Let $H$ be a Hilbert space. An operator $x\in B(H)$ is called a partial isometry if it is an isometry on Ker$x^{\perp}$.
Two well-behaved classes of partial isometries are maximal partial isometries (i.e., isometries and co-isometries) and power partial isometries. Indeed we have that:
(Wold-decomposition theorem 1954.) Every isometry is a direct sum of unitary and unilateral shifts.
(Halmos and Wallen 1970.) Every power partial isometry is a direct sum of unitary, isometry, co-isometry and truncated shifts.
B. Fishel also obtained a characterization of those partial isometries which are sums of shifts:
- (Fishel 1975.) A necessary and sufficient condition that a partial isometry $x$ (without zero or unitary parts) be the orthogonal direct sum of left and right shifts is that $x^nxH^{\perp}\subseteq x^nH$ or $x^{*n}xH^{\perp}\subseteq x^{*n}H$.
Q. Does there exist any other factorization(s) of a partial isometry into some nicer operators?