Maximal numerical ranges of certain classes of operators and approximation

Abstract

Let \(\mathcal {B(H)}\) be the collection of bounded linear operators on a complex separable Hilbert space \(\mathcal {H}\). For \(T\in \mathcal {B(H)}\), its numerical range and maximal numerical range are denoted by W(T) and \(W_0(T)\), respectively. First, we give in this paper a characterization of the maximal numerical range and, as applications, we determine maximal numerical ranges of weighted shifts, partial isometries, the Volterra integral operator and classical Toeplitz operators. Second, we study the universality of maximal numerical ranges, showing that any nonempty bounded convex closed subset of \(\mathbb {C}\) is the maximal numerical range of some operator. Finally, we discuss the relations among the numerical range, the maximal numerical range and the spectrum. It is shown that the collection of those operators T with \(W_0(T)\cap \sigma (T)=\emptyset \) is a nonempty open subset of \(\mathcal {B(H)}\) precisely when \(\dim \mathcal {H}>1\), and is dense precisely when \(1<\dim \mathcal {H}<\infty \). We also show that those operators T with \(W_0(T)= W(T)\) constitute a nowhere dense subset of \(\mathcal {B(H)}\) precisely when \(\dim \mathcal {H}>1\)