Translator Disclaimer
December 2020 Maps preserving the $c$-numerical radius of products for operators in $\mathfrak{B}(H)$
Yanfang Zhang, Xiaochun Fang
Rocky Mountain J. Math. 50(6): 2265-2280 (December 2020). DOI: 10.1216/rmj.2020.50.2265

Abstract

Let 𝔅(β„‹) and 𝔅s(β„‹) be the algebra of all bounded linear operators on a complex Hilbert space β„‹ and the real Jordan algebra of all self-adjoint operators on β„‹, respectively. Suppose 𝒲 and 𝒱 are subsets of 𝔅(β„‹) or 𝔅s(β„‹) and F(β‹…) is the c-numerical radius or the diameter of an operator. Under an assumption on 𝒲 and 𝒱, characterizations are obtained for surjective maps Ξ¦:𝒲→𝒱 satisfying

F ( A B ) = F ( Ξ¦ ( A ) Ξ¦ ( B ) ) ( A , B ∈ 𝒲 ) .

When dimβ„‹β‰₯3, to establish the proofs, some general results are obtained for functions F:𝔅(β„‹)β†’[0,+∞) satisfying (P1) F(UAUβˆ—)=F(A) for all Aβˆˆπ”…(β„‹) and unitary U on β„‹; (P2) For Aβˆˆπ”…(β„‹), F(A)=0 if and only if A is a multiple of the identity; (P3) There are nonnegative real numbers Ξ±,Ξ² with Ξ±2+Ξ²2β‰ 0 such that F(T)=Ξ±βˆ₯Tβˆ₯+Ξ²|tr(T)| for each rank-one Tβˆˆπ”…(β„‹).

Citation

Download Citation

Yanfang Zhang. Xiaochun Fang. "Maps preserving the $c$-numerical radius of products for operators in $\mathfrak{B}(H)$." Rocky Mountain J. Math. 50 (6) 2265 - 2280, December 2020. https://doi.org/10.1216/rmj.2020.50.2265

Information

Received: 29 May 2019; Revised: 29 May 2020; Accepted: 30 May 2020; Published: December 2020
First available in Project Euclid: 5 January 2021

Digital Object Identifier: 10.1216/rmj.2020.50.2265

Subjects:
Primary: 47A12, 47B49

Rights: Copyright Β© 2020 Rocky Mountain Mathematics Consortium

JOURNAL ARTICLE
16 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.50 β€’ No. 6 • December 2020
Back to Top