zbMATH — the first resource for mathematics

Factorization theorems and the structure of operators on Hilbert space. (English) Zbl 0706.47010
A contraction T, without a unitary part, on a Hilbert space is said to belong to the class \({\mathfrak A}\) if the Sz.-Nagy and Foiaş functional calculus \(f\mapsto f(T)\) is isometric from the Hardy space \(H^{\infty}\) onto \({\mathcal A}_ T\), the ultraweakly closed unstarred subalgebra generated by T and I. The author settles affirmatively the conjecture that for T of class \({\mathfrak A}\) every ultraweakly continuous linear functional \(\psi\) on \({\mathcal A}_ T\) admits a representation \(\psi (A)=(Ax,y)\) for some vectors x and y. More precisely, he shows that for any \(\lambda >1\) the vectors x and y can be chosen such that \(\| x\| \cdot \| y\| <\lambda \| \psi \|\). The proof uses a factorization theorem of \(L^ 1\)-functions and an extension of results of B. Chevreau and C. Pearcy. This result implies that every contraction whose spectrum includes the unit circle has a nontrivial invariant subspace.

47A65 Structure theory of linear operators
47A45 Canonical models for contractions and nonselfadjoint linear operators
47A15 Invariant subspaces of linear operators
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
Full Text: DOI