- Tytuł:
- Boundedness properties of resolvents and semigroups of operators
- Autorzy:
- van Casteren, J.
- Powiązania:
- https://bibliotekanauki.pl/articles/1358667.pdf
- Data publikacji:
- 1997
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
operator Poisson kernel
bounded semigroup
power bounded operator
square bounded in average - Opis:
- 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.
- Źródło:
-
Banach Center Publications; 1997, 38, 1; 59-74
0137-6934 - Pojawia się w:
- Banach Center Publications
- Dostawca treści:
- Biblioteka Nauki
Artykuł