Let {b (n)} ¥n=0 be a sequence of positive numbers and 1<p<¥. We consider the weighted HARDY space Hp (b). We investigate the relation between the generating function and the functional of pointe valuations. Also, under a sufficient condition we determine the structure of all non-zero multiplicative linear functionals on Hp (b).

Let T be a bounded operator on the Banach space X and j be an analytic self-map of the unit disk D. We investigate some operator theoretic properties of bilateral composition operator Cj, T: f®T o f o j on the VECTOR-VALUED HARDY space Hp (X) for 1£p£+¥. Compactness and weak compactness of Cj, T on Hp (X) are characterized and when p = 2, a concrete formula for its adjoint is given.

Let p ³ 1 and ( Wn) n w be a sequence with non-negative entries. If (tn,k) ³0, denote by ||T||p,x the infimum of those U satisfying the following inequality:(S¥n=1wn(S¥k=1 tn,k ak)p)1/p £U (S¥k=1 WKAPK)1/P ,whenever ( an) ∈lp (w ) n p a ∈l w . The purpose of this paper is to give an upper bound for the norm of operator T on weighted sequence SPACES d(w,p) and lp(w) and also e(w,¥). We considered this problem for certain matrix operators such as Norlund, Weighted mean, Ceasaro and Copson matrices. This problem is considered by some authors like Bennett, Jamson and the first author on sequence SPACES lp and weighted sequence SPACES for some kind of matrix operators. Also, this study is an extension of paper by Chang-Pao Chen, Dah-Chin Luor and Zong-Yin Ou.

We investigate the class Bloc(Rn) of exponents p(.) for which the local HARDY–Littlewood maximal operator is bounded in variable exponent Lebesgue SPACES Lp(·)(Rn). Littlewood–Paley square function characterization of Lp(·)(Rn) SPACES with the above class of exponent are also obtained.

An operator T on a Banach space X is said to be weakly hypercyclic if there exists a vector x Î X whose orbit under T is weakly dense in X. We show that every weakly hypercyclic composition operator on classic HARDY space H2 is norm hypercyclic.