On p-dependent local spectral properties of certain linear differential operators in $L^{p}(ℝ^{N})

E. Albrecht; W. Ricker

ArticleOriginal scientific text

Title

On p-dependent local spectral properties of certain linear differential operators in !$!L^{p}(ℝ^{N})

Authors ¹, ²

Affiliations

Fachbereich Mathematik, Universität des Saarlandes, Postfach 151150, D-66141 Saarbrücken, Germany
School of Mathematics, University of New South Wales, Sydney, N.S.W., 2025, Australia

Abstract

The aim is to investigate certain spectral properties, such as decomposability, the spectral mapping property and the Lyubich-Matsaev property, for linear differential operators with constant coefficients ( and more general Fourier multiplier operators) acting in

L^{p} (ℝ^{N})

. The criteria developed for such operators are quite general and p-dependent, i.e. they hold for a range of p in an interval about 2 (which is typically not (1,∞)). The main idea is to construct appropriate functional calculi: this is achieved via a combination of methods from the theory of Fourier multipliers and local spectral theory.

E. Albrecht and R. D. Mehta, Some remarks on local spectral theory, J. Operator Theory 12 (1984), 285-317.
E. Albrecht and W. J. Ricker, Local spectral properties of constant coefficient differential operators in $L^{p} (ℝ^{N})$ , ibid. 24 (1990), 85-103.
E. Albrecht and W. J. Ricker, Functional calculi and decomposability of unbounded multiplier operators in $L^{p} (ℝ^{N})$ , Proc. Edinburgh Math. Soc. 38 (1995), 151-166.
E. Albrecht and W. J. Ricker, Local spectral properties of certain matrix differential operators in $L^{p} {(ℝ^{N})}^{m}$ , J. Operator Theory 35 (1996), 3-37.
C. Apostol, Spectral decompositions and functional calculus, Rev. Roumaine Math. Pures Appl. 13 (1968), 1481-1528.
Y.-C. Chang and P. A. Tomas, Invertibility of some second order differential operators, Studia Math. 79 (1984), 289-296.
I. Colojoară and C. Foiaş, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968.
N. Dunford and J. T. Schwartz, Linear Operators I: General Theory, Interscience, New York, 1964.
R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier Theory, Springer, Berlin, 1977.
C. Foiaş, Spectral maximal spaces and decomposable operators, Arch. Math. (Basel) 14 (1963), 341-349.
L. Hörmander, On interior regularity of the solutions of partial differential equations, Comm. Pure Appl. Math. 11 (1958), 197-218.
L. Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93-140.
L. Hörmander, The Analysis of Linear Partial Differential Operators II. Differential Operators with Constant Coefficients, Springer, Berlin, 1983.
F. T. Iha and C.F. Schubert, The spectrum of partial differential operators on !$!L^p(ℝ^n), Trans. Amer. Math. Soc. 152 (1970), 215-226
M. Jodeit, Jr., A note on Fourier multipliers, Proc. Amer. Math. Soc. 27 (1971), 423-424.
C. E. Kening and P. A. Tomas, On conjectures of Rivière Strichartz, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 694-697).
C. E. Kening and P. A. Tomas, $L^{p}$ behaviour of certain second order partial differential operators, Trans. Amer. Math. Soc. 262 (1980), 521-531.
H. König and R. Reader, Vorlesung über die Theorie der Distributionen, Ann. Univ. Sarav. Ser. Math. 6 (1995), 1-213.
K. de Leeuw, On $L^{p}$ multipliers, Ann. of Math. 81 (1965), 364-379.
W. Littman, Multipliers in $L^{p}$ and interpolation, Bull. Amer. Math. Soc. 71 (1965), 764-766.
Yu. I. Lyubich and V. I. Matsaev, On operators with separable spectrum, Mat. Sb. 56 (1962), 433-468.
J. Peetre, Applications de la théorie des espaces d'interpolation dans l'Analyse Harmonique, Ricerche Mat. 15 (1966), 3-36.
A. Ruiz, Multiplicadores asociados a curvas en el plano y teoremas de restricción de la transformada de Fourier a curvas en $ℝ^{2}$ y ℝ ^3!$!, Tesis doctoral, Universidad Complutense de Madrid, 1980.
A. Ruiz, $L^{p}$ - boundedness of a certain class of multipliers associated with curves on the plane. I, Proc. Amer. Math. Soc. 87 (1983), 271-276.
A. Ruiz, $L^{p}$ - boundedness of a certain class of multipliers associated with curves on the plane. II, ibid., 277-282.
M. Schechter, The spectrum of operators on $L^{p} (E^{n})$ , Ann. Scoula Norm. Sup. Pisa 24 (1970), 201-207.
M. Schechter, Spectra of Partial Differential Operators, 2nd ed., North-Holland, Amsterdam, 1986.
F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, D. Reidel, Dordrecht, and Editura Academiei, Bucureşti, 1982.

Title

On p-dependent local spectral properties of certain linear differential operators in !$!L^{p}(ℝ^{N})

Affiliations

Abstract

Bibliography