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We characterize the boundedness and compactness of the weighted composition operator on the Zygmund space Z {f ∈ H D : sup z∈D 1 − |z| 2 |f z | < ∞} and the little Zygmund space Z 0 .

Let µ be a positive Borel measure on the interval [0, 1). The Hankel matrix H µ = (µ n,k ) n,k≥0 with entries µ n,k = µ n+k , where µ n = [0,1) t n dµ(t), induces formally the operator(1−tz) 2 dµ(t) for all in Hardy spaces H p (0 < p < ∞), and among them we describe those for which DH µ is a bounded(resp.,compact) operator from H p (0 < p < ∞) into H q (q > p and q ≥ 1). We also study the analogous problem in Hardy spaces H p (1 ≤ p ≤ 2).

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Abstract. We characterize the boundedness and compactness of the weighted composition operator on the logarithmic Bloch space)|f (z)| < +∞} and the little logarithmic Bloch space LB 0 . The results generalize the known corresponding results on the composition operator and the pointwise multiplier on the logarithmic Bloch space LB and the little logarithmic Bloch space LB 0 .

This paper aims at studying the boundedness and compactness of weighted composition operator between spaces of analytic functions. We characterize boundedness and compactness of the weighted composition operator from the Hardy spaces to the Zygmund type spaces Z = { ∈ ( ) : sup ∈ (1 − | | 2 ) | ( )| < ∞} and the little Zygmund type spaces Z ,0 in terms of function theoretic properties of the symbols and .

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