# Boundedness properties of resolvents and semigroups of operators

• Volume: 38, Issue: 1, page 59-74
• ISSN: 0137-6934

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## Abstract

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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 $1/\left(n+1\right){\sum }_{j=0}^{n}\parallel {T}^{j}x{\parallel }^{2}\le M{\left(T\right)}^{2}\parallel x{\parallel }^{2}$ 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 $sup\parallel {T}^{i}n\parallel :n\in ℕ$ 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 ${\left(I-\lambda S\right)}^{-1}$ exists and that the expression $sup\left(1-|\lambda |\right)\parallel {\left(I-\lambda S\right)}^{-1}\parallel :|\lambda |<1$ 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.

## How to cite

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van Casteren, J.. "Boundedness properties of resolvents and semigroups of operators." Banach Center Publications 38.1 (1997): 59-74. <http://eudml.org/doc/208649>.

@article{vanCasteren1997,
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 $1/(n+1) ∑_\{j=0\}^n ∥T^\{j\}x∥^2 ≤ M(T)^\{2\} ∥x∥^\{2\}$ 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: ∥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 $(I-λS)^\{-1\}$ 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. },
author = {van Casteren, J.},
journal = {Banach Center Publications},
keywords = {operator Poisson kernel; bounded semigroup; power bounded operator; square bounded in average; square bounded average; power bounded; similar to a unitary operator; strongly continuous semigroups},
language = {eng},
number = {1},
pages = {59-74},
title = {Boundedness properties of resolvents and semigroups of operators},
url = {http://eudml.org/doc/208649},
volume = {38},
year = {1997},
}

TY - JOUR
AU - van Casteren, J.
TI - Boundedness properties of resolvents and semigroups of operators
JO - Banach Center Publications
PY - 1997
VL - 38
IS - 1
SP - 59
EP - 74
AB - 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: ∥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 $(I-λS)^{-1}$ 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.
LA - eng
KW - operator Poisson kernel; bounded semigroup; power bounded operator; square bounded in average; square bounded average; power bounded; similar to a unitary operator; strongly continuous semigroups
UR - http://eudml.org/doc/208649
ER -

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