On the generation of Beurling type Carleman ultradifferentiable $C_0$-semigroups by scalar type spectral operators
DOI:
Keywords:
Scalar type spectral operator, $C_0$-semigroup of linear operators, Carleman classes of functions and vectorsAbstract
A characterization of the scalar type spectral generators of Beurling type Carleman ultradifferentiable $C_0$-semigroups is established, the important case of the Gevrey ultradifferentiability is considered in detail, the implementation of the general criterion corresponding to a certain rapidly growing defining sequence is observed.References
J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), no. 2, 370-373. MathSciNet CrossRef
Earl Berkson, Semi-groups of scalar type operators and a theorem of Stone, Illinois J. Math. 10 (1966), 345-352. MathSciNet
T. Carleman, Edition Complete des Articles de Torsten Carleman, Institut Mathematique Mittag-Leffler, Djursholm, Suede, 1960.
Nelson Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958), 217-274. MathSciNet
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Interscience Publishers, New York, 1958. MathSciNet
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers, New York, 1963. MathSciNet
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III: Spectral operators, Interscience Publishers, New York, 1971. MathSciNet
Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. MathSciNet
M. Gevrey, Sur la nature analytique des solutions des equations aux derivees partielles, Ann. Ec. Norm. Sup. Paris 35 (1918), 129-196.
Roe W. Goodman, Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc. 143 (1969), 55-76. MathSciNet
V. I. Gorbachuk, Spaces of infinitely differentiable vectors of a nonnegative self-adjoint operator, Ukrainian Math. J. 35 (1983), no. 5, 531-534. MathSciNet CrossRef
V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Kluwer Academic Publishers, Dordrecht-Boston-London, 1991. MathSciNet CrossRef
V. I. Gorbachuk and A. V. Knyazyuk, Boundary values of solutions of operator-differential equations, Russian Math. Surveys 44 (1989), no. 3, 67-111. MathSciNet CrossRef
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. MathSciNet
Hikosaburo Komatsu, Ultradistributions. I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25-105. MathSciNet
S. Mandelbrojt, Series de Fourier et Classes Quasi-Analytiques de Fonctions, Gauthier-Villars, Paris, 1935.
Marat V. Markin, On an abstract evolution equation with a spectral operator of scalar type, Int. J. Math. Math. Sci. 32 (2002), no. 9, 555-563. MathSciNet CrossRef
Marat V. Markin, A note on the spectral operators of scalar type and semigroups of bounded linear operators, Int. J. Math. Math. Sci. 32 (2002), no. 10, 635-640. MathSciNet CrossRef
Marat V. Markin, On scalar type spectral operators, infinite differentiable and Gevrey ultradifferentiable $C_ 0$-semigroups, Int. J. Math. Math. Sci. (2004), no. 45-48, 2401-2422. MathSciNet CrossRef
Marat V. Markin, On the Carleman classes of vectors of a scalar type spectral operator, Int. J. Math. Math. Sci. 2004 (2004), 3219-3235. MathSciNet CrossRef
M. V. Markin, On scalar-type spectral operators and Carleman ultradifferentiable $C_ 0$-semigroups, Ukrainian Math. J. 60 (2008), no. 9, 1418-1436. MathSciNet CrossRef
Marat V. Markin, On the Carleman ultradifferentiability of weak solutions of an abstract evolution equation, Modern analysis and applications. The Mark Krein Centenary Conference. Vol. 2: Differential operators and mechanics, Oper. Theory Adv. Appl., vol. 191, Birkhauser Verlag, Basel, 2009, pp. 407-443. MathSciNet CrossRef
M. V. Markin, On the Carleman ultradifferentiable vectors of a scalar type spectral operator, Methods Funct. Anal. Topol. 21 (2015), no. 4, 360-369. MFAT Article
Edward Nelson, Analytic vectors, Ann. of Math. 70 (1959), no. 3, 572-615. MathSciNet CrossRef
T. V. Panchapagesan, Semi-groups of scalar type operators in Banach spaces, Pacific J. Math. 30 (1969), 489-517. MathSciNet
A. Pazy, On the differentiability and compactness of semigroups of linear operators, J. Math. Mech. 17 (1968), 1131-1141. MathSciNet
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MathSciNet CrossRef
A. I. Plesner, Spectral theory of linear operators, Izdat. ``Nauka'', Moscow, 1965. MathSciNet
Ya. V. Radyno, The space of vectors of exponential type, Dokl. Akad. Nauk BSSR 27 (1983), no. 9, 791-793. MathSciNet
John Wermer, Commuting spectral measures on Hilbert space, Pacific J. Math. 4 (1954), 355-361. MathSciNet
Kosaku Yosida, On the differentiability of semigroups of linear operators, Proc. Japan Acad. 34 (1958), 337-340. MathSciNet
Kosaku Yosida, Functional analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MathSciNet
