Spectral Approximations of a Normal Operator

Abstract

If $\Lambda$ is a closed convex set in the complex plane then $\mathfrak {N}(\Lambda ;H)$ denotes all the normal (bounded linear) operators on the fixed separable Hilbert space H with spectrum contained in $\Lambda$. The fixed operator A has N as an $\mathfrak {N}(\Lambda ;H)$-approximant provided N belongs to $\mathfrak {N}(\Lambda ;H)$ and the operator norm $\left \| {A - N} \right \|$ equals ${\rho _\Lambda }(A)$, the distance from A to $\mathfrak {N}(\Lambda ;H)$. With some hypothesis on $\Lambda$, this note proves that the dimension of the convex set of all $\mathfrak {N}(\Lambda ;H)$-approximants of normal operator A is ${(\dim {H_0})^2}$ where ${H_0}$ is the orthogonal complement of $\ker (|A - F(A)| - {\rho _\Lambda }(A))$ and $F(z)$ is the unique distaince minimizing retract of the complex plane onto $\Lambda$.