Hostname: page-component-6bf8c574d5-w79xw Total loading time: 0 Render date: 2025-02-17T18:08:36.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Equisummability for linear operators in Banach spaces

Published online by Cambridge University Press:  14 November 2011

M. A. Kon ,
A. G. Ramm  and
L. A. Raphael
M. A. Kon
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, U.S.A.
A. G. Ramm
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, KS 66506, U.S.A.
L. A. Raphael
Affiliation:
Department of Mathematics, Howard University, Washington, DC 20059, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let A and B be closed linear operators on a Banach space X. Assume that ε(εI – A)−1f→f as |ε|→ ∞ for all f in X, ζ∊∑ ⊂ℂ. Under what conditions on B − A does the same relationship hold for B? When does [ε(εI − A)−1 − ε(εI − B)−1 ] f→ 0 in some stronger norm than that of X? The questions are discussed in an abstract setting and the results are generalised to other analytic functions of A. Applications are given to second order elliptic operators.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 106 , Issue 3-4 , 1987 , pp. 315 - 325
Copyright
Copyright © Royal Society of Edinburgh 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Benzinger, H.. Green's function for ordinary differential operators. J. Differential Equations 7 (1970), 478–496.Google Scholar
2Berezanskii, M. Jr, Expansions in Eigenfunctions of Self-Adjoint Operators (Providence, R.I.: A.M.S. Translations, 1968).Google Scholar
3Birkhoff, G. D.. Boundary value and expansion problems of ordinary linear differential equations. Trans. Amer. Math. Soc. 9 (1908), 373–395.Google Scholar
4Eidelman, S. D.. Estimates for solutions of parabolic systems and some applications. Math. Sbornik 33 (1953), 359–382.Google Scholar
5Eidelman, S. D.. On fundamental solutions of parabolic systems. Math. Sbornik 38 (1956), 51–92.Google Scholar
6Fefferman, C.. The new multiplier theorem for the ball. Ann. of Math. 94 (1971), 330–336.Google Scholar
7Gelfand, I. and Shilov, G.. Generalized Functions, Vol 3 (New York: Academic Press, 1967).Google Scholar
8Gohberg, I. C. and Krein, M. G.. Introduction to the theory of linear non self-adjoint operators (Providence, R.I.: A.M.S. Translations, 1969).Google Scholar
9Gurarie, D. and Kon, M.. Radial bounds for perturbations of elliptic operators. J. Fund. Anal. 56 (1984), 99–123.Google Scholar
10Gurarie, D. and Kon, M.. Resolvents and regularity properties of elliptic operators. In Operator Theory: Advances and Applications, pp. 151–162. Basel: Birkhaiiser, 1983).Google Scholar
11Haar, A.. Zur theorie der orthogonalen Funktionensysteme. Math. Ann. 69 (1910), 331–371.Google Scholar
12Kenig, C. E., Stanton, R. J. and Tomas, P. A.. Divergence of eigenfunction expansions. J. Funct. Anal. 46 (1982), 28–44.Google Scholar
13Komornik, V.. An equiconvergence theorem for the Schrodinger operator. Ada Math. Hungar. 44 (1984), 101–114.Google Scholar
14Kon, M.. Regularity properties of Schrödinger operators on a domain of ℝn. In Differential Equations, eds Knowles, I. W. and Lewis, R. T., pp. 359–366 (Amsterdam: Elsevier Science Publishers, 1984).Google Scholar
15Levitan, B. and Sargsjan, I.. Introduction to the Spectral Theory; Self-Adjoint Differential Operators (Providence, R.I.: A.M.S. Translations, 1975).Google Scholar
16Pazy, A.. Semigroups of Linear Operators and Applications to Partial Differential Equations (New York: Springer-Verlag, 1983).Google Scholar
17Raphael, L. A.. Equisummability of eigenfunction expansions under analytic multipliers. J. Math. Anal. Appl. 115 (1986), 93–104.Google Scholar
18Stone, M. H.. A comparison of the series of Fourier and Birkhoff. Trans. Amer. Math. Soc. 28 (1926), 695–761.Google Scholar
19Tamarkin, J. D.. Sur quelques points de la theorie des equations differentielles lineaires ordinaires et sur la generalisation de la serie de Fourier. Rend. Circ. Math. Palermo 34 (1912), 345–395.Google Scholar
20Titchmarsh, E. C.. Eigenfunction Expansions, parts I and II (Oxford: Oxford University Press, 1962).Google Scholar
21Walsh, J. L.. On the convergence of the Sturm-Liouville series. Ann. of Math. 24 (1022–23), 109–120.Google Scholar