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 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 is finite. In fact the following inequality is valid for all n ∈ ℕ: ∥Tn∥ ≤ 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 exists and that the expression 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.
@article{bwmeta1.element.bwnjournal-article-bcpv38i1p59bwm,
author = {van Casteren, J.},
title = {Boundedness properties of resolvents and semigroups of operators},
journal = {Banach Center Publications},
volume = {38},
year = {1997},
pages = {59-74},
zbl = {0892.47009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv38i1p59bwm}
}
van Casteren, J. Boundedness properties of resolvents and semigroups of operators. Banach Center Publications, Tome 38 (1997) pp. 59-74. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv38i1p59bwm/
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