Let g be an analytic function on the unit disc and consider the integration operator of the form Tgf (z) = z 0 fg dζ. We derive estimates for the essential and weak essential norms of Tg on the spaces H p and BMOA. In particular, on H 1 and BMOA the operator Tg is weakly compact if and only if it is compact. Mathematics Subject Classification (2010). Primary 47B38;Secondary 30H10, 30H35, 47B07.

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Abstract. We prove that a Volterra-type integral operatordefined on Hardy spaces H p , 1 ≤ p < ∞, fixes an isomorphic copy of ℓ p , if the operator T g is not compact. In particular, this shows that the strict singularity of the operator T g coincides with the compactness of the operator T g on spaces H p . As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of the operator T g on H 1 .

Very recently, Božin and Karapetrović [4] solved a conjecture by proving that the norm of the Hilbert matrix operator H on the Bergman space A p is equal to π sin( 2π p ) for 2 < p < 4. In this article we present a partly new and simplified proof of this result. Moreover, we calculate the exact value of the norm of H defined on the Korenblum spaces H ∞ α for 0 < α ≤ 2/3 and an upper bound for the norm on the scale 2/3 < α < 1.