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## Studia Mathematica

1992 | 103 | 2 | 143-159
Tytuł artykułu

### Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform

Autorzy
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Języki publikacji
EN
Abstrakty
EN
Suppose A is a (possibly unbounded) linear operator on a Banach space. We show that the following are equivalent. (1) A is well-bounded on [0,∞). (2) -A generates a strongly continuous semigroup ${e^{-sA}}_{s≤0}$ such that ${(1/s^2)e^{-sA}}_{s>0}$ is the Laplace transform of a Lipschitz continuous family of operators that vanishes at 0. (3) -A generates a strongly continuous differentiable semigroup ${e^{-sA}}_{s≥0}$ and ∃ M < ∞ such that $∥H_n(s)∥ ≡ ∥(∑_{k=0}^n (s^k A^{k})/k!) e^{-sA}∥ ≤ M$, ∀s > 0, n ∈ ℕ ∪ {0}. (4) -A generates a strongly continuous holomorphic semigroup ${e^{-zA}}_{Re(z)>0}$ that is O(|z|) in all half-planes Re(z) > a > 0 and $K(t) ≡ ʃ_{1+iℝ} e^{zt} e^{-zA} dz/(2πiz^3)$ defines a differentiable function of t, with Lipschitz continuous derivative, with K'(0) = 0. We may then construct a decomposition of the identity, F, for A, from K(t) or $H_n(s)$. For ϕ ∈ X*, x ∈ X, $(F(t)ϕ)(x) = (d/dt)^2 (ϕ(K(t)x)) = lim_{n→∞} ϕ(H_n(n/t)x)$, for almost all t.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
143-159
Opis fizyczny
Daty
wydano
1992
otrzymano
1991-01-16
poprawiono
1992-01-22
Twórcy
autor
• Department of Mathematics, Ohio University, Athens, Ohio 45701, U.S.A.
Bibliografia
• [1] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352.
• [2] H. Benzinger, E. Berkson and T. A. Gillespie, Spectral families of projections, semigroups, and differential operators, Trans. Amer. Math. Soc. 275 (1983), 431-475.
• [3] E. Berkson, Semigroups of scalar type operators and a theorem of Stone, Illinois J. Math. 10 (1966), 345-352.
• [4] R. deLaubenfels, Scalar-type spectral operators and holomorphic semigroups, Semigroup Forum 33 (1986), 257-263.
• [5] H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, 1978.
• [6] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Interscience, New York 1971.
• [7] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, 1985.
• [8] F. Neubrander and B. Hennig, On representations, inversions and approximations of Laplace transforms in Banach spaces, Resultate Math., to appear.
• [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York 1983.
• [10] D. J. Ralph, Semigroups of well-bounded operators and multipliers, Thesis, Univ. of Edinburgh, 1977.
• [11] W. Ricker, Spectral properties of the Laplace operator in $L^p(ℝ)$, Osaka J. Math. 25 (1988), 399-410.
• [12] A. R. Sourour, Semigroups of scalar type operators on Banach spaces, Trans. Amer. Math. Soc. 200 (1974), 207-232.
• [13] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton 1970.
• [14] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton 1946.
Typ dokumentu
Bibliografia
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