The joint essential numerical range of operators: convexity and related results

Chi-Kwong Li, and Yiu-Tung Poon. "The joint essential numerical range of operators: convexity and related results." Studia Mathematica 194.1 (2009): 91-104. <http://eudml.org/doc/285316>.

@article{Chi2009,
abstract = {Let W(A) and $W_\{e\}(A)$ be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that $W_\{e\}(A)$ is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, $W_\{e\}(A)$ can be obtained as the intersection of all sets of the form $cl(W(A₁, ..., A_\{i+1\}, A_\{i\} + F, A_\{i+1\}, ..., Aₘ))$, where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in $W_\{e\}(A)$ as star centers. Although cl(W(A)) is usually not convex, an analog of the separation theorem is obtained, namely, for any element d ∉ cl(W(A)), there is a linear functional f such that f(d) > supf(a): a ∈ cl(W(Ã)), where à is obtained from A by perturbing one of the components $A_\{i\}$ by a finite rank self-adjoint operator. Other results on W(A) and $W_\{e\}(A)$ extending those on a single operator are obtained.},
author = {Chi-Kwong Li, Yiu-Tung Poon},
journal = {Studia Mathematica},
keywords = {joint numerical range; joint essential numerical range; convexity; star-shapedness},
language = {eng},
number = {1},
pages = {91-104},
title = {The joint essential numerical range of operators: convexity and related results},
url = {http://eudml.org/doc/285316},
volume = {194},
year = {2009},
}

TY - JOUR
AU - Chi-Kwong Li
AU - Yiu-Tung Poon
TI - The joint essential numerical range of operators: convexity and related results
JO - Studia Mathematica
PY - 2009
VL - 194
IS - 1
SP - 91
EP - 104
AB - Let W(A) and $W_{e}(A)$ be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that $W_{e}(A)$ is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, $W_{e}(A)$ can be obtained as the intersection of all sets of the form $cl(W(A₁, ..., A_{i+1}, A_{i} + F, A_{i+1}, ..., Aₘ))$, where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in $W_{e}(A)$ as star centers. Although cl(W(A)) is usually not convex, an analog of the separation theorem is obtained, namely, for any element d ∉ cl(W(A)), there is a linear functional f such that f(d) > supf(a): a ∈ cl(W(Ã)), where à is obtained from A by perturbing one of the components $A_{i}$ by a finite rank self-adjoint operator. Other results on W(A) and $W_{e}(A)$ extending those on a single operator are obtained.
LA - eng
KW - joint numerical range; joint essential numerical range; convexity; star-shapedness
UR - http://eudml.org/doc/285316
ER -