Advances in Operator Theory

Pseudospectra of elements of reduced Banach algebras

Arundhathi Krishnan and S. H. Kulkarni

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Abstract

Let $A$ be a Banach algebra with identity $1$ and $p \in A$ be a non-trivial idempotent. Then $q=1-p$ is also an idempotent. The subalgebras $pAp$ and $qAq$ are Banach algebras, called reduced Banach algebras, with identities $p$ and $q$ respectively. For $a \in A$ and $\varepsilon > 0$, we examine the relationship between the $\varepsilon$-pseudospectrum $\Lambda_{\varepsilon}(A,a)$ of $a \in A$, and $\varepsilon$-pseudospectra of $pap \in pAp$ and $qaq \in qAq$. We also extend this study by considering a finite number of idempotents $p_{1},\cdots,p_{n}$, as well as an arbitrary family of idempotents satisfying certain conditions.

Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 475-493.

Dates
Received: 3 February 2017
Accepted: 12 July 2017
First available in Project Euclid: 4 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aot/1512431723

Digital Object Identifier
doi:10.22034/aot.1702-1112

Mathematical Reviews number (MathSciNet)
MR3730042

Zentralblatt MATH identifier
06804223

Subjects
Primary: 47A10: Spectrum, resolvent
Secondary: 46H05: General theory of topological algebras 47A12: Numerical range, numerical radius

Keywords
direct sum reduced Banach algebra idempotent pseudospectrum spectrum

Citation

Krishnan, Arundhathi; Kulkarni, S. H. Pseudospectra of elements of reduced Banach algebras. Adv. Oper. Theory 2 (2017), no. 4, 475--493. doi:10.22034/aot.1702-1112. https://projecteuclid.org/euclid.aot/1512431723


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References

  • J. Alaminos, J. Extremera, and A. R. Villena, Approximately spectrum-preserving maps, J. Funct. Anal. 261 (2011), no. 1, 233–266.
  • F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, New York, 1973.
  • F. F. Bonsall and J. Duncan, Numerical ranges II (Vol. 2), Cambridge University Press, 1973.
  • E. O. Cevik and Z. I. Ismailov, Spectrum of the direct sum of operators, Electron. J. Differential Equations (2012), no. 210, 1–8.
  • J. B. Conway, A Course in Functional Analysis (Vol. 96), Springer Science and Business Media, 2013.
  • P. R. Halmos, A Hilbert Space Problem Book, second edition, Springer-Verlag, New York, 1982.
  • P. Harmand, D. Werner, and W. Werner, M-ideals in Banach spaces and Banach algebras, Springer-Verlag, Berlin, 1993.
  • M. Z. Kolundžija, (p,q)- outer generalized inverse of block matrices in Banach algebras, Banach J. Math. Anal. 8 (2014), no. 1, 98–108.
  • A. Krishnan and S. H. Kulkarni, Pseudospectrum of an element of a Banach algebra, Oper. Matrices 11 (2017), no. 1, 263–287.
  • C. S. Kubrusly, Elements of Operator Theory, Birkhäuser, Boston, 2001.
  • P. Lancaster and H. K. Farahat, Norms on direct sums and tensor products, Math. Comp. 26 (1972), no. 118, 401–414.
  • C. R. Putnam, Operators satisfying a $G_{1}$ condition, Pacific J. Math. 84 (1979), no. 2, 413–426.
  • E. Shargorodsky, On the level sets of the resolvent norm of a linear operator, Bull. London Math. Soc. 40 (2008), no. 3, 493–504.
  • E. Shargorodsky, On the definition of pseudospectra, Bull. London Math. Soc. 41 (2009), no. 3, 524–534.
  • S. Shkarin, Norm attaining operators and pseudospectrum, Integral Equations Operator Theory 64 (2009), no. 1, 115–136.
  • A. M. Sinclair, The norm of a Hermitian element in a Banach algebra, Proc. Amer. Math. Soc. 28 (1971), 446–450.
  • L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton Univ. Press, Princeton, NJ, 2005.