Partial isometries closed under multiplication on Hilbert spaces

Factorization of Matrices into Partial Isometries

“…Products of partial isometries have also been considered before by Erdelyi [3]. However his concern is different from ours.…”

A structure theorem for operators with closed range

“…(2) implies ( 3 ) . An immediate consequence of (2) and the identities v J _ yO-1^ VV*)V*' } '~1 and V^V 3…”

“…Write g = T*h + k where h is in H and k is in the kernel of T . Then, by ( l ) , Next note that T* 1 ker T is orthogonal to T* 3 ker T for all N 0 5 i + j £ N , and £ ® T* 1 ker T is contained in the kernel of IT* i=0 N by (l). Let / be in ker T^"*" 1 0 f, ® T* % ker T .…”

“…Fix j (1 S 3 5 N) . By the representation of T and the definition of generalized truncated shift, i t follows that ker T. = ker T n ker T* 3 n range T 3 . 3…”

On partial isometries with no isometric part

“…In the next section, a direct proof of the above theorem is obtained. To prove sufficiency, note that V 1 V* J = I on the kernel of V for every / = 1,2, , N. Therefore by Theorem 2 in [5], it follows that V' is a partial isometry for every / = 1,2, , N + 1. Uniqueness of the above representation and the form of projections commuting with T follow from the explicit nature of the decomposition in Theorem 3.2.…”