The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set

Shown: for every bounded operator $A$ on any complex Hilbert space (no separability assumed) and every matrix polynomial $P$, $\|P[A]\|\le2\sup_{W(A)}\|P\|$, with 2 sharp; the same bound holds for finite matrix-valued functions holomorphic near $\overline{W(A)}$ and for rational functions with poles outside it. For a matrix and an admissible convex domain containing $W(A)$, an optimal similarity with condition number at most 2 makes the conformal image contractive, with one positive boundary density representing all matrix-valued analytic evaluations. It contains the scalar Crouzeix theorem as the case $m=1$.

Shown: for every bounded operator $A$ on any complex Hilbert space (no separability assumed) and every matrix polynomial $P$, $\|P[A]\|\le2\sup_{W(A)}\|P\|$, with 2 sharp; the same bound holds for finite matrix-valued functions holomorphic near $\overline{W(A)}$ and for rational functions with poles outside it. For a matrix and an admissible convex domain containing $W(A)$, an optimal similarity with condition number at most 2 makes the conformal image contractive, with one positive boundary density representing all matrix-valued analytic evaluations. It contains the scalar Crouzeix theorem as the case $m=1$.