Which operators are similar to partial isometries?

Abstract

Let H \mathcal {H} denote a separable, infinite dimensional complex Hilbert space and let L ( H ) \mathcal {L}(\mathcal {H}) denote the algebra of all bounded linear operators on H \mathcal {H} . Let P = { T in L ( H ) | r ( T ) > 1 and T is similar to a partial isometry with infinite rank } \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 S = { S in L ( H ) | r ( S ) > 1 , range ( S ) is closed, and rank ( S ) = nullity ( S ) = corank ( S ) = ℵ 0 } \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 P = S \mathcal {P} = \mathcal {S} and it is proved that P ⊂ S ⊂ P − \mathcal {P} \subset \mathcal {S} \subset {\mathcal {P}^ - } .