Numerical Information of Hilbert Space Operators

For a Hilbert space operator $A\in B(H)$, the numerical range map of $A$, denoted ${\phi }_A$ is the map sending unit vectors $x$ in $H$ to $⟨Ax,x⟩$. From this map we define the Numerical Information of $A$, denoted ${\mathrm{\Sigma }}_A$, 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 ${\phi }_B=f\circ {\phi }_A$, then $f$ is affine. Our other main result is showing that ${\mathrm{\Sigma }}_B\subset {\mathrm{\Sigma }}_A$ if and only if $B=\alpha Re(A)+\beta Im(A)+\gamma I$ for some complex numbers $\alpha ,\beta ,\gamma$.