Quasisimilarity of Invariant Subspaces for Uniform Jordan Operators of Infinite Multiplicity

LetTbe a bounded linear operator on a separable Hilbert spaceH, and letMandNbe two invariant subspaces forT. We will say thatMandNare quasisimilar if there exist quasiaffinities (i.e. bounded one-to-one operators with dense ranges)XandY, commuting withT, such that (XM)−=Nand (YN)−=M. Recall that operatorsTandT′ are quasisimilar if there exist quasiaffinitiesX: H→H′ andY: H′→Hsuch thatXT=T′XandYT′=TY. We show that if T is a uniform Jordan operator of infinite multiplicity, thenMis quasisimilar toNif and only ifT∣Mis quasisimilar toT∣Nand (T*∣M⊥)* is quasisimilar to (T*∣N⊥)*.