On the growth of the resolvent operators for power bounded operators

Olavi Nevanlinna

Banach Center Publications (1997)

  • Volume: 38, Issue: 1, page 247-264
  • ISSN: 0137-6934

Abstract

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Outline. In this paper I discuss some quantitative aspects related to power bounded operators T and to the decay of T n ( T - 1 ) . For background I refer to two recent surveys J. Zemánek [1994], C. J. K. Batty [1994]. Here I try to complement these two surveys in two different directions. First, if the decay of T n ( T - 1 ) is as fast as O(1/n) then quite strong conclusions can be made. The situation can be thought of as a discrete version of analytic semigroups; I try to motivate this in Section 1 by demonstrating the similarity and lack of it between power boundedness of T and uniform boundedness of e t ( c T - 1 ) where c is a constant of modulus 1 and t > 0. Section 2 then contains the main result in this direction. I became interested in studying the quantitative aspects of the decay of T n ( T - 1 ) since it can be used as a simple model for what happens in the early phase of an iterative method (O. Nevanlinna [1993]). Secondly, the so called Kreiss matrix theorem relates bounds for the powers to bounds for the resolvent. The estimate is proportional to the dimension of the space and thus has as such no generalization to operators. However, qualitatively such a result holds in Banach spaces e.g. for Riesz operators: if the resolvent satisfies the resolvent condition, then the operator is power bounded operator (but without an estimate). I introduce in Section 3 a growth function for bounded operators. This allows one to obtain a result of the form: if the resolvent condition holds and if the growth function is finite at 1, then the powers are bounded and can be estimated. In Section 4 in addition to the Kreiss matrix theorem, two other applications of the growth function are given.

How to cite

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Nevanlinna, Olavi. "On the growth of the resolvent operators for power bounded operators." Banach Center Publications 38.1 (1997): 247-264. <http://eudml.org/doc/208633>.

@article{Nevanlinna1997,
abstract = {Outline. In this paper I discuss some quantitative aspects related to power bounded operators T and to the decay of $T^\{n\}(T-1)$. For background I refer to two recent surveys J. Zemánek [1994], C. J. K. Batty [1994]. Here I try to complement these two surveys in two different directions. First, if the decay of $T^\{n\}(T-1)$ is as fast as O(1/n) then quite strong conclusions can be made. The situation can be thought of as a discrete version of analytic semigroups; I try to motivate this in Section 1 by demonstrating the similarity and lack of it between power boundedness of T and uniform boundedness of $e^\{t(cT-1)\}$ where c is a constant of modulus 1 and t > 0. Section 2 then contains the main result in this direction. I became interested in studying the quantitative aspects of the decay of $T^\{n\}(T-1)$ since it can be used as a simple model for what happens in the early phase of an iterative method (O. Nevanlinna [1993]). Secondly, the so called Kreiss matrix theorem relates bounds for the powers to bounds for the resolvent. The estimate is proportional to the dimension of the space and thus has as such no generalization to operators. However, qualitatively such a result holds in Banach spaces e.g. for Riesz operators: if the resolvent satisfies the resolvent condition, then the operator is power bounded operator (but without an estimate). I introduce in Section 3 a growth function for bounded operators. This allows one to obtain a result of the form: if the resolvent condition holds and if the growth function is finite at 1, then the powers are bounded and can be estimated. In Section 4 in addition to the Kreiss matrix theorem, two other applications of the growth function are given.},
author = {Nevanlinna, Olavi},
journal = {Banach Center Publications},
keywords = {power bounded operators on a Banach space; growth of the resolvent; growth functions of the resolvent; value distribution theory for analytic functions; semigroups of operators; iterative matrix methods},
language = {eng},
number = {1},
pages = {247-264},
title = {On the growth of the resolvent operators for power bounded operators},
url = {http://eudml.org/doc/208633},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Nevanlinna, Olavi
TI - On the growth of the resolvent operators for power bounded operators
JO - Banach Center Publications
PY - 1997
VL - 38
IS - 1
SP - 247
EP - 264
AB - Outline. In this paper I discuss some quantitative aspects related to power bounded operators T and to the decay of $T^{n}(T-1)$. For background I refer to two recent surveys J. Zemánek [1994], C. J. K. Batty [1994]. Here I try to complement these two surveys in two different directions. First, if the decay of $T^{n}(T-1)$ is as fast as O(1/n) then quite strong conclusions can be made. The situation can be thought of as a discrete version of analytic semigroups; I try to motivate this in Section 1 by demonstrating the similarity and lack of it between power boundedness of T and uniform boundedness of $e^{t(cT-1)}$ where c is a constant of modulus 1 and t > 0. Section 2 then contains the main result in this direction. I became interested in studying the quantitative aspects of the decay of $T^{n}(T-1)$ since it can be used as a simple model for what happens in the early phase of an iterative method (O. Nevanlinna [1993]). Secondly, the so called Kreiss matrix theorem relates bounds for the powers to bounds for the resolvent. The estimate is proportional to the dimension of the space and thus has as such no generalization to operators. However, qualitatively such a result holds in Banach spaces e.g. for Riesz operators: if the resolvent satisfies the resolvent condition, then the operator is power bounded operator (but without an estimate). I introduce in Section 3 a growth function for bounded operators. This allows one to obtain a result of the form: if the resolvent condition holds and if the growth function is finite at 1, then the powers are bounded and can be estimated. In Section 4 in addition to the Kreiss matrix theorem, two other applications of the growth function are given.
LA - eng
KW - power bounded operators on a Banach space; growth of the resolvent; growth functions of the resolvent; value distribution theory for analytic functions; semigroups of operators; iterative matrix methods
UR - http://eudml.org/doc/208633
ER -

References

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  1. A. Atzmon [1980], Operators which are annihilated by analytic functions and invariant subspaces, Acta Math. 144, 27-63. Zbl0449.47007
  2. C. J. K. Batty [1994], Asymptotic behaviour of semigroups of operators, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 35-52. Zbl0818.47034
  3. M. Berkani [1983], Inégalites et propriétés spectrales dans les algèbres de Banach, Thèse, Université de Bordeaux I. 
  4. R. P. Boas Jr. [1954], Entire Functions, Academic Press. 
  5. Ph. Brenner, V. Thomée and L. Wahlbin [1975], Besov Spaces and Applications to Difference Methods for Initial Value Problems, Lecture Notes in Math. 434, Springer. 
  6. J. L. M. van Doersselaer, J. F. B. M. Kraaijevanger and M. N. Spijker [1993], Linear stability analysis in the numerical solution of initial value problems, Acta Numer., 199-237. Zbl0796.65091
  7. J. Esterle [1983], Quasimultipliers, representations of H ∞ , and the closed ideal problem for commutative Banach algebras, in: Lecture Notes in Math. 975, Springer, 66-162. 
  8. W. Feller [1968], An Introduction to Probability Theory and its Applications, Vol. I, Wiley, p. 184 of 3rd edition. Zbl0155.23101
  9. A. G. Gibson [1972], A discrete Hille-Yosida-Phillips theorem, J. Math. Anal. Appl. 39 761-770. Zbl0213.14504
  10. W. Hayman [1956], A generalisation of Stirling's formula, J. Reine Angew. Math. 196, 65-97. Zbl0072.06901
  11. Y. Katznelson and L. Tzafriri [1986], On power bounded operators, J. Funct. Anal. 68, 313-328. Zbl0611.47005
  12. H.-O. Kreiss [1962], Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT 2, 153-181. Zbl0109.34702
  13. R. L. LeVeque and L. N. Trefethen [1984], On the resolvent condition in the Kreiss matrix theorem, ibid. 24, 584-591. Zbl0559.15018
  14. Ch. Lubich and O. Nevanlinna [1991], On resolvent conditions and stability estimates, ibid. 31, 293-313. Zbl0731.65043
  15. C. A. McCarthy [1971], A strong resolvent condition does not imply power-boundedness, Chalmers Institute of Technology and the University of Gothenburg, preprint no. 15. 
  16. O. Nevanlinna [1993], Convergence of Iterations for Linear Equations, Birkhäuser. Zbl0846.47008
  17. A. Pazy [1983], Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer. Zbl0516.47023
  18. A. Pokrzywa [1994], On an infinite-dimensional version of the Kreiss matrix theorem, in: Numerical Analysis and Mathematical Modelling, Banach Center Publ. 29, Inst. Math., Polish Acad. Sci., 45-50. Zbl0814.47036
  19. A. L. Shields [1978], On Möbius bounded operators, Acta Sci. Math. (Szeged) 40, 371-374. Zbl0358.47025
  20. M. N. Spijker [1991], On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem, BIT 31, 551-555. Zbl0736.15015
  21. J. C. Strikwerda and B. A. Wade [1991], Cesàro means and the Kreiss matrix theorem, Linear Algebra Appl. 145, 89-106. Zbl0724.15021
  22. J. Zemánek [1994], On the Gelfand-Hille theorems, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., 369-385. Zbl0822.47005

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