On the eleventh question of Allen Shields

Bahmann Yousefi,Ali Iloon Kashkooly

doi:10.1186/1687-1812-2011-16

Bahmann Yousefi, Ali Iloon Kashkooly

Open Access

https://doi.org/10.1186/1687-1812-2011-16

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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.

Highlights

Let {β(n)}∞ n=−∞ be a sequence of positive numbers satisfying b(0) = 1
{0}, the space Lp(b) only contains formal power series f (z) = ∞ f(n)zn, and it is usually denoted n=0 by Hp(b). These spaces are called as weighted Hardy spaces
We investigate the fixed points of some weighted composition operators acting on weighted Hardy spaces

Summary

When n just runs over

Shields showed a close relation between injective weighted shifts and the multiplication operator Mz acting on L2(b) or H2(b) (see [[1], Proposition 7]) These are reflexive Banach spaces with the norm ||·||b. We want to investigate those weighted Hardy spaces that admit the analytic projection as a bounded linear operator on Lp∞(β). This answers the following question that has been considered by Shields in [[1], p. By the same method used in the proof of Lemma 1, we can see that J1 ∈ B(Lp∞(β)) and this implies that J is a bounded operator from Lp∞(β) into Lp∞(β). By using the relation (*) with replacing z by 1 z and

Mz by

Let φ

By taking k

Also suppose that is a multiplier of

This implies that