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

deLaubenfels, Ralph

Studia Mathematica, Tome 103 (1992), p. 143-159 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ${e^{- s A}}_{s \leq 0}$ such that ${(1 / s^{2}) e^{- s A}}_{s > 0}$ is the Laplace transform of a Lipschitz continuous family of operators that vanishes at 0. (3) -A generates a strongly continuous differentiable semigroup ${e^{- s A}}_{s \geq 0}$ and ∃ M < ∞ such that $∥ H_{n} (s) ∥ \equiv ∥ (\sum_{k = 0}^{n} (s^{k} A^{k}) / k!) e^{- s A} ∥ \leq M$ , ∀s > 0, n ∈ ℕ ∪ 0. (4) -A generates a strongly continuous holomorphic semigroup ${e^{- z A}}_{R e (z) > 0}$ that is O(|z|) in all half-planes Re(z) > a > 0 and $K (t) \equiv ʃ_{1 + i ℝ} e^{z t} e^{- z A} d z / (2 π i z^{3})$ 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, $(F (t) ϕ) (x) = {(d / d t)}^{2} (ϕ (K (t) x)) = l i m_{n \to \infty} ϕ (H_{n} (n / t) x)$ , for almost all t.

Publié le : 1992-01-01

Zbl 0811.47012

EUDML-ID : urn:eudml:doc:215942

@article{bwmeta1.element.bwnjournal-article-smv103i2p143bwm,
     author = {Ralph deLaubenfels},
     title = {Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform},
     journal = {Studia Mathematica},
     volume = {103},
     year = {1992},
     pages = {143-159},
     zbl = {0811.47012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv103i2p143bwm}
}

deLaubenfels, Ralph. Unbounded well-bounded operators, strongly continuous semigroups and the Laplace transform. Studia Mathematica, Tome 103 (1992) pp. 143-159. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv103i2p143bwm/

Bibliographie

[00000] [1] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352. | Zbl 0637.44001

[00001] [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. | Zbl 0509.47028

[00002] [3] E. Berkson, Semigroups of scalar type operators and a theorem of Stone, Illinois J. Math. 10 (1966), 345-352. | Zbl 0141.13004

[00003] [4] R. deLaubenfels, Scalar-type spectral operators and holomorphic semigroups, Semigroup Forum 33 (1986), 257-263. | Zbl 0583.47040

[00004] [5] H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, 1978. | Zbl 0384.47001

[00005] [6] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Interscience, New York 1971.

[00006] [7] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, 1985. | Zbl 0592.47034

[00007] [8] F. Neubrander and B. Hennig, On representations, inversions and approximations of Laplace transforms in Banach spaces, Resultate Math., to appear.

[00008] [9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York 1983.

[00009] [10] D. J. Ralph, Semigroups of well-bounded operators and multipliers, Thesis, Univ. of Edinburgh, 1977.

[00010] [11] W. Ricker, Spectral properties of the Laplace operator in $L^{p} (ℝ)$ , Osaka J. Math. 25 (1988), 399-410. | Zbl 0706.47028

[00011] [12] A. R. Sourour, Semigroups of scalar type operators on Banach spaces, Trans. Amer. Math. Soc. 200 (1974), 207-232. | Zbl 0304.47034

[00012] [13] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton 1970. | Zbl 0207.13501

[00013] [14] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton 1946. | Zbl 0060.24801