Opuscula Math. 36, no. 6 (2016), 787-797

Opuscula Mathematica

Elementary operators - still not elementary?

Martin Mathieu

Abstract. Properties of elementary operators, that is, finite sums of two-sided multiplications on a Banach algebra, have been studied under a vast variety of aspects by numerous authors. In this paper we review recent advances in a new direction that seems not to have been explored before: the question when an elementary operator is spectrally bounded or spectrally isometric. As with other investigations, a number of subtleties occur which show that elementary operators are still not elementary to handle.

Keywords: spectral isometries, elementary operators, Jordan isomorphisms.

Mathematics Subject Classification: 47B47, 46H99, 47A10, 47A65, 47B48.

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  1. J. Alaminos, J. Extremera, A.R. Villena, Spectral preservers and approximate spectral preservers on operator algebras, Linear Alg. Appl. 496 (2016), 36-70.
  2. P. Ara, M. Mathieu, Local Multipliers of C*-Algebras, Springer Monographs in Mathematics, Springer-Verlag, London, 2003.
  3. B. Aupetit, A Primer on Spectral Theory, Springer-Verlag, New York, 1991.
  4. B. Aupetit, Spectral characterization of the radical in Banach or Jordan-Banach algebras, Math. Proc. Cambridge Phil. Soc. 114 (1993), 31-35.
  5. B. Aupetit, M. Mathieu, The continuity of Lie homomorphisms, Studia Math. 138 (2000), 193-199.
  6. N. Boudi, M. Mathieu, Elementary operators that are spectrally bounded, Oper. Theory Adv. Appl. 212 (2011), 1-15.
  7. N. Boudi, M. Mathieu, Locally quasi-nilpotent elementary operators, Oper. Matrices 8 (2014), 785-798.
  8. N. Boudi, M. Mathieu, More elementary operators that are spectrally bounded, J. Math. Anal. Appl. 428 (2015), 471-489.
  9. M. Brešar, M. Mathieu, Derivations mapping into the radical, III, J. Funct. Anal. 133 (1995), 21-29.
  10. C. Costara, D. Repovš, Spectral isometries onto algebras having a separating family of finite-dimensional irreducible representations, J. Math. Anal. Appl. 365 (2010), 605-608.
  11. R.E. Curto, M. Mathieu, Spectrally bounded generalized inner derivations, Proc. Amer. Math. Soc. 123 (1995), 2431-2434.
  12. R.E. Curto, M. Mathieu (eds), Elementary Operators and Their Applications, Proc. 3rd Int. Workshop (Belfast, 14-17 April 2009), Operator Theory Adv. Appl. 212, Springer-Verlag, Basel, 2011.
  13. R.J. Fleming, J.J. Jamison, Isometries on Banach Spaces: Function Algebras, Monographs and Surveys in Pure and Appl. Maths. 129, Chapman and Hall, Boca Raton, 2003.
  14. K. Jarosz (ed.), Function Spaces in Analysis, Proc. Seventh Conf. Function Spaces, Contemp. Math. 645 (2015).
  15. R.V. Kadison, Isometries of operator algebras, Annals of Math. 54 (1951), 325-338.
  16. Y.-F. Lin, M. Mathieu, Jordan isomorphism of purely infinite C*-algebras, Quart. J. Math. 58 (2007), 249-253.
  17. M. Mathieu (ed.), Proc. Int. Workshop on Elementary Operators and Applications, Blaubeuren, 9-12 June 1991, World Scientific, Singapore, 1992.
  18. M. Mathieu, Where to find the image of a derivation, Banach Center Publ. 30 (1994), 237-249.
  19. M. Mathieu, Spectrally bounded operators on simple C*-algebras II, Irish Math. Soc. Bull. 54 (2004), 33-40.
  20. M. Mathieu, Towards a non-selfadjoint version of Kadison's theorem, Ann. Math. Inf. 32 (2005), 87-94.
  21. M. Mathieu, A collection of problems on spectrally bounded operators, Asian-Eur. J. Math. 2 (2009), 487-501.
  22. M. Mathieu, G.J. Schick, First results on spectrally bounded operators, Studia Math. 152 (2002), 187-199.
  23. M. Mathieu, G.J. Schick, Spectrally bounded operators from von Neumann algebras, J. Operator Theory 49 (2003), 285-293.
  24. M. Mathieu, A.R. Sourour, Hereditary properties of spectral isometries, Arch. Math. 82 (2004), 222-229.
  25. M. Mathieu, A.R. Sourour, Spectral isometries on non-simple C*-algebras, Proc. Amer. Math. Soc. 142 (2014), 129-145.
  26. M. Mathieu, M. Young, Spectral isometries into commutative Banach algebras, Contemp. Math. 645 (2015), 217-222.
  27. M. Mathieu, M. Young, Spectrally isometric elementary operators, Studia Math. (to appear).
  28. V. Pták, Derivations, commutators and the radical, Manuscripta Math. 23 (1978), 355-362.
  • Martin Mathieu
  • Queen's University Belfast, Pure Mathematics Research Centre, Belfast BT7 1NN, Northern Ireland
  • Communicated by P.A. Cojuhari.
  • Received: 2016-04-17.
  • Accepted: 2016-07-28.
  • Published online: 2016-10-29.
Opuscula Mathematica - cover

Cite this article as:
Martin Mathieu, Elementary operators - still not elementary?, Opuscula Math. 36, no. 6 (2016), 787-797, http://dx.doi.org/10.7494/OpMath.2016.36.6.787

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