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On the Carleman ultradifferentiable vectors of a scalar type spectral operator


Abstract

A description of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a reflexive complex Banach space is shown to remain true without the reflexivity requirement. A similar nature description of the entire vectors of exponential type, known for a normal operator in a complex Hilbert space, is generalized to the case of a scalar type spectral operator in a complex Banach space.

Key words: Scalar type spectral operator, normal operator, Carleman classes of vectors.


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Article Information

TitleOn the Carleman ultradifferentiable vectors of a scalar type spectral operator
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 4, 361-369
MathSciNet MR3469533
zbMATH 06630279
MilestonesReceived 04/06/2015
CopyrightThe Author(s) 2015 (CC BY-SA)

Authors Information

Marat V. Markin
Department of Mathematics, California State University, Fresno 5245 N. Backer Avenue, M/S PB 108 Fresno, CA 93740-8001


Citation Example

Marat V. Markin, On the Carleman ultradifferentiable vectors of a scalar type spectral operator, Methods Funct. Anal. Topology 21 (2015), no. 4, 361-369.


BibTex

@article {MFAT838,
    AUTHOR = {Markin, Marat V.},
     TITLE = {On the Carleman ultradifferentiable vectors of a scalar type spectral operator},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {4},
     PAGES = {361-369},
      ISSN = {1029-3531},
  MRNUMBER = {MR3469533},
 ZBLNUMBER = {06630279},
       URL = {http://mfat.imath.kiev.ua/article/?id=838},
}


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References

  1. Nelson Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958), 217-274.  MathSciNet
  2. Nelson Dunford, Jacob T. Schwartz, Linear operators. Part I, John Wiley & Sons, Inc., New York, 1988.  MathSciNet
  3. Nelson Dunford, Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons\ New York-London, 1963.  MathSciNet
  4. Nelson Dunford, Jacob T. Schwartz, Linear operators. Part III, John Wiley & Sons, Inc., New York, 1988.  MathSciNet
  5. Reinhard Farwig, Erich Marschall, On the type of spectral operators and the nonspectrality of several differential operators on $L^p$, Integral Equations Operator Theory 4 (1981), no. 2, 206-214.  MathSciNet CrossRef
  6. Roe W. Goodman, Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc. 143 (1969), 55-76.  MathSciNet
  7. M. L. Gorbachuk, V. I. Gorbachuk, On the approximation of smooth vectors of a closed operator by entire vectors of exponential type, Ukrain. Mat. Zh. 47 (1995), no. 5, 616-628.  MathSciNet CrossRef
  8. Myroslav L. Gorbachuk, Valentyna I. Gorbachuk, On the well-posed solvability in some classes of entire functions of the Cauchy problem for differential equations in a Banach space, Methods Funct. Anal. Topology 11 (2005), no. 2, 113-125.  MathSciNet
  9. Myroslav L. Gorbachuk, Valentyna I. Gorbachuk, On completeness of the set of root vectors for unbounded operators, Methods Funct. Anal. Topology 12 (2006), no. 4, 353-362.  MathSciNet
  10. V. I. Gorbachuk, Spaces of infinitely differentiable vectors of a nonnegative selfadjoint operator, Ukrain. Mat. Zh. 35 (1983), no. 5, 617-621.  MathSciNet
  11. V. I. Gorbachuk, M. L. Gorbachuk, Boundary value problems for operator differential equations, Kluwer Academic Publishers Group, Dordrecht, 1991.  MathSciNet CrossRef
  12. V. I. Gorbachuk, A. V. Knyazyuk, Boundary values of solutions of operator-differential equations, Uspekhi Mat. Nauk 44 (1989), no. 3(267), 55-91, 208.  MathSciNet CrossRef
  13. S. Mandelbrojt, Series de Fourier et Classes Quasi-Analytiques de Fonctions, Gauthier-Villars, Paris, 1935.
  14. 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
  15. Marat V. Markin, On the Carleman classes of vectors of a scalar type spectral operator, Int. J. Math. Math. Sci. (2004), no. 57-60, 3219-3235.  MathSciNet CrossRef
  16. M. V. Markin, On scalar-type spectral operators and Carleman ultradifferentiable $C_ 0$-semigroups, Ukrain. Mat. Zh. 60 (2008), no. 9, 1215-1233.  MathSciNet CrossRef
  17. Marat V. Markin, On the Carleman ultradifferentiability of weak solutions of an abstract evolution equation, in: Modern analysis and applications. The Mark Krein Centenary Conference. Vol. 2: Differential operators and mechanics, Birkhauser Verlag, Basel, 2009.  MathSciNet CrossRef
  18. Marat V. Markin, On the generation of Beurling type Carleman ultradifferentiable $C_0$-semigroups by scalar type spectral operators, Methods Funct. Anal. Topology (to appear).
  19. Edward Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572-615.  MathSciNet
  20. Raymond E. A. C. Paley, Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society, Providence, RI, 1987.  MathSciNet
  21. A. I. Plesner, Spektralnaya teoriya lineinykh operatorov, Izdat. ``Nauka'', Moscow, 1965.  MathSciNet
  22. Ya. V. Radyno, The space of vectors of exponential type, Dokl. Akad. Nauk BSSR 27 (1983), no. 9, 791-793.  MathSciNet
  23. John Wermer, Commuting spectral measures on Hilbert space, Pacific J. Math. 4 (1954), 355-361.  MathSciNet


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