Some results about Beurling algebras with applications to operator theory
Vils Pedersen, Thomas
Studia Mathematica, Tome 113 (1995), p. 39-52 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

We prove that certain maximal ideals in Beurling algebras on the unit disc have approximate identities, and show the existence of functions with certain properties in these maximal ideals. We then use these results to prove that if T is a bounded operator on a Banach space X satisfying ∥Tn∥=O(nβ) as n → ∞ for some β ≥ 0, then ∑n=1∞∥(1-T)nx∥/∥(1-T)n-1x∥ diverges for every x ∈ X such that (1-T)[β]+1x≠0.

Publié le : 1995-01-01
EUDML-ID : urn:eudml:doc:216197 
@article{bwmeta1.element.bwnjournal-article-smv115i1p39bwm,
     author = {Thomas Vils Pedersen},
     title = {Some results about Beurling algebras with applications to operator theory},
     journal = {Studia Mathematica},
     volume = {113},
     year = {1995},
     pages = {39-52},
     zbl = {0831.46058},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv115i1p39bwm}
}

Vils Pedersen, Thomas. Some results about Beurling algebras with applications to operator theory. Studia Mathematica, Tome 113 (1995) pp. 39-52. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv115i1p39bwm/

[00000] [1] A. Atzmon, Operators which are annihilated by analytic functions and invariant subspaces, Acta Math. 144 (1980), 27-63. | Zbl 0449.47007

[00001] [2] C. Bennett and J. E. Gilbert, Homogeneous algebras on the circle: I. Ideals of analytic functions, Ann. Inst. Fourier (Grenoble) 22 (3) (1972), 1-19. | Zbl 0228.46046

[00002] [3] J. G. Caughran, Factorization of analytic functions with Hp derivative, Duke Math. J. 36 (1969), 153-158. | Zbl 0176.37601

[00003] [4] J. Esterle et F. Zouakia, Rythme de décroissance des itérés de certains opérateurs de l'espace de Hilbert, J. Operator Theory 21 (1989), 387-396. | Zbl 0703.47001

[00004] [5] I. M. Gelfand, D. A. Raikov and G. E. Shilov, Commutative Normed Rings, Chelsea, Bronx, 1964.

[00005] [6] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. | Zbl 0117.34001

[00006] [7] J.-P. Kahane, Idéaux primaires fermés dans certaines algèbres de Banach de fonctions analytiques, in: E. J. Akutowicz (ed.), L'analyse harmonique dans le domaine complexe, Lecture Notes in Math. 336, Springer, Berlin, 1973, 5-14.

[00007] [8] Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968.

[00008] [9] T. V. Pedersen, Norms of powers in Banach algebras, Bull. London Math. Soc., to appear. | Zbl 0835.46045

[00009] [10] H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford Univ. Press, London, 1968. | Zbl 0165.15601