Boundedness properties of resolvents and semigroups of operators

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 ${\displaystyle \frac{1}{n+1}{\sum }_{j=0}^n\parallel T^{j}x{\parallel }^2\le M{\left(T\right)}^2\parallel x{\parallel }^2}$ 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 ${\displaystyle \supset \{\parallel T^i\left\{n\right\}\parallel :n\in \mathbb{N}\}}$ is finite. In fact the following inequality is valid for all n ∈ ℕ: ∥T^n∥ ≤ e M(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 ${\displaystyle {(I-\lambda S)}^{-1}}$ exists and that the expression ${\displaystyle \supset \{(1-|\lambda |)\parallel {(I-\lambda S)}^{-1}\parallel :|\lambda |<1\}}$ 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 operatorsimilarity to 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 continuous semigroups instead of (powers) of a single operator. Some results are also given in the more general Banach space context.