Regular Factorizations of Contractions

Equivalent conditions are given for the regularity of a factorization of a contraction, two of which exhibit immediately the duality property of this notion. The concept of regular factorization of contractions of Hubert spaces1 was introduced in [1] by the authors in connection with their investigations on the invariant subspace problem; cf. [2, §VIL3]. Let A0 be a contraction of a Hubert space 31 into a Hubert space 31,, and let (F) A0 = A2AX be a factorization of A0 as a product of a contraction Ax of 31 into some "intermediate" Hubert space 33, and of a contraction A2 of 33 into 31*: 33 Define the corresponding "defect operators" by £>, = (/, A*A,)112, D^ = (/„, A.A*)1'2 (j = 0, 1, 2), where I} and /*, denote the identity operators on the space of definition of Aj and A*, respectively. The factorization (F) was defined to be regular if condition (i) given in the Proposition below holds. Some basic arithmetical properties of such factorizations were established in [2], in particular it was proven that (F) is regular if and only if its dual