Numerical ranges of composition operators on \(l^2\) (Q2505141)

For any mapping \(T\) of the set of all positive integers into itself, the composition operator \(C_T\) on \(\ell^2\) is defined as \(C_Tf= f\circ T\) for \(f\) in \(\ell^2\). It is known that \(C_T\) is bounded if and only if the cardinalities of the inverse images \(T^{-1}(\{n\})\) of the singletons \(\{n\}\) are bounded for \(n\geq 1\). The purpose of this paper is to obtain the numerical ranges of such composition operators. For injective \(T\), this is completely determined. In particular, if \(T\) is such that there are some positive integers \(n\) and \(m\) with \(T^r(n)= n\) for some \(r\geq 1\) and \(T^s(m)\neq m\) for all \(s\geq 1\), then \(W(C_T)\) equals the convex hull of \(\{\lambda\in\mathbb{C}: |\lambda|< 1\}\cup\{\lambda\in \mathbb{C}: \lambda^k= 1\) for some period \(k\) of \(T\) at some \(n\geq 1\}\). On the other hand, it is also shown that if \(C_T\) is a densely-defined unbounded operator on \(\ell^2\), then \(W(C_T)\) equals the complex plane. (This can be compared with the fact that the same is true for any unbounded operator on \(\ell^2\).) Another case considered is when \(T^2= T\) and \(T\) is not the identity map, in which case \(W(C_T)\) is an elliptic disc. Finally, it is proved that \(W(C_T)\) always contains \(0\) whenever \(T\) is not the identity map. Several examples are given to illustrate these phenomena.