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## Studia Mathematica

1999 | 132 | 2 | 151-172
Tytuł artykułu

### Functional calculi, regularized semigroups and integrated semigroups

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We characterize closed linear operators A, on a Banach space, for which the corresponding abstract Cauchy problem has a unique polynomially bounded solution for all initial data in the domain of $A^n$, for some nonnegative integer n, in terms of functional calculi, regularized semigroups, integrated semigroups and the growth of the resolvent in the right half-plane. We construct a semigroup analogue of a spectral distribution for such operators, and an extended functional calculus: When the abstract Cauchy problem has a unique $O(1 + t^k)$ solution for all initial data in the domain of $A^n$, for some nonnegative integer n, then a closed operator f(A) is defined whenever f is the Laplace transform of a derivative of any order, in the sense of distributions, of a function F such that $t → (1 + t^k)F(t)$ is in $L^{1}([0,∞))$. This includes fractional powers. In general, A is neither bounded nor densely defined.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
151-172
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-03-25
Twórcy
autor
• Scientia Research Institute, P.O. Box 988, Athens, Ohio 45701, U.S.A.
autor
• Département de Mathématiques, Université Libanaise, Hadeth Liban B.P. 11-4015, Beyrouth, Liban
Bibliografia
• [Balak] A. V. Balakrishnan, An operational calculus for infinitesimal generators of semigroups, Trans. Amer. Math. Soc. 91 (1959), 330-353.
• [Balab-E-J] M. Balabane, H. Emamirad and M. Jazar, Spectral distributions and generalization of Stone's theorem to the Banach space, Acta Appl. Math. 31 (1993), 275-295.
• [d1] R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Math. 1570, Springer, 1994.
• [d2] R. deLaubenfels, Automatic extensions of functional calculi, Studia Math. 114 (1995), 237-259.
• [d3] R. deLaubenfels, Functional calculi, semigroups of operators, and Hille-Yosida operators, Houston J. Math. 22 (1996), 787-805.
• [d-H-W-W] R. deLaubenfels, Z. Huang, S. Wang and Y. Wang, Laplace transforms of polynomially bounded vector-valued functions and semigroups of operators, Israel J. Math. 98 (1997), 189-207.
• [d-Y-W] R. deLaubenfels, F. Yao and S. Wang, Fractional powers of operators of regularized type, J. Math. Appl. 199 (1996), 910-933.
• [E-J] H. Emamirad and M. Jazar, Applications of spectral distributions to some Cauchy problems in $L^p(ℝ^n)$, in: Semigroup Theory and Evolution Equations: the Second International Conference, Delft 1989, Lecture Notes in Pure and Appl. Math. 135, Marcel Dekker, 1991, 143-151.
• [G] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, 1985.
• [L] Y. C. Li, Integrated C-semigroups and C-cosine functions of operators on locally convex spaces, Ph.D. dissertation, National Central University, 1991.
• [L-Sha] Y. C. Li and S. Y. Shaw, N-times integrated C-semigroups and the abstract Cauchy problem, Taiwanese J. Math. 1 (1997), 75-102.
• [Nee-St] J. M. A. M. van Neerven and B. Straub, On the existence and growth of mild solutions of the abstract Cauchy problem for operators with polynomially bounded resolvents, Houston J. Math. 24 (1998), 137-171.
• [Nel] E. Nelson, A functional calculus using singular Laplace integrals, Trans. Amer. Math. Soc. 88 (1958), 400-413.
• [P] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
• [Sho-T] J. A. Shohat and J. D. Tamarkin, The Problem of Moments, Amer. Math. Soc. Math. Surveys 1, Amer. Math. Soc., 1943.
• [St] B. Straub, Fractional powers of operators with polynomially bounded resolvent and the semigroups generated by them, Hiroshima Math. J. 24 (1994), 529-548.
• [W] S. Wang, Mild integrated C-existence families, Studia Math. 112 (1995), 251-266.
• [Q-Liu] Q. Zheng and L. Liu, Almost periodic C-groups, C-semigroups, and C-cosine functions, J. Math. Anal. Appl. 197 (1996), 90-112.
Typ dokumentu
Bibliografia
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