On contractions satisfying 𝐴𝑙𝑔𝑇={𝑇}’

Author:

Wu Pei Yuan

Abstract

For a bounded linear operator T on a Hilbert space let { T } \{ T\} ’ and Alg T {\operatorname {Alg}}\;T denote the commutant, the double commutant and the weakly closed algebra generated by T and 1, respectively. Assume that T is a completely nonunitary contraction with a scalar-valued characteristic function ψ ( λ ) \psi (\lambda ) . In this note we prove the equivalence of the following conditions: (i) | ψ ( e i t ) | = 1 |\psi ({e^{it}})| = 1 on a set of positive Lebesgue measure; (ii) Alg T = { T } {\operatorname {Alg}}\;T = \{ T\} ’ ; (iii) every invariant subspace for T is hyperinvariant. This generalizes the well-known fact that compressions of the shift satisfy Alg T = { T } {\operatorname {Alg}}\;T = \{T\}’ .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference7 articles.

1. Commutants and cyclic vectors;Deddens, James A.;Proc. Amer. Math. Soc.,1974

2. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77;Radjavi, Heydar,1973

3. Generalized interpolation in 𝐻^{∞};Sarason, Donald;Trans. Amer. Math. Soc.,1967

4. The invariant subspaces of almost unitary operators;Sickler, Sheldon O.;Indiana Univ. Math. J.,1974

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