c-numerical range of operator products on B(H)

TL;DR

This paper characterizes surjective maps on operator algebras that preserve the c-numerical range of operator products, extending to finite-dimensional matrices, under conditions involving weak zero product preservation.

Contribution

It provides a complete description of structure-preserving maps for c-numerical ranges on B(H) and M_n(C), including new propositions about operators in B(H).

Findings

Surjective maps preserving c-numerical range are characterized.

Results apply to both infinite-dimensional B(H) and finite-dimensional M_n(C).

New operator propositions of independent interest are established.

Abstract

Let H be a complex Hilbert space of dimension no less than 2 and B(H) be the algebra of all bounded linear operators on H. We give the form of surjective maps on B(H) preserving c-numerical range of operator products when the maps satisfy preserving weak zero products. As a result, we obtain the characterization of surjective maps on Mn(C) preserving c-numerical range of operator products. The proof of the results depends on some propositions of operators in B(H), which are of different interest.