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 $1/(n+1){\sum }_{j=0}^n\parallel T^{j}x{\parallel }^2\le M{\left(T\right)}^2\parallel x{\parallel }^2$ 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 $sup\parallel T^{i}n\parallel :n\in \mathbb{N}$ is finite. In fact the following inequality is valid for all n ∈ ℕ: ∥Tn∥ ≤ e M(T)M(T*). Suppose...