Abstract
Let X and Y be infinite dimensional Banach spaces over the real or complex field 𝔽, and let 𝒜 and ℬ be standard operator algebras on X and Y, respectively. In this paper, the structures of surjective maps from 𝒜 onto ℬ that completely preserve involutions in both directions and that completely preserve Drazin inverse in both direction are determined, respectively. From the structures of these maps, it is shown that involutions and Drazin inverse are invariants of isomorphism in complete preserver problems.
Highlights
In the last decades, the study of preserver problems is an active topic in operator algebra or operator space theory see 1
These results showed that involutions and Drazin inverse are invariants of isomorphism in preserver problems
Since completely positive linear maps and completely bounded linear maps are very important in operator algebra or operator space theory 4, and the concept of completely rank nonincreasing linear maps was introduced by Hadwin and Larson in 5, many mathematicians began to focus on complete preserver problems, that is, characterizations of maps on operator spaces subsets that preserve some property or invariant completely 6
Summary
The study of preserver problems is an active topic in operator algebra or operator space theory see 1. In 2 , the form of involutivity-preserving maps was given by using the known results of idempotence-preserving maps, and in 3 , the authors gave the characterization of additive maps preserving Drazin inverse. These results showed that involutions and Drazin inverse are invariants of isomorphism in preserver problems. Since involutions and Drazin inverse are closely related to idempotents, it is interesting to consider whether the involutions and Drazin inverse are still invariants of isomorphism in complete preserver problems. the characterization of surjective maps that completely preserve involutions between standard operator algebras on Banach spaces;. the characterization of surjective maps that completely preserve Drazin inverse between standard operator algebras on Banach spaces. Given P, Q ∈ PA, we say P and Q are orthogonal if P Q QP 0, where 0 is the zero operator in A
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