Numerical Range Inclusion, Dilation, and Operator Systems

TL;DR

This paper characterizes m-tuples of matrices whose numerical range inclusion ensures joint dilations of operators and the complete positivity of associated unital positive maps, advancing the understanding of operator systems.

Contribution

It identifies conditions on matrix m-tuples that guarantee joint dilations and complete positivity of maps, extending previous results to multi-operator settings.

Findings

Characterization of matrix m-tuples for joint dilation

Conditions ensuring complete positivity of unital positive maps

New techniques relating numerical range inclusion 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 ${\rm span} \{I, A, A^*\}$ to ${\cal B}({\cal H})$, the algebra of bounded linear operators acting on a Hilbert space ${\cal H}$, is completely positive. In this paper, we identify $m$-tuple of matrices ${\bf A} = (A_1, \dots, A_m)$ such that any $m$-tuple of operators ${\bf B} = (B_1, \dots, B_m)$ satisfying the joint numerical range inclusion $W({\bf B}) \subseteq {\rm conv} W({\bf A})$ will have a joint dilation of the form $(A_1\otimes I, \dots, A_m\otimes I)$. Consequently, every unital positive map from ${\rm span} \{I, A_1, A_1^*, \dots, A_m, A_m^*\}$ to ${\cal B}({\cal H})$ is completely positive. New results and techniques are obtained relating to the study of numerical range inclusion, dilation, and maximal operator systems.