The joint essential numerical range, compact perturbations, and the Olsen problem

Volume 197 / 2010

Vladimír Müller Studia Mathematica 197 (2010), 275-290 MSC: Primary 47A12; Secondary 47A13. DOI: 10.4064/sm197-3-5

Abstract

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.

Authors

  • Vladimír MüllerMathematical Institute
    Czech Academy of Sciences
    Žitna 25, 115 67 Praha 1, Czech Republic
    e-mail

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