Abstract. Completing research made in [K2], we characterize vectors of a Hilbert space, which have a canonical, inner--outer-type factorization in relation to a given operator $T$. Special emphasis is put on the case, when $T$ is an operator admitting an $H^{\infty }$-functional calculus, and, in particular, when $T$ is quasisimilar to the unilateral shift.

AMS Subject Classification (1991): 47A16, 47A45, 47A60, 47B35

Keyword(s): Paley type inequalities, multipliers, Walsh system, spline functions, Ciesielski system, Hardy spaces, atomic decomposition

Received October 10, 2003. (Registered under 5815/2009.)