Necessary and sufficient conditions for a scalar-type spectral operator in a Banach space to be a generator of a Carleman ultradifferentiable C 0-semigroup are found. The conditions are formulated exclusively in terms of the spectrum of the operator.
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M. V. Markin, “A note on the spectral operators of scalar type and semigroups of bounded linear operators,” Int. J. Math. Math. Sci., 32, No. 10, 635–640 (2002).
M. V. Markin, “On scalar-type spectral operators, infinite differentiable and Gevrey ultradifferentiable C 0-semigroups,” Int. J. Math. Math. Sci., No. 45, 2401–2422 (2004).
E. Hille and R. S. Phillips, “Functional analysis and semigroups,” Amer. Math. Soc. Colloq. Publ., Amer. Math. Soc., RI, 31 (1957).
K. Yosida, “On the differentiability of semi-groups of linear operators,” Proc. Jpn. Acad., 34, 337–340 (1958).
K. Yosida, “Functional analysis,” Grundlehren Math., Acad. Press, Wissenschaften, New York, 123 (1965).
A. Pazy, “On the differentiability and compactness of semi-groups of linear operators,” J. Math. Mech., 17, No. 12, 1131–1141 (1968).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983).
K.-J. Engel and R. Nagel, “One-parameter semigroups for linear evolution equations,” Grad. Texts Math., 194 (2000).
J. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York (1985).
M. V. Markin, “On the ultradifferentiability of weak solutions of a first-order operator-differential equation in Hilbert space,” Dopov. Akad. Nauk Ukr., No. 6, 22–26 (1996).
M. V. Markin, “On the smoothness of weak solutions of an abstract evolution equation. I. Differentiability,” Appl. Anal., 73, No. 3–4, 573–606 (1999).
M. V. Markin, “On the smoothness of weak solutions of an abstract evolution equation. II. Gevrey ultradifferentiability,” Appl. Anal., 78, No. 1–2, 97–137 (2001).
M. V. Markin, “On the smoothness of weak solutions of an abstract evolution equation. III. Gevrey ultradifferentiability of orders less than one,” Appl. Anal., 78, No. 1–2, 139–152 (2001).
N. Dunford, “Survey of the theory of spectral operators,” Bull. Amer. Math. Soc., 64, 217–274 (1958).
N. Dunford, Linear Operators. Part III: Spectral Operators, Int. Publ., New York (1971).
J. Wermer, “Commuting spectral measures on Hilbert space,” Pacif. J. Math., 4, 355–361 (1954).
N. Dunford, Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space, Int. Publ., New York (1963).
A. I. Plesner, Spectral Theory of Linear Operators [in Russian], Nauka, Moscow (1965).
N. Dunford and J. T. Schwartz, “Linear operators. I. General theory,” Pure Appl. Math., 7 (1958).
M. V. Markin, “On the Carleman classes of vectors of a scalar-type spectral operator,” Int. J. Math. Math. Sci., No. 60, 3219–3235 (2004).
J. M. Ball, “Strongly continuous semigroups, weak solutions, and the variation of constants formula, ” Proc. Amer. Math. Soc., 63, 370–373 (1977).
M. V. Markin, “On an abstract evolution equation with a spectral operator of scalar type,” Int. J. Math. Math. Sci., 32, No. 9, 555–563 (2002).
T. Carleman, Édition Complète des Articles de Torsten Carleman, Inst. Math. Mittag-Leffler, Djursholm, Suède (1960).
H. Komatsu, “Ultradistributions. I. Structure theorems and characterization,” J. Fac. Sci. Univ. Tokyo, 20, 25–105 (1973).
S. Mandelbrojt, Series de Fourier et Classes Quasi-Analytiques de Fonctions, Gauthier-Villars, Paris (1935).
M. Gevrey, “Sur la nature analytique des solutions des équations aux dérivées partielles,” Ann. Econe Norm. Super. Paris, 35, 129–196 (1918).
M. L. Gorbachuk and V. I. Gorbachuk, Boundary-Value Problems for Operator Differential Equations, Kluwer, Dordrecht (1991).
V. I. Gorbachuk, “Spaces of infinitely differentiable vectors of a nonnegative self-adjoint operator, ” Ukr. Math. J., 1983, 35, No. 5, 531–535.
V. I. Gorbachuk and A. V. Knyazyuk, “Boundary values of solutions of operator differential equations, ” Russ. Math. Surv., 44, 67–111 (1989).
E. Nelson, “Analytic vectors,” Ann. Math., 70, 572–615 (1959).
R. Goodman, “Analytic and entire vectors for representations of Lie groups,” Trans. Amer. Math. Soc., 143, 55–76 (1969).
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Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1215–1233, September, 2008.
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Markin, M.V. On scalar-type spectral operators and Carleman ultradifferentiable C 0-semigroups. Ukr Math J 60, 1418–1436 (2008). https://doi.org/10.1007/s11253-009-0141-0
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DOI: https://doi.org/10.1007/s11253-009-0141-0