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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.

##### MSC:
 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
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