Products of Laurent operators and fields of values (Q306983)

The paper is devoted to the study of the numerical range of a Laurent (multiplication) operator \(L_\varphi\) with symbol \(\varphi\in L^\infty({\mathbb T})\) acting on the space \(L^2({\mathbb T})\) over the unit circle \({\mathbb T}\). It is shown that the closure of the numerical range of \(L_\varphi\) coincides with the convex hull of the spectrum of \(L_\varphi\) and with the convex hull of the essential range of the symbol \(\varphi\). Several consequences of this result are listed. The case of matrix-valued symbols is briefly discussed.