Maps commuting with the λ-Aluthge transform for the semistar Jordan product

Abstract

Let H and K be two complex separable Hilbert spaces, such that $\dim (H)\ge 2$ and $B(H)$ the algebra of bounded linear operators of H on itself. For every $A,B\in B(H)$, the semistar Jordan product is denoted by $A▹B=\frac{1}{2}(\mathrm{AB}+B^{\ast }A)$ and for every $\lambda \in [0,1]$, the λ-Aluthge transform of A is denoted by ${\mathrm{\Delta }}_{\lambda }(A)$. We show that a bijective map $\mathrm{\Phi }:B(H)⟶B(K)$ satisfies the following condition for some $\lambda \in (0,1)$, ${\mathrm{\Delta }}_{\lambda }(\mathrm{\Phi }(A)▹\mathrm{\Phi }(B))=\mathrm{\Phi }({\mathrm{\Delta }}_{\lambda }(A▹B)),\mathrm{forall}A,B\in B(H),$if and only if there exists a unitary or anti-unitary operator $U:H⟶K$, such that $\mathrm{\Phi }(A)=\mathrm{UA}{U}^{\ast },\mathrm{forall}A\in B(H).$

Keywords:

• Non linear preservers
• λ-Aluthge transform
• Jordan product of operators
• Hilbert spaces
• spectrum and trace
• orthogonal projections

2020 Mathematics Subject Classifications:

• 47A10
• 47A25
• 47A56
• 47B02
• 47B20
• 47B25
• 47B49

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Our manuscript has no associated data.

Additional information

Funding

This work is made possible with the support of the CNRST of Morocco under the Research Excellence Grant Program from which Mr. Yassine Labbane benefits.