Maps preserving the local spectrum of triple product of operators

Abstract

Let denote the algebra of all bounded linear operators on a Banach space . Let and be infinite-dimensional complex Banach spaces, and fix two nonzero vectors and . Then we show that a surjective map satisfiesif and only if there exists a bijective mapping such that andwhere is a third root of unity.

Keywords:

• linear preservers
• local spectrum
• local spectral radius
• the single-valued extension property

AMS Subject Classifications:

• Primary: 47B49
• Secondary: 47A10
• 47A11

Acknowledgements

The authors profoundly thank the anonymous referee for his/her remarks and suggestions. Upon applying his/her comments, we could enormously simplify our proofs.

Notes

1 This work was supported by a grant from NSERC.