Stable norms –from theory to applications and back. (English) Zbl 1078.15022
This survey paper gives a brief review on certain aspects of stability of norms and subnorms acting on algebras over \({\mathbb C}\) or \({\mathbb R}\). Several results regarding norm stability, including conditions under which norms on certain algebras are stable, are considered. The second part of the paper is devoted to applications, where the notion of norm stability is employed to obtain criteria for the convergence of a well-known family of finite-difference schemes for the initial-value problem associated with a certain parabolic system. The third and last part of the paper deals with the question of stability for subnorms acting on subsets of power-associative algebras that are closed under scalar multiplication and under raising to powers.
Reviewer: Alexey Alimov (Moskva)
MSC:
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 17A05 | Power-associative rings |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
Keywords:
associative algebra; norm; seminorm; stable norm; parabolic system; power-associative algebras; alternative algebras; finite-difference schemes; stability; survey paper; convergenceSoftware:
RODASReferences:
| [1] | Arens, R.; Goldberg, M., Multiplicativity factors for seminorms, J. Math. Anal. Appl., 146, 469-481 (1990) · Zbl 0701.46026 |
| [2] | Arens, R.; Goldberg, M., A class of seminorms on function algebras, J. Math. Anal. Appl., 162, 592-609 (1991) · Zbl 0769.46035 |
| [3] | Arens, R.; Goldberg, M., Weighted \(ℓ_∞\) norms for matrices, Linear Algebra Appl., 201, 155-163 (1994) · Zbl 0809.15011 |
| [4] | Arens, R.; Goldberg, M., Homotonic mappings, J. Math. Anal. Appl., 194, 414-427 (1995) · Zbl 0869.15018 |
| [5] | Arens, R.; Goldberg, M., Weighted \(ℓ_1\) norms for matrices, Linear and Multilinear Algebra, 40, 229-234 (1996) · Zbl 0865.15023 |
| [6] | Arens, R.; Goldberg, M.; Luxemburg, W. A.J., Multiplicativity factors for seminorms. II, J. Math. Anal. Appl., 170, 401-413 (1992) · Zbl 0796.46034 |
| [7] | Arens, R.; Goldberg, M.; Luxemburg, W. A.J., Multiplicativity factors for function norms, J. Math. Anal. Appl., 177, 368-385 (1993) · Zbl 0801.46026 |
| [8] | Arens, R.; Goldberg, M.; Luxemburg, W. A.J., Multiplicativity factors for Orlicz space function norms, J. Math. Anal. Appl., 177, 386-411 (1993) · Zbl 0799.46022 |
| [9] | Arens, R.; Goldberg, M.; Luxemburg, W. A.J., Stable seminorms revisited, Math. Inequal. Appl., 1, 31-40 (1998) · Zbl 0892.47008 |
| [10] | Arens, R.; Goldberg, M.; Luxemburg, W. A.J., Stable norms on complex numbers and quaternions, J. Algebra, 219, 1-15 (1999) · Zbl 0943.47007 |
| [11] | Baez, J. C., The octonions, Bull. Amer. Math. Soc. (N.S.), 39, 145-205 (2002) · Zbl 1026.17001 |
| [12] | C.A. Berger, On the numerical range of powers of an operator, Abstract No. 625-152, Notices Amer. Math. Soc. 12 (1965) 590; C.A. Berger, On the numerical range of powers of an operator, Abstract No. 625-152, Notices Amer. Math. Soc. 12 (1965) 590 |
| [13] | Crank, J.; Nicholson, P., A practical method for numerical integration of solutions of partial differential equations of heat-conduction type, Proc. Cambridge Philos. Soc., 43, 50-67 (1947) · Zbl 0029.05901 |
| [14] | Dickson, L. E., On quaternions and their generalization and the history of the Eight Square Theorem, Ann. of Math., 20, 155-171 (1918-1919) · JFM 47.0099.01 |
| [15] | Friedland, S.; Zenger, C., All spectral dominant norms are stable, Linear Algebra Appl., 58, 97-107 (1984) · Zbl 0543.15018 |
| [16] | Friedman, A., Generalized Functions and Partial Differential Equations (1963), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0116.07002 |
| [17] | Goldberg, M., Stable difference schemes for parabolic systems-a numerical radius approach, SIAM J. Numer. Anal., 35, 478-493 (1998) · Zbl 0915.65100 |
| [18] | Goldberg, M., Stable difference schemes for parabolic systems-a numerical radius approach II, SIAM J. Numer. Anal., 35, 1995-2003 (1998) · Zbl 0941.65088 |
| [19] | Goldberg, M.; Guralnick, R.; Luxemburg, W. A.J., Not all quadrative norms are strongly stable, Indag. Mathem. N.S., 12, 469-476 (2001) · Zbl 1021.15017 |
| [20] | Goldberg, M.; Guralnick, R.; Luxemburg, W. A.J., Stable subnorms II, Linear and Multilinear Algebra, 51, 209-219 (2003) · Zbl 1046.15029 |
| [21] | Goldberg, M.; Luxemburg, W. A.J., Stable subnorms, Linear Algebra Appl., 307, 89-101 (2000) · Zbl 0997.15021 |
| [22] | Goldberg, M.; Luxemburg, W. A.J., Discontinuous subnorms, Linear and Multilinear Algebra, 49, 1-24 (2001) · Zbl 0982.15032 |
| [23] | Goldberg, M.; Luxemburg, W. A.J., Stable subnorms revisited, Pac. J. Math., 215, 15-27 (2004) · Zbl 1068.17002 |
| [24] | Goldberg, M.; Pidgirniak, A., Stability criteria for finite difference approximations to parabolic systems-an update, J. Sci. Comput., 17, 423-435 (2002) · Zbl 1002.65097 |
| [25] | Halmos, P. R., A Hilbert Space Problem Book (1967), Van Nostrand: Van Nostrand New York · Zbl 0144.38704 |
| [26] | Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0576.15001 |
| [27] | Hardy, H.; Littlewood, J. E.; Pólya, G., Inequalities (1934), Cambridge University Press · JFM 60.0169.01 |
| [28] | Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0859.65067 |
| [29] | Kreiss, H.-O., Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT, 2, 153-181 (1962) · Zbl 0109.34702 |
| [30] | Laasonen, P., Über eine Methode zur Lösung der Wärmeleitungsgleichung, Acta Math., 81, 309-317 (1949) · Zbl 0040.35802 |
| [31] | Luxemburg, W. A.J.; Zaanen, A. C., Notes on Banach function spaces, I, Indag. Math., 25, 135-147 (1963) · Zbl 0117.08002 |
| [32] | Pearcy, C., An elementary proof of the power inequality for the numerical radius, Michigan Math. J., 13, 289-291 (1966) · Zbl 0143.16205 |
| [33] | Richtmyer, R. D.; Morton, K. W., Difference Methods for Initial-Value Problems (1967), Interscience: Interscience New York · Zbl 0155.47502 |
| [34] | Sun, W.; Yuan, G., Stability condition for difference schemes for parabolic systems, SIAM J. Numer. Anal., 38, 548-555 (2000) · Zbl 0970.65098 |
| [35] | Zaanen, A. C., Integration (1967), North-Holland: North-Holland Amsterdam · Zbl 0175.05002 |
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