If $$\mu $$ μ is a positive Borel measure on the interval [0, 1) we let $$\mathcal H_\mu $$ H μ be the Hankel matrix $$\mathcal H_\mu =(\mu _{n, k})_{n,k\ge 0}$$ H μ = ( μ n , k ) n , k ≥ 0 with entries $$\mu _{n, k}=\mu _{n+k}$$ μ n , k = μ n + k , where, for $$n\,=\,0, 1, 2, \dots $$ n = 0 , 1 , 2 , ⋯ , $$\mu _n$$ μ n denotes the moment of order n of $$\mu $$ μ . This matrix induces formally the operator $$\begin{aligned} \mathcal {H}_\mu (f)(z)= \sum _{n=0}^{\infty }\left( \sum _{k=0}^{\infty } \mu _{n,k}{a_k}\right) z^n \end{aligned}$$ H μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n , k a k z n on the space of all analytic functions $$f(z)=\sum _{k=0}^\infty a_kz^k$$ f ( z ) = ∑ k = 0 ∞ a k z k , in the unit disc $${\mathbb D}$$ D . This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators $$\mathcal {H}_\mu $$ H μ on Hardy spaces and on Möbius invariant spaces.
Integral Equations and Operator Theory – Springer Journals
Published: Nov 2, 2017
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