Resolvent algebra of finite rank operators

Abstract

Let $${\mathscr {H}}$$ be a Hilbert space. Suppose that $$A\in {\mathbb {B}}({\mathscr {H}})$$ and the operators $$I+mA$$ are invertible for all integers $$m \ge 1$$ . We characterize the resolvent algebra $$\begin{aligned} R_A:= \left\{ T \in {\mathbb {B}}({\mathscr {H}}) : \sup _{m \ge 1}\Vert (I+mA)T(I+mA)^{-1}\Vert < \infty \right\} , \end{aligned}$$ when A is a finite rank operator with $$\mathrm{cov}(A)\ne 0$$ . Moreover, we determine the elements of $$\{A\}'$$ and $$R_A^{c_0}$$ and prove that $$R_A = R_A^c = \{A\}'\oplus R_A^{c_0}$$ , where $$\{A\}'$$ is the commutant A and both $$R_A^c$$ and $$R_A^{c_0}$$ are subclasses of $$R_A$$ defined by $$\begin{aligned} R_A^c = \left\{ T \in R_A : \lim _{m \rightarrow \infty } \Vert (I+mA)T(I+mA)^{-1}\Vert ~ \mathrm {exists} \right\} \end{aligned}$$ and $$\begin{aligned} R_A^{c_0} = \left\{ T \in R_A^c : \lim _{m \rightarrow \infty } \Vert (I+mA)T(I+mA)^{-1}\Vert =0 \right\} . \end{aligned}$$ We provide a counterexample showing that $$R_A^c= \{A\}'\oplus R_A^{c_0}$$ is not true for some compact operators.