The joint essential numerical range, compact perturbations, and the Olsen problem
Let $T_1,\ldots,T_n$ be bounded linear operators on a complex Hilbert space $H$. Then there are compact operators $K_1,\dots,K_n\in B(H)$ such that the closure of the joint numerical range of the $n$-tuple $(T_1-K_1,\ldots,T_n-K_n)$ equals the joint essential numerical range of $(T_1,\ldots,T_n)$. This generalizes the corresponding result for $n=1$. We also show that if $S\in B(H)$ and $n\in\mathbb N$ then there exists a compact operator $K\in B(H)$ such that $\|(S-K)^n\|=\|S^n\|_e$. This generalizes results of C. L. Olsen.