On the singular unitary part of a contraction. (English) Zbl 0723.47007

Zajac, Michal

Rev. Roum. Math. Pures Appl. 35, No. 4, 379-384 (1990).

If T is a contraction on a separable Hilbert space, then it can be decomposed as \(T_ a\oplus T_ s\), where \(T_ a\) is the direct sum of the completely nonunitary part and the absolutely continuous unitary part of T and \(T_ s\) is its singular unitary part. The paper under review concerns the splitting of various algebras and lattices associated with T along this decomposition. The algebras considered include Alg T (the weakly closed algebra generated by T and I), \(\{T\}''\) (the double commutant of T) and \(\{T\}'\) (the commutant of T); their associated invariant subspace lattices. The results contained herein are all standard routine ones, easy to verify and well-known among experts. One sample: \(Alg (T_ a\oplus T_ s)=Alg T_ a\oplus Alg T_ s\).

Reviewer: Wu Pei Yuan (Hsinchu)

MSC:

47A45

Canonical models for contractions and nonselfadjoint linear operators

Keywords:

absolutely continuous contraction; singular unitary operator; reflexive algebra; contraction on a separable Hilbert space; direct sum of the completely nonunitary part and the absolutely continuous unitary part; singular unitary part