Amenable Operators of the Form Normal Plus Compact

An open question, raised independently by several authors, asks if a closed amenable subalgebra of ${\mathfrak{B}(\mathfrak{H})}$ must be similar to an C ^*-algebra. Recently, Choi, Farah and Ozawa have found a counter-example to this question, but their example is neither separable nor commutative, which leaves the question open for singly-generated algebras. In this paper we continue this line of investigation for special singly-generated algebras. It is shown that if an amenable operator T = N + K, where N is a normal operator, K is a compact operator and σ _e (N) has only finite accumulation points, then T is similar to a normal operator; if an amenable operator T = N + K, where N is a normal operator, ${K\in\mathcal{C}_p}$ for some p > 1 and ${\sigma(T)\cup\sigma(N)}$ is included in a smooth Jordan curve, then T is similar to a normal operator; if an amenable operator T = N + Q, where N is a normal operator, Q is a polynomial compact operator and NQ = QN, then T is similar to a normal operator; if there exists p, 1 < p < ∞, such that an amenable operator T satisfies one of the following conditions, then T is similar to a normal operator: (i) ${T-T^*\in\mathcal{C}_p}$ ; (ii) ${I-TT^*\in\mathcal{C}_p}$ ; (iii) ${I-T^*T\in\mathcal{C}_p}$ .