×

Additive maps on standard operator algebras preserving parts of the spectrum. (English) Zbl 1042.47027

It has been proved by A. A. Jafarian and A. R. Sourour [J. Funct. Anal. 66, 255–261 (1986; Zbl 0589.47003)] that a surjective linear map preserving the spectrum from \({\mathcal B}(X)\) onto \({\mathcal B}(Y)\) is either an isomorphism or an anti-isomorphism, where \(X\) and \(Y\) are infinite-dimensional Banach spaces. This result was extended by B. Aupetit and H. du T. Mouton [Stud. Math. 109, 91–100 (1994; Zbl 0829.46039)] to primitive Banach algebras with minimal ideals, and by the authors to spectrum compressing linear maps [A characterization of homomorphisms between Banach algebras, Acta Math. Sinica, to appear].
In the paper under review, the authors improve the previous results in two ways: they consider maps between standard operator algebras (closed subalgebras of the algebra of all bounded operators acting on a Banach space that contain the identity and the ideal of finite rank operators); and instead of the spectrum they consider a variety of parts of it.
The main result shows that, given any unital additive surjection between standard operator algebras on infinite dimensional complex Banach spaces, there are thirteen different parts of the spectrum (not all disjoint) such that it is enough for the map to preserve one of those parts to guarantee that the map is either an isomorphism or an anti-isomorphism.

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aupetit, B., Spectrum-preserving linear mappings between Banach algebras or Jordan-Banach algebras, J. London Math. Soc. (2), 62, 917-924 (2000) · Zbl 1070.46504
[2] Aupetit, B.; Mouton, H.du T., Spectrum preserving linear mappings in Banach algebras, Studia Math., 109, 91-100 (1994) · Zbl 0829.46039
[3] Conway, J. B., A Course in Functional Analysis. A Course in Functional Analysis, Graduate Texts in Mathematics, 96 (1986), Springer-Verlag · Zbl 0706.46003
[4] J. Cui, J. Hou, A characterization of homomorphisms between Banach algebras, Acta Math. Sinica, to appear; J. Cui, J. Hou, A characterization of homomorphisms between Banach algebras, Acta Math. Sinica, to appear
[5] J. Cui, J. Hou, Linear maps between semi-simple Banach algebras compressing certain spectral functions, Rocky Mountain J. Math., to appear; J. Cui, J. Hou, Linear maps between semi-simple Banach algebras compressing certain spectral functions, Rocky Mountain J. Math., to appear · Zbl 1071.47037
[6] Hou, J., Spectrum-preserving elementary operators on \(B(X)\), Chinese Ann. Math. Ser. B, 19, 511-516 (1998) · Zbl 0918.47006
[7] Hou, J., Rank preserving linear maps on \(B(X)\), Sci. China Ser. A., 32, 929-940 (1989) · Zbl 0686.47030
[8] Jafarian, A. A.; Sourour, A. R., Spectrum-preserving linear maps, J. Funct. Anal., 66, 255-261 (1986) · Zbl 0589.47003
[9] Lindenstrauss, J., On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc., 72, 967-970 (1966) · Zbl 0156.36403
[10] Omladic, M.; Semrl, P., Spectrum-preserving additive maps, Linear Algebra Appl., 153, 67-72 (1991) · Zbl 0736.47001
[11] Ovsepian, R. I.; Pelczynski, A., Existences of a fundamental total and bounded biorthogonal sequence, Studia Math., 54, 149-159 (1975) · Zbl 0317.46019
[12] Semrl, P., Two characterizations of automorphisms on \(B(X)\), Studia Math., 105, 143-149 (1993) · Zbl 0810.47001
[13] Wang, Q., Additive maps and elementary operators on \(B(X)\) that preserve the point spectrum, J. Math. (Wuhan), 17, 468-472 (1997) · Zbl 0934.47022
[14] Wang, Q.; Hou, J., Point-spectrum preserving elementary operators on \(B(H)\), Proc. Amer. Math. Soc., 126, 2083-2088 (1998) · Zbl 0894.47027
[15] Zhang, X.; Hou, J., Positive elementary operators compressing spectrum, Chinese Sci. Bull., 42, 270-273 (1997) · Zbl 0939.47030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.