A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions
Strikwerda, John ; Wade, Bruce
Banach Center Publications, Tome 38 (1997), p. 339-360 / Harvested from The Polish Digital Mathematics Library

We survey results related to the Kreiss Matrix Theorem, especially examining extensions of this theorem to Banach space and Hilbert space. The survey includes recent and established results together with proofs of many of the interesting facts concerning the Kreiss Matrix Theorem.

Publié le : 1997-01-01
EUDML-ID : urn:eudml:doc:208640
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     title = {A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions},
     journal = {Banach Center Publications},
     volume = {38},
     year = {1997},
     pages = {339-360},
     zbl = {0877.15029},
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Strikwerda, John; Wade, Bruce. A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions. Banach Center Publications, Tome 38 (1997) pp. 339-360. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv38i1p339bwm/

[000] [1] N. K. Bari, A Treatise on Trigonometric Series, Volume II, 2nd ed., Pergamon Press (1964) | Zbl 0154.06103

[001] [2] C. A. Berger and J. G. Stampfli, Mapping theorems for the numerical range, Amer. J. Math. 89 (1967), 1047-1055 | Zbl 0164.16602

[002] [3] B. Bollobás, The power inequality on Banach spaces, Proc. Cambridge Philos. Soc. 69 (1971), 411-415 | Zbl 0216.16404

[003] [4] F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, Cambridge University Press (1971) | Zbl 0207.44802

[004] [5] F. F. Bonsall and J. Duncan, Numerical Ranges II, Cambridge University Press (1973) | Zbl 0262.47001

[005] [6] P. Brenner and V. Thomée, Stability and convergence rates in Lp for certain difference schemes, Math. Scand. 27 (1970), 5-23 | Zbl 0208.16201

[006] [7] P. Brenner and V. Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979), 683-694 | Zbl 0413.41011

[007] [8] P. Brenner, V. Thomée, and L. B. Wahlbin, Besov Spaces and Applications to Difference Methods for Initial Value Problems, Lecture Notes in Math. 434, Springer, New York (1975) | Zbl 0294.35002

[008] [9] M. L. Buchanon, A necessary and sufficient condition for stability of difference schemes for initial value problems, SIAM J. Appl. Math. 11 (1963), 919-935

[009] [10] M. J. Crabb, Numerical range estimates for the norms of iterated operators, Glasgow Math. J. 11 (1970), 85-87 | Zbl 0244.47002

[010] [11] M. J. Crabb, The power inequality on normed spaces, Proc. Edinburgh Math. Soc. 17 (1971), 237-240 | Zbl 0219.47004

[011] [12] A. J. Chorin, T. J. R. Hughes, M. F. McCracken, and J. E. Marsden, Product formulas and numerical algorithms, Comm. Pure Appl. Math. 31 (1978), 205-256 | Zbl 0358.65082

[012] [13] M. Crouzeix, S. Larsson, S. Piskarev and V. Thomée, The stability of rational approximations of analytic semigroups, BIT 33 (1993), 74-84 | Zbl 0783.65050

[013] [14] G. Dahlquist, H. Mingyou, and R. LeVeque, On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods, Numer. Math. 42 (1983), 1-13 | Zbl 0526.65051

[014] [15] E. B. Davies, One-Parameter Semigroups, Academic Press (1980) | Zbl 0457.47030

[015] [16] J. L. M. van Dorsselaer, J. F. B. M. Kraaijevanger, and M. N. Spijker, Linear stability analysis in the numerical solution of initial value problems, Acta Numerica (1993), 199-237 | Zbl 0796.65091

[016] [17] S. R. Foguel, A counterexample to a problem of Sz.-Nagy, Proc. Amer. Math. Soc. 15 (1964), 788-790 | Zbl 0124.06602

[017] [18] S. Friedland, A generalization of the Kreiss matrix theorem, SIAM J. Math. Anal. 12 (1981), 826-832 | Zbl 0467.65013

[018] [19] M. Goldberg and E. Tadmor, On the numerical radius and its applications, Linear Algebra Appl. 42 (1982), 263-284 | Zbl 0479.47002

[019] [20] M. Gorelick and H. Kranzer, An extension of the Kreiss stability theorem to families of matrices of unbounded order, Linear Algebra Appl. 14 (1976), 237-256 | Zbl 0339.65020

[020] [21] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM (1977) | Zbl 0412.65058

[021] [22] P. R. Halmos, On Foguel's answer to Nagy's question, Proc. Amer. Math. Soc. 15 (1964), 791-793 | Zbl 0123.09701

[022] [23] P. R. Halmos, Ten Problems in Hilbert Space, Bull. Amer. Math. Soc. 76 (1970), 887-933 | Zbl 0204.15001

[023] [24] P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer (1982)

[024] [25] G. Hedstrom, Norms of powers of absolutely convergent Fourier series, Michigan Math. J. 13 (1966), 393-416

[025] [26] R. Hersh and T. Kato, High-accuracy stable difference schemes for well-posed initial-value problems, SIAM J. Numer. Anal. 16 (1979), 670-682 | Zbl 0419.65036

[026] [27] T. Kato, Estimation of iterated matrices, with applications to the von Neumann condition, Numer. Math. 2 (1960), 22-29 | Zbl 0119.32001

[027] [28] T. Kato, Some mapping theorems for the numerical range, Proc. Japan Acad. 41 (1965), 652-655 | Zbl 0143.36702

[028] [29] J. F. B. M. Kraaijevanger, Two counterexamples related to the Kreiss matrix theorem, BIT 34 (1994), 113-119 | Zbl 0842.15013

[029] [30] H.-O. Kreiss, Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, Nord. Tidskr. Inf. (BIT) 2 (1962), 153-181 | Zbl 0109.34702

[030] [31] H.-O. Kreiss, Über sachgemässe Cauchyprobleme, Math. Scand. 13 (1963), 109-128 | Zbl 0145.13303

[031] [32] H.-O. Kreiss, On difference approximations of the dissipative type for hyperbolic differential equations, Comm. Pure Appl. Math. 17 (1964), 335-353 | Zbl 0279.35059

[032] [33] E. G. Landau, Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie, in: Das Kontinuum, und andere Monographien, 2nd ed., Chelsea Publishing Company, 1929 | Zbl 55.0171.03

[033] [34] P. D. Lax and L. Nirenberg, On stability of difference schemes; a sharp form of Gå rding's inequality, Comm. Pure Appl. Math. 19 (1966), 473-492 | Zbl 0185.22801

[034] [35] H. W. J. Lenferink and M. N. Spijker, A generalization of the numerical range of a matrix, Linear Algebra Appl. 140 (1990), 251-266 | Zbl 0712.15027

[035] [36] H. W. J. Lenferink and M. N. Spijker, On a generalization of the resolvent condition in the Kreiss Matrix Theorem, Math. Comp. 57 (1991), 211-220 | Zbl 0726.15020

[036] [37] H. W. J. Lenferink and M. N. Spijker, On the use of stability regions in the numerical analysis of initial value problems, ibid. 57 (1991), 221-237 | Zbl 0727.65072

[037] [38] R. L. LeVeque and L. N. Trefethen, On the resolvent condition in the Kreiss matrix theorem, Nord. Tidskr. Inf. Beh. (BIT) 24 (1984), 584-591 | Zbl 0559.15018

[038] [39] C. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), 293-313 | Zbl 0731.65043

[039] [40] C. A. McCarthy, A strong resolvent condition does not imply power-boundedness, Chalmers Inst. of Tech. and Univ. of Göteborg, preprint # 15 (1971)

[040] [41] C. A. McCarthy and J. Schwartz, On the norm of a finite boolean algebra of projections, and applications to theorems of Kreiss and Morton, Comm. Pure Appl. Math. 18 (1965), 191-201 | Zbl 0151.19401

[041] [42] D. Michelson, Stability theory of difference approximations for multi-dimensional initial-boundary value problems, Math. Comp. 40 (1983), 1-45 | Zbl 0563.65064

[042] [43] J. J. H. Miller, On power bounded operators and operators satisfying a resolvent condition, Numer. Math. 10 (1967), 389-396 | Zbl 0166.41504

[043] [44] J. Miller and G. Strang, Matrix theorems for partial differential equations, Math. Scand. 18 (1966), 113-123 | Zbl 0144.13404

[044] [45] K. W. Morton, On a matrix theorem due to H.-O. Kreiss, Comm. Pure Appl. Math. 17 (1964), 375-380 | Zbl 0146.13702

[045] [46] K. W. Morton and S. Schechter, On the stability of finite difference matrices, SIAM J. Numer. Anal. Ser. B 2 (1965), 119-128 | Zbl 0133.38101

[046] [47] O. Nevanlinna, Convergence of Iterations for Linear Equations, Lectures in Math., Birkhäuser, Basel (1993) | Zbl 0846.47008

[047] [48] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer (1983)

[048] [49] C. Pearcy, An elementary proof of the power inequality for the numerical radius, Michigan Math. J. 13 (1966), 289-291 | Zbl 0143.16205

[049] [50] A. Pokrzywa, 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., Warszawa, 1994, 45-50 | Zbl 0814.47036

[050] [51] S. C. Reddy and L. N. Trefethen, Stability of the method of lines, Numer. Math. 62 (1992), 235-267 | Zbl 0734.65077

[051] [52] R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed., Wiley Interscience (1967) | Zbl 0155.47502

[052] [53] A. L. Shields, On Möbius bounded operators, Acta Sci. Math. (Szeged) 40 (1978), 371-374 | Zbl 0358.47025

[053] [54] H. Shintani and K. Toemeda, Stability of difference schemes for nonsymmetric linear hyperbolic systems with variable coefficients, Hiroshima Math. J. 7 (1977), 309-78

[054] [55] M. N. Spijker, On a conjecture by LeVeque and Trefethen, BIT 31 (1991), 551-555

[055] [56] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks/Cole, Pacific Grove, Calif. (1989)

[056] [57] J. C. Strikwerda and B. A. Wade, An extension of the Kreiss matrix theorem, SIAM J. Numer. Anal. 25 (1988), 1272-1278 | Zbl 0667.65074

[057] [58] J. C. Strikwerda and B. A. Wade, Cesàro means and the Kreiss matrix theorem, Linear Algebra Appl. 145 (1991), 89-106 | Zbl 0724.15021

[058] [59] B. Sz.-Nagy and C. Foiaş, On certain classes of power-bounded operators in Hilbert space, Acta Sci. Math. (Szeged) 27 (1966), 17-25 | Zbl 0141.32201

[059] [60] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, North-Holland (1970)

[060] [61] E. Tadmor, The equivalence of L2-stability, the resolvent condition, and strict H-stability, Linear Algebra Appl. 41 (1981), 151-159 | Zbl 0469.15011

[061] [62] E. Tadmor, Complex symmetric matrices with strongly stable iterates, ibid. 78 (1986), 65-77 | Zbl 0591.15018

[062] [63] E. Tadmor, Stability analysis of finite-difference, pseudospectral and Fourier-Galerkin approximations for time dependent problems, SIAM Rev. 29 (1987), 525-555 | Zbl 0646.65072

[063] [64] V. Thomée, Stability theory for partial differential operators, ibid. 11 (1969), 152-195

[064] [65] E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press (1979) | Zbl 0005.21004

[065] [66] B. A. Wade, Stability and sharp convergence estimates for symmetrizable difference operators, Ph.D. dissertation, University of Wisconsin-Madison, 1987

[066] [67] B. A. Wade, Symmetrizable finite difference operators, Math. Comp. 54 (1990), 525-543 | Zbl 0697.65069

[067] [68] O. B. Widlund, On the stability of parabolic difference schemes, ibid. 19 (1965), 1-13 | Zbl 0125.07402

[068] [69] M. Yamaguti and T. Nogi, An algebra of pseudo difference schemes and its applications, Publ. Res. Inst. Math. Sci. Kyoto University 3 (1967), 151-66 | Zbl 0182.18601

[069] [70] K. Yosida, Functional Analysis, 6th ed., Springer, New York, 1980

[070] [71] J. Zemánek, On the Gelfand-Hille theorems, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 369-385 | Zbl 0822.47005

[071] [72] A. Zygmund, Trigonometric Series, Volume I, 2nd ed., Cambridge University Press (1968) | Zbl 0157.38204