Banach Journal of Mathematical Analysis

Linear maps between operator algebras preserving certain spectral functions

Xiaohong Cao and Shizhao Chen

Full-text: Open access

Abstract

Let $H$ be an infinite dimensional complex Hilbert space and let $\phi$ be a surjective linear map on $B(H)$ with $\phi(I)-I\in{\mathcal{K}}(H)$, where $\mathcal{K}(H)$ denotes the closed ideal of all compact operators on $H$. If $\phi$ preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then $\phi$ is an automorphism of the algebra $B(H)$. Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

Article information

Source
Banach J. Math. Anal., Volume 8, Number 1 (2014), 39-46.

Dates
First available in Project Euclid: 14 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1381782085

Digital Object Identifier
doi:10.15352/bjma/1381782085

Mathematical Reviews number (MathSciNet)
MR3161680

Zentralblatt MATH identifier
1290.47038

Subjects
Primary: 47B48: Operators on Banach algebras
Secondary: 47A10: Spectrum, resolvent 46H05: General theory of topological algebras

Keywords
Calkin algebra upper semi-Weyl operator linear preservers left invertible

Citation

Cao, Xiaohong; Chen, Shizhao. Linear maps between operator algebras preserving certain spectral functions. Banach J. Math. Anal. 8 (2014), no. 1, 39--46. doi:10.15352/bjma/1381782085. https://projecteuclid.org/euclid.bjma/1381782085


Export citation

References

  • B. Aupetit, Spectrum-preserving linear mapping between Banach algebra or Jordan Banach algebras, J. London Math. Soc. 62 (2000), 917–924.
  • B. Aupetit, A Primer on spectral theory, Springer-Verlag, 1990.
  • M. Bendaoud, A. Bourhim, M. Burgos and M. Sarih, Linear maps preservig Fredholm and Atkinson elements of $C^*$-algebra, Linear Multilinear Algebra 57 (2009), no. 8, 823–838.
  • M. Bendaoud, A. Bourhim and M. Sarih, Linear maps preservig the essential spectral radius, Linear Algebra Appl. 428 (2008), 1041–1045.
  • J. Cui and J. Hou, Linear maps between Banach algebras compressing certain spectral functions, Rocky Mountain J. Math. 34 (2004), no. 2, 565–585.
  • J. Cui and J. Hou, Additive maps on standard operator algebras preserving parts of the spectrum, J. Math. Anal. Appl. 282 (2003), 266–278.
  • J. Dieudonně, Sur une g${\mathrm{\acute{e}}}$n${\mathrm{\acute{e}}}$ralisation du groupe orthogonal ${\mathrm{\grave{a}}}$ quatre variables, Arch. Math. (Basel) 1 (1949), 282–287.
  • R.E. Harte, Invertibility and singularity for bounded linear operators, Dekker, New York, 1988.
  • R.E. Harte, Fredholm, Weyl and Browder theory, Proc. Royal Irish Acad. 85 (1985), A(2), 151-176.
  • I.N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331–341.
  • N. Jacobson and C.E. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479–502.
  • A.A. Jafarian and A.R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), 255–261.
  • I. Kaplansky, Algebraic and analytic aspects of opertors algebras, Amer. Math. Soc. Providence, 1970.
  • I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, 1954.
  • M. Marcus and R. Purves, Linear transformations on algebras of matrices: The invariance of the elementary symmetric functions, Canad. J. Math. 11 (1959), 383–396.
  • M. Mbekhta and P. Šemrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Algebra 57 (2009), 55–64.
  • M. Mbekhta, Linear maps preserving the set of Fredholm operators, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3613-3619.
  • M. Mbekhta, L. Rodman and P.Šemrl, Linear maps preserving generalized invertibility, Integral Equations Operator Theory 55 (2006), 93–109.
  • M. Omladič, On operators preserving commutativity, J. Funct. Anal. 66 (1986), 105–122.
  • C. Pearcy and D.Topping, Sums of small number of idempotents, Michigan Math. J. 14 (1967), 453–465.
  • P. Šemrl, Two characterizations of automorphisms on $B(X)$, Studia Math. 105 (1993), no. 2, 143–149.
  • A.R. Sourour, Invertibility preserving linear maps on $L(H)$, Trans. Amer. Math. Soc. 348 (1996), no. 1, 13–30.
  • W. Żelazko, A characterization of multiplicative linear functionals in complex Banach algebras, Studia Math. 30 (1968), 83–85.