Numerical Information of Hilbert Space Operators

Abstract

For a Hilbert space operator $A\in B(H)$, the numerical range map of $A$, denoted $ϕ_A$ is the map sending unit vectors $x$ in $H$ to $\langle Ax,x\rangle$. From this map we define the Numerical Information of $A$, denoted $Σ_A$, which is the sigma algebra induced on the unit sphere by the numerical range map. One of the main results of the paper are that if $A,B\in B(H)$, and $ϕ_B=f\circ ϕ_A$, then $f$ is affine. Our other main result is showing that $Σ_B\subsetΣ_A$ if and only if $B=αRe(A)+βIm(A)+γI$ for some complex numbers $α,β,γ$.