# [unknown]

Studia Mathematica (1998)

- Volume: 130, Issue: 1, page 23-52
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topAlbrecht, E., and Ricker, W.. "null." Studia Mathematica 130.1 (1998): 23-52. <http://eudml.org/doc/216539>.

@article{Albrecht1998,

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.},

author = {Albrecht, E., Ricker, W.},

journal = {Studia Mathematica},

keywords = {decomposable operators; Fourier multipliers; local spectral theory; functional calculi; linear differential operators with constant coefficients; Fourier multiplier operators; decomposability; spectral mapping property; Lyubich-Matsaev property},

language = {eng},

number = {1},

pages = {23-52},

url = {http://eudml.org/doc/216539},

volume = {130},

year = {1998},

}

TY - JOUR

AU - Albrecht, E.

AU - Ricker, W.

JO - Studia Mathematica

PY - 1998

VL - 130

IS - 1

SP - 23

EP - 52

AB - 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.

LA - eng

KW - decomposable operators; Fourier multipliers; local spectral theory; functional calculi; linear differential operators with constant coefficients; Fourier multiplier operators; decomposability; spectral mapping property; Lyubich-Matsaev property

UR - http://eudml.org/doc/216539

ER -

## References

top- [1] E. Albrecht and R. D. Mehta, Some remarks on local spectral theory, J. Operator Theory 12 (1984), 285-317. Zbl0583.47039
- [2] E. Albrecht and W. J. Ricker, Local spectral properties of constant coefficient differential operators in ${L}^{p}\left({\mathbb{R}}^{N}\right)$, ibid. 24 (1990), 85-103.
- [3] E. Albrecht and W. J. Ricker, Functional calculi and decomposability of unbounded multiplier operators in ${L}^{p}\left({\mathbb{R}}^{N}\right)$, Proc. Edinburgh Math. Soc. 38 (1995), 151-166. Zbl0820.47037
- [4] E. Albrecht and W. J. Ricker, Local spectral properties of certain matrix differential operators in ${L}^{p}{\left({\mathbb{R}}^{N}\right)}^{m}$, J. Operator Theory 35 (1996), 3-37. Zbl0851.47010
- [5] C. Apostol, Spectral decompositions and functional calculus, Rev. Roumaine Math. Pures Appl. 13 (1968), 1481-1528. Zbl0176.43701
- [6] Y.-C. Chang and P. A. Tomas, Invertibility of some second order differential operators, Studia Math. 79 (1984), 289-296. Zbl0569.35009
- [7] I. Colojoară and C. Foiaş, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968. Zbl0189.44201
- [8] N. Dunford and J. T. Schwartz, Linear Operators I: General Theory, Interscience, New York, 1964. Zbl0084.10402
- [9] R. E. Edwards and G. I. Gaudry, Littlewood-Paley and Multiplier Theory, Springer, Berlin, 1977.
- [10] C. Foiaş, Spectral maximal spaces and decomposable operators, Arch. Math. (Basel) 14 (1963), 341-349. Zbl0176.43802
- [11] L. Hörmander, On interior regularity of the solutions of partial differential equations, Comm. Pure Appl. Math. 11 (1958), 197-218. Zbl0081.31501
- [12] L. Hörmander, Estimates for translation invariant operators in ${L}^{p}$ spaces, Acta Math. 104 (1960), 93-140. Zbl0093.11402
- [13] L. Hörmander, The Analysis of Linear Partial Differential Operators II. Differential Operators with Constant Coefficients, Springer, Berlin, 1983. Zbl0521.35002
- [15] M. Jodeit, Jr., A note on Fourier multipliers, Proc. Amer. Math. Soc. 27 (1971), 423-424. Zbl0214.13301
- [16] C. E. Kening and P. A. Tomas, On conjectures of Rivière Strichartz, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 694-697). Zbl0418.42007
- [17] 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.
- [18] H. König and R. Reader, Vorlesung über die Theorie der Distributionen, Ann. Univ. Sarav. Ser. Math. 6 (1995), 1-213.
- [19] K. de Leeuw, On ${L}^{p}$ multipliers, Ann. of Math. 81 (1965), 364-379.
- [20] W. Littman, Multipliers in ${L}^{p}$ and interpolation, Bull. Amer. Math. Soc. 71 (1965), 764-766. Zbl0156.36504
- [21] Yu. I. Lyubich and V. I. Matsaev, On operators with separable spectrum, Mat. Sb. 56 (1962), 433-468.
- [22] J. Peetre, Applications de la théorie des espaces d'interpolation dans l'Analyse Harmonique, Ricerche Mat. 15 (1966), 3-36. Zbl0154.15302
- [24] 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. Zbl0523.42012
- [25] A. Ruiz, ${L}^{p}$ - boundedness of a certain class of multipliers associated with curves on the plane. II, ibid., 277-282. Zbl0523.42013
- [26] M. Schechter, The spectrum of operators on ${L}^{p}\left({E}^{n}\right)$, Ann. Scoula Norm. Sup. Pisa 24 (1970), 201-207.
- [27] M. Schechter, Spectra of Partial Differential Operators, 2nd ed., North-Holland, Amsterdam, 1986.
- [28] F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, D. Reidel, Dordrecht, and Editura Academiei, Bucureşti, 1982.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.