Which operators are similar to partial isometries?

Abstract:

Let $\mathcal {H}$ denote a separable, infinite dimensional complex Hilbert space and let $\mathcal {L}(\mathcal {H})$ denote the algebra of all bounded linear operators on $\mathcal {H}$. Let $\mathcal {P} = \{ T{\text { in }}\mathcal {L}(\mathcal {H})|r(T) < 1{\text { and }}T{\text {is similar to a partial isometry with infinite rank} \}}$; let $\mathcal {S} = \{ S{\text { in }}\mathcal {L}(\mathcal {H})|r(S) < 1,{\text {range}}(S){\text { is closed, and rank}}(S)= {\text {nullity}}(S)= {\text {corank}}(S)={\aleph _0}\}$. It is conjectured that $\mathcal {P} = \mathcal {S}$ and it is proved that $\mathcal {P} \subset \mathcal {S} \subset {\mathcal {P}^ - }$.