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## Banach Center Publications

1997 | 38 | 1 | 247-264
Tytuł artykułu

### On the growth of the resolvent operators for power bounded operators

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Treść / Zawartość
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Języki publikacji
EN
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EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
247-264
Opis fizyczny
Daty
wydano
1997
Twórcy
autor
• Institute of Mathematics, Helsinki University of Technology, Otakaari 1, FIN-02150 Espoo, Finland
Bibliografia
• A. Atzmon [1980], Operators which are annihilated by analytic functions and invariant subspaces, Acta Math. 144, 27-63.
• 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.
• M. Berkani [1983], Inégalites et propriétés spectrales dans les algèbres de Banach, Thèse, Université de Bordeaux I.
• R. P. Boas Jr. [1954], Entire Functions, Academic Press.
• 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.
• 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.
• 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.
• W. Feller [1968], An Introduction to Probability Theory and its Applications, Vol. I, Wiley, p. 184 of 3rd edition.
• A. G. Gibson [1972], A discrete Hille-Yosida-Phillips theorem, J. Math. Anal. Appl. 39 761-770.
• W. Hayman [1956], A generalisation of Stirling's formula, J. Reine Angew. Math. 196, 65-97.
• Y. Katznelson and L. Tzafriri [1986], On power bounded operators, J. Funct. Anal. 68, 313-328.
• H.-O. Kreiss [1962], Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT 2, 153-181.
• R. L. LeVeque and L. N. Trefethen [1984], On the resolvent condition in the Kreiss matrix theorem, ibid. 24, 584-591.
• Ch. Lubich and O. Nevanlinna [1991], On resolvent conditions and stability estimates, ibid. 31, 293-313.
• 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.
• O. Nevanlinna [1993], Convergence of Iterations for Linear Equations, Birkhäuser.
• A. Pazy [1983], Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer.
• 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.
• A. L. Shields [1978], On Möbius bounded operators, Acta Sci. Math. (Szeged) 40, 371-374.
• M. N. Spijker [1991], On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem, BIT 31, 551-555.
• J. C. Strikwerda and B. A. Wade [1991], Cesàro means and the Kreiss matrix theorem, Linear Algebra Appl. 145, 89-106.
• 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.
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Bibliografia
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