The Matricial Range of $E_{21}$

TL;DR

This paper describes the matricial range of the 2x2 unilateral shift, showing it consists of matrices with numerical radius at most 1/2, and reviews historical proofs involving dilation theory.

Contribution

It provides a clear overview of the matricial range of E_{21} and summarizes two classical proofs, making the results more accessible.

Findings

Matricial range of E_{21} is all matrices with numerical radius ≤ 1/2

Historical proofs rely on dilation theory and are complex

The paper offers an accessible introduction and recap of these proofs

Abstract

The matricial range of the $2\times2$ matrix $E_{21}$ (i.e., the $2\times 2$ unilateral shift) is described very simply: it consists of all matrices with numerical radius at most $1/2$. The known proofs of this simple statement, however, are far from trivial and they depend on subtle results on dilations. We offer here a brief introduction to the matricial range and a recap of those two proofs, following independent work of Arveson and Ando in the early 1970s.