Nonlinear maps commuting with the λ-Aluthge transform under (n,m)–Jordan-triple product

ABSTRACT

In this paper, we give a complete form of bijective (not necessarily linear) maps $\mathrm{\Phi }:\mathcal{B}\left(H\right)\to \mathcal{B}\left(K\right)$ where H,K are Hilbert spaces with $\dim H\ge 2$ that satisfy ${\mathrm{\Delta }}_{\lambda }\left(\mathrm{\Phi }\right(A)\star \mathrm{\Phi }(B\left)\right)=\mathrm{\Phi }\left({\mathrm{\Delta }}_{\lambda }\right(A\star B\left)\right)\mathrm{for}\mathrm{all}A,B\in \mathcal{B}\left(H\right),$ where ${\mathrm{\Delta }}_{\lambda }\left(T\right)$ is the λ-Aluthge transform of T and the operation $A\star B=A^{n}B{A}^{m}n+m\ge 1$ is the $(n,m)$-Jordan-triple product. We show that there exists a unitary or anti-unitary operator $U:H\to K$ and a constant $\alpha \in \mathbb{C}$, with ${\alpha }^{n+m}=1$ such that $\mathrm{\Phi }\left(T\right)=\alpha UT{U}^{\ast }\mathrm{for}\mathrm{every}T\in \mathcal{B}\left(H\right).$

KEYWORDS:

• Jordan-triple product
• quasi-normal operators
• polar decomposition
• λ-Aluthge transform

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 47B49
• 47L99
• 47A55
• 47B37

Acknowledgments

The authors would like to thank the referee for carefully reading our manuscript and making many valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by the Labex CEMPI [grant number ANR-11-LABX-0007-01].