Mean\(_2\)-bounded operators on Hilbert space and weight sequences of positive operators (Q1303817)

Let \(H\) be a complex Hilbert space. An operator-valued weight sequence on \(H\) is a bilateral sequence \(W= \{W_k\}^\infty_{k= -\infty}\) whose terms are bounded, positive, invertible self-adjoint operators. The authors introduce and study the class of \(\text{mean}_2\)-bounded operators on Hilbert space. These operators, which are characterized by a uniform boundedness condition on the ``quadratic averages'' of their powers, are intimately related to operator-valued weight sequences and weighted norm inequalities. Under certain circumstances, \(\text{mean}_2\)-bounded operators have a rich spectral theory. Much of the paper is taken up with a discussion of relationships between the classes of trigonometrically well-bounded operators and the \(\text{mean}_2\)-bounded operators. The former class was introduced by the authors in AC functions on the circle and spectral families [J. Oper. Theory 13, 33-47 (1985; Zbl 0566.47011)]. The present paper develops further ideas in the papers by \textit{S. Treil} and \textit{A. Volberg} [J. Funct. Analysis 143, 269-308 (1997; Zbl 0876.42027)].