Multipliers of weighted spaces and reflexivity property

Dussau, Xavier

doi:10.1090/S0002-9939-04-07640-3

Multipliers of weighted spaces and reflexivity property
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Proc. Amer. Math. Soc. 133 (2005), 1379-1386 Request permission

Abstract:

We prove for some translation-invariant weighted spaces $E$ the following characterization: $T$ is a multiplier of $E$ if and only if $T$ leaves invariant every translation-invariant subspace of $E$. This result is equivalent with the reflexivity of the multiplier algebra of $E$.

References

Aharon Atzmon, The existence of translation invariant subspaces of symmetric self-adjoint sequence spaces on $\textbf {Z}$, J. Funct. Anal. 178 (2000), no. 2, 372–380. MR 1802899, DOI 10.1006/jfan.2000.3660
Isabelle Chalendar and Jean Esterle, Le problème du sous-espace invariant, Development of mathematics 1950–2000, Birkhäuser, Basel, 2000, pp. 235–267 (French). MR 1796843
B. Chevreau, A survey of recent reflexivity results, Operator algebras and operator theory (Craiova, 1989) Pitman Res. Notes Math. Ser., vol. 271, Longman Sci. Tech., Harlow, 1992, pp. 46–61. MR 1189165
Yngve Domar, Translation invariant subspaces of weighted $l^{p}$ and $L^{p}$ spaces, Math. Scand. 49 (1981), no. 1, 133–144. MR 645094, DOI 10.7146/math.scand.a-11926
Xavier Dussau, Les shifts à poids dissymétriques sont hyper-réflexifs, Bull. Soc. Math. France 130 (2002), no. 4, 573–585 (French, with English and French summaries). MR 1947453, DOI 10.24033/bsmf.2430
Xavier Dussau, Les shifts à poids de croissance polynomiale sont hyper-réflexifs, Arch. Math. (Basel) 80 (2003), no. 1, 71–78 (French, with English and French summaries). MR 1968289, DOI 10.1007/s000130300007
Jean Esterle, Singular inner functions and biinvariant subspaces for dissymmetric weighted shifts, J. Funct. Anal. 144 (1997), no. 1, 64–104. MR 1430716, DOI 10.1006/jfan.1996.2996
Jean Esterle, Apostol’s bilateral weighted shifts are hyper-reflexive, Recent advances in operator theory and related topics (Szeged, 1999) Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 243–266. MR 1902804
V. V. Kapustin, Reflexivity of operators: general methods and a criterion for almost isometric contractions, Algebra i Analiz 4 (1992), no. 2, 141–160 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 2, 319–335. MR 1182398
Ronald Larsen, The multiplier problem, Lecture Notes in Mathematics, Vol. 105, Springer-Verlag, Berlin-New York, 1969. MR 0435737, DOI 10.1007/BFb0059711
D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517.
Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899
Béla Sz.-Nagy and Ciprian Foiaş, Analyse harmonique des opérateurs de l’espace de Hilbert, Masson et Cie, Paris; Akadémiai Kiadó, Budapest, 1967 (French). MR 0225183
U. B. Tewari, The multiplier problem, Math. Student 65 (1996), no. 1-4, 45–56. MR 1618307