Numerical Range Inclusion, Dilation, and Operator Systems

Abstract

Researchers have identified complex matrices $A$ such that a bounded linear operator $B$ acting on a Hilbert space will admit a dilation of the form $A\otimes I$ whenever the numerical range inclusion relation $W(B)\subseteq W(A)$ holds. Such an operator $A$ and the identity matrix will span a maximal operator system, i.e., every unital positive map from $\mathrm{span}\{I,A,A^{\ast }\}$ to $\mathcal{B}(\mathcal{H})$, the algebra of bounded linear operators acting on a Hilbert space $\mathcal{H}$, is completely positive. In this paper, we identify $m$-tuple of matrices $\mathbf{A}=(A_1,\dots ,A_m)$ such that any $m$-tuple of operators $\mathbf{B}=(B_1,\dots ,B_m)$ satisfying the joint numerical range inclusion $W(\mathbf{B})\subseteq \mathrm{conv}W(\mathbf{A})$ will have a joint dilation of the form $(A_1\otimes I,\dots ,A_m\otimes I)$. Consequently, every unital positive map from $\mathrm{span}\{I,A_1,A_1^{\ast },\dots ,A_m,A_m^{\ast }\}$ to $\mathcal{B}(\mathcal{H})$ is completely positive. New results and techniques are obtained relating to the study of numerical range inclusion, dilation, and maximal operator systems.

13 pages, 3 figures

References in corpus (1)

• Numerical Range Inclusion, Dilation, and completely positive maps

Cited by in corpus (1)

• On the matrix range of random matrices