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1997 | 38 | 1 | 59-74
Tytuł artykułu

Boundedness properties of resolvents and semigroups of operators

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $1/(n+1) ∑_{j=0}^n ∥T^{j}x∥^2 ≤ M(T)^{2} ∥x∥^{2}$ 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 $sup{∥T^i{n}∥: n ∈ ℕ}$ 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 $(I-λS)^{-1}$ 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 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.
Rocznik
Tom
38
Numer
1
Strony
59-74
Opis fizyczny
Daty
wydano
1997
Twórcy
  • University of Antwerp, Department of Mathematics and Computer Science, Universiteitsplein 1, 2610 Antwerp/Wilrijk, Belgium
Bibliografia
  • [1] J. M. Anderson, J. G. Clunie and Ch. Pommerenke, On Bloch functions and normal families, J. Reine Angew. Math. 270 (1974), 12-37.
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  • [24] B. Sz.-Nagy and C. Foiaş, Sur les contractions de l'espace de Hilbert X; contractions similaires à des transformations unitaires, ibid. 26 (1965), 79-91.
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Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-bcpv38i1p59bwm
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