Constant size control in stability estimates under some resolvent conditions

Authors

  • N.Yu. Bakaev

Keywords:

оценки устойчивости
резольвентные условия
константы устойчивости

Abstract

The paper deals with the question of stability of a discrete semigroup under certain resolvent conditions on its generator. The main objective is to examine the behavior of the stability constants as functions of the constant in the original resolvent estimate.

New computational technologies

Downloads

Published

2003-10-18

Issue

Vol. 4 (2003): Issue 1.

Section

Section 1. Numerical methods and applications

Author Biography

N.Yu. Bakaev

Russian State Social University

References

  1. Bakaev N.Yu. Stability theory of difference schemes in arbitrary norms // Dokl. Akad. Nauk SSSR. 1987. 297. 275-279 (in Russian); English transl. in Soviet Math. Dokl. 1988. 36.
  2. Bakaev N.Yu. Stability theory of difference schemes for parabolic equations in arbitrary norms. Part 1 // Vopr. Atomn. Nauki i Tekhniki, ser. Metodi i Progr. Chisl. Reshen. Zadach Matem. Fiz. 1987. Iss. 1. 69-75 (in Russian).
  3. Bakaev N.Yu. Stability theory of difference schemes for parabolic equations in arbitrary norms. Part 2 // Vopr. Atomn. Nauki i Tekhniki, ser. Metodi i Progr. Chisl. Reshen. Zadach Matem. Fiz. 1987. Iss. 2. 29-34 (in Russian).
  4. Bakaev N.Yu. Stability estimates for a certain general discretization method // Dokl. Akad. Nauk SSSR. 1989. 309. 11-15 (in Russian); English transl. in Soviet Math. Dokl. 1990. 40.
  5. Bakaev N.Yu. Stability estimates of difference schemes for a differential equation with constant operator. I // Partial Diff. Eqs. Novosibirsk: SO AN SSSR, 1989. 3-14 (in Russian).
  6. Bakaev N.Yu. Analysis of linear discrete parabolic problems. Part 1. Well-posedness of a Cauchy problem for the evolution equation in discrete time // Comput. Math. Reports. Valladolid, Spain: Departamento de Matem’atica Aplicada y Computaci’on, Universidad de Valladolid, 1997. Report 1997/1.
  7. Borovykh N., Drissi D., Spijker M.N. A note about Ritt’s condition, related resolvent conditions and power bounded operators // Numer. Funct. Anal. and Optimiz. 2000. 21. 425-438.
  8. El-Fallah O., Ransford T. Extremal growth of powers of operators satisfying resolvent conditions of Kreiss-Ritt type // J. Fuction. Anal. 2002. 196. 135-154.
  9. Hille E., Phillips R.S. Functional analysis and semi-groups. Providence: AMS, Colloquium Publs, Vol. XXXI, 1957.
  10. Kalton N., Montgomery-Smith S., Oleszkiewicz K., Tomilov Yu. Power-bounded operators and related norm estimates. To appear.
  11. Komatsu H. Fractional powers of operators // Pacific J. Math. 1966. 19. 285-346.
  12. Kreiss H.O. Über die stabilitätsdefinition für differenzengleichungen die partielle differentialgleichungen approximieren // BIT. 1962. 2. 153-181.
  13. Le Veque R.J., Trefethen L.N. On the resolvent condition in the Kreiss matrix theorem // BIT. 1984. 24. 584-591.
  14. Lubich C., Nevanlinna O. On resolvent conditions and stability estimates // BIT. 1991. 31. 293-313.
  15. Lyubich Yu. Spectral localization, power boundedness and invariant subspaces under Ritt’s type condition // Studia Math. 1999. 134. 153-167.
  16. Miller J., Strang G. Matrix theorems for partial differential and difference equations // Math. Scand. 1966. 18. 113-123.
  17. Morton K.W. On a matrix theorem due to H.-O. Kreiss // Comm. Pure Appl. Math. 1964. 17. 375-379.
  18. Sz.-Nagy B., Foiacs C. Analyse Harmonique des Opérateurs de l’Espace de Hilbert. Masson et mboxC愦灭;circmboxie: Académiai Kiad’o, 1967.
  19. Nagy B., Zem’anek J. A resolvent condition implying power boundedness // Studia Math. 1999. T. 134(2). 143-151.
  20. Nevanlinna O. Converegence of Iterations of Linear Equations. Birkhäuser Verlag: Basel-Boston-Berlin, 1993.
  21. Nevanlinna O. On the growth of the resolvent operators for power bounded operators // Banach Center Publ. Issue 38. Warsaw: Inst. Math., Polish Acad. Sci., 1997. 247-264.
  22. Nevanlinna O. Resolvent conditions and powers of operators // Studia Math. 2001. T. 145(2). 113-134.
  23. Ritt R.K. A condition that limlimitsn ightarrow infty n愦灭;circ-1 T愦灭;circn =0 // Proc. Amer. Math. Soc. 1953. 4. 898-899.
  24. Spijker M.N. On a conjuncture by LeVeque and Trefethen related to the Kreiss matrix theorem // BIT. 1991. 31. 551-555.
  25. Spijker M.N., Straetemans F.A. J. Stability estimates for families of matrices of nonuniformly bounded order // Linear Algebra Appl. 1996. 239. 77-102.
  26. Tadmor E. The equivalence of L_2-stability, the resolvent condition, and strict H-stability // Linear Algebra Appl. 1981. 40. 151-159.
  27. Tadmor E. The resolvent condition and uniform power boundedness // Linear Algebra Appl. 1986. 80. 250-252.
  28. Vitse P. Functional calculus under the Tadmor-Ritt condition, and free interpolation by polynomials of a given degree. Submitted.
  29. Vitse P. The Riesz turndown theorem giving an asymptotic estimate of the powers of an operator under the Ritt condition. Submitted.
  30. Yuan Zh. On the Resolvent and Tauberian Conditions for Bounded Linear Operators. Licentiate’s Thesis, Helsinki University of Technology, January 2002.