A contribution to the structure theory of operators in the class \({\mathbb{A}}\)
Let \({\mathcal H}\) be a separable Hilbert space, T be an absolutely continuous contraction on \({\mathcal H}\), \(\phi_ T:{\mathcal H}^{\infty}\to L({\mathcal H})\) be the corresponding contractive homomorphism given by the Sz. Nagy-Foias functional calculus, \({\mathbb{A}}\) the class of all T sucht that \(\phi_ T\) is an isometry and that \({\mathbb{A}}_ 1\) the set of all \(T\epsilon\) \({\mathbb{A}}\) such that for each weak * continuous functional \(\phi\) on the weak * closed algebra \({\mathcal A}_ T\) generated by T, there exists x,y\(\epsilon\) \({\mathcal H}\) such that \(\phi (A)=(Ax,y)\) for all \(A\epsilon\) \({\mathcal A}_ T\). This paper is a contribution to the conjecture that \({\mathbb{A}}={\mathbb{A}}_ 1\), and to the invariant subspace problem for contractions T such that \(\sigma\) (T) contains all z of modulus 1.
invariant subspace problem for contractions
scientific article; zbMATH DE number 4063838