabad.

Footnotes

References

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[1]Aliprantis, C. D. and Border, K. C., Infinite Dimensional Analysis, 3rd edn (Springer, Berlin, 2006).
[2]Aubin, J.-P. and Frankowska, H., Set-Valued Analysis, reprint of the 1990 edition, Modern Birkhäuser Classics (Birkhäuser, Boston, MA, 2009).
[3]Bhatia, R., Notes on Functional Analysis, Texts and Readings in Mathematics, 50 (Hindustan Book Agency, New Delhi, 2009).
[4]Burlando, L., ‘Continuity of spectrum and spectral radius in Banach algebras’, in: Functional Analysis and Operator Theory (Warsaw, 1992), Banach Center Publications, 30 (Institute of Mathematics Polish Academy of Sciences, Warsaw, 1994), 53100.
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[6]Horn, R. A. and Johnson, C. R., Matrix Analysis, 2nd edn (Cambridge University Press, Cambridge, 2013).
[7]Krishnan, A. and Kulkarni, S. H., ‘Pseudospectrum of an element of a Banach algebra’, Oper. Matrices 11(1) (2017), 263287.
[8]Kulkarni, S. H. and Sukumar, D., ‘The condition spectrum’, Acta Sci. Math. (Szeged) 74(3–4) (2008), 625641.
[9]Newburgh, J. D., ‘The variation of spectra’, Duke Math. J. 18 (1951), 165176.
[10]Rickart, C. E., General Theory of Banach Algebras, The University Series in Higher Mathematics (van Nostrand, Princeton, NJ, 1960).
[11]Shargorodsky, E., ‘On the level sets of the resolvent norm of a linear operator’, Bull. Lond. Math. Soc. 40(3) (2008), 493504.
[12]Shargorodsky, E., ‘On the definition of pseudospectra’, Bull. Lond. Math. Soc. 41(3) (2009), 524534.
[13]Sukumar, D. and Veeramani, S., ‘Level sets of the condition spectrum’, Ann. Funct. Anal. 8(3) (2017), 314328.
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CONTINUITY OF A CONDITION SPECTRUM AND ITS LEVEL SETS

  • D. SUKUMAR (a1) and S. VEERAMANI (a2)

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