Boundedness properties of resolvents and semigroups of operators

Abstract. Let T: H → H be an operator in the complex Hilbert space H. Suppose that T is square bounded in average in the sense that there exists a constant M(T) with the property that, for all natural numbers n and for all x ∈ H, the inequality 1 n+ 1 n∑ j=0 ‖T jx‖2 ≤M(T)2‖x‖2 is satisfied. Also suppose that the adjoint T ∗ of the operator T is square bounded in average with constant M(T ∗). Then the operator T is power bounded in the sense that sup{‖Tn ‖ : n ∈ N} is finite. In fact the following inequality is valid for all n ∈ N: ‖Tn ‖ ≤ eM(T)M(T ∗). Suppose that T has a bounded everywhere defined inverse S with the property that for λ in the open unit disc of C the operator (I − λS)−1 exists and that the expression sup{(1 − |λ|)‖(I − λS)−1 ‖ : |λ | < 1} is finite. If T is power bounded, then so is S and hence in such a situation the operator T is similar to a unitary operator. If both the operators T ∗ and S are square bounded in average, then again the operator T is similar to a unitary operator. Similar results hold for strongly 1991 Mathematics Subject Classification: 47A30, 47D05. Key words and phrases: power bounded operator, bounded semigroup, operator Poisson kernel, square bounded in average. The author is grateful to a number of people who in one way or another were involved during the preparation of this paper: F. Delbaen, L. Waelbroeck (Brussels), J. Zemánek (Warsaw). The author also wants to thank the National Fund for Scientific Research (NFWO) and the University of Antwerp (UIA) for their material support. The author is indebted to the referee for pointing out some errors in an earlier draft of the paper. Finally, the author is obliged to the “International Scientific and Technical Cooperation BLEU-Poland ” for making it possible to visit the Banac