On the eleventh question of Allen Shields

Yousefi Bahmann, Kashkooly, Kashkooly Ali

Description

In this article, we will give sufficient conditions for the boundedness of the analytic projection on the set of multipliers of the formal Laurent series spaces. This answers a question that has been raised by A. L. Shields. Also, we will characterize the fixed points of some weighted composition operators acting on weighted Hardy spaces. AMS Subject Classification: Primary 47B37; Secondary 47B38.

Sujets

• Spectral set
• Weak operator topology

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 6
• Langue: English

Extrait

Yousefi and KashkoolyFixed Point Theory and Applications2011,2011:16 http://www.fixedpointtheoryandapplications.com/content/2011/1/16

R E S E A R C H

On the eleventh question 1* 2 Bahmann Yousefi and Ali Iloon Kashkooly

* Correspondence: b_yousefi@pnu. ac.ir 1 Department of Mathematics, Payame Noor University, P.O. Box 719551368, Shiraz, Iran Full list of author information is available at the end of the article

of

Allen

Shields

Open Access

Abstract In this article, we will give sufficient conditions for the boundedness of the analytic projection on the set of multipliers of the formal Laurent series spaces. This answers a question that has been raised by A. L. Shields. Also, we will characterize the fixed points of some weighted composition operators acting on weighted Hardy spaces. AMS Subject Classification:Primary 47B37; Secondary 47B38. Keywords:Banach space of Laurent series associated with a sequenceβ, bounded point evaluation; spectral set, weak operator topology

1 Introduction ∞ Let {β(n)} be a sequence of positive numbers satisfyingb(0) = 1. If 1< p <∞, then =− ∞  ˆ p n the spaceL(b) consists of all formal Laurent series f(z) =f(n)z such that the =−∞  p ∞ ˆ p p norm ||f||=||f||=|f(n)|β(n) is finite. Whennjust runs overN∪{0}, the n=−∞ ∞  ˆ p spaceL(b) only contains formal power series f(z) =f(n)z , and it is usually denoted = p byH(b). These spaces are also called as weighted Hardy spaces. Ifp= 2, such spaces were introduced by Allen L. Shields to study weighted shift operators in his article [1] which is one of basic studies in this area, and is a pretty large study that contains a num ber of interesting results, and indeed it is mainly of auxiliary nature. Actually, Shields showed a close relation between injective weighted shifts and the multiplication operator 2 2 M z acting onL(b) orH(b) (see [[1], Proposition 7]). These are reflexive Banach spaces k p ˆ with the norm ||∙|| . Let . Hence,f(z) =zand {fa basis for} is L(b) b f k n=δ k n k k kÎℤ p such that ||f k || =b(k). ClearlyM z , the operator of multiplication byzonL(b), shifts the basis {f k } k . The operatorM z is bounded, if and only if {b(k+ 1)/b(k)} k is bounded, and in n this case ||M||= sup [β(k+n)/β(k) for allnÎN∪{0}. k p We say that a complex numberlis a bounded point evaluation onL(b) if the func tionale(l) : Lβ→ defined bye(l)(f) =f(l) is bounded. LetXbe a Banach space. It is convenient and helpful to introduce the notation< x, x* >to stand forx*(x), forxÎXandx*ÎX*. Also, the set of bounded linear operators onXis denoted byB(X). IfAÎB(X), then bys(A) we mean the spectrum ofAand by r(A) we mean the spectral radius ofA.

© 2011 Yousefi and Kashkooly; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.