On the spacel P(β)

Reflexive operators on analytic function spaces

“…We denote the set of multipliers fu 2 L p ðbÞ : uL p ðbÞ L p ðbÞg by L p 1 ðbÞ and the linear operator of multiplication by u on L p ðbÞ by M u . Also, note that L p ðbÞ Ã ¼ L q ðb p q Þ, where 1 p þ 1 q ¼ 1 ( [21,23]). Note that the operator of multiplication M z is bounded on L p ðbÞ if and only if sup n bðnþ1Þ bðnÞ \1 ( [15,19]).…”

“…we have \f ; g [ ¼ P 1 n¼À1 f ðnÞ ĝðnÞ where the notation \f ; g [ stands for g(f). For a source in formal series, we refer the reader to [19,[21][22][23]25].…”

“…Assume that 0\r 0 \r 1 ¼ 1 and put X ¼ fz 2 C : r 0 \jzj\r 1 g: Then X is an annulus with X 1 ¼ D and L p ðbÞ & HðXÞ. Also note that each point of X is a bounded point evaluation on L p ðbÞ ( [21,23]). Clearly, ff k g k2Z is a basis for both L p ðbÞ and it's dual L q ðbÞ, where 1 p þ 1 q ¼ 1 (note that here ðf j ; f k Þ¼ : f j ðf k Þ ¼ d j ðkÞ where f j 2 L p ðbÞ Ã ¼ L q ðb À1 Þ and f k 2 L p ðbÞ).…”

On the Weighted Composition Operators on Hilbert Spaces of Formal Power Series

“…We will use the notations H(U ) and C(U ) to denote the set of analytic functions on U and the set of continuous functions on U , the closure of U . Some sources on formal power series and composition operators include [1][2][3][4][5][6][7].…”

On the Weighted Composition Operators on Weighted Banach Spaces

“…In this paper we will give a definition for the numerical range of an operator acting on Banach spaces and then we will investigate it's closedness for weighted composition operators acting on the weighted Hardy spaces H p (β). For some sources on the topics of this paper one can refer to [1][2][3][4][5][6][7].…”