Let \(\mu\) be a finite positive measure on the real interval \((0,1)\), for \(1\leq p<\infty\), let \(H^p\) denote the Hardy space of analytic functions on the unit disc \(\mathbb{D}= \{z\in\mathbb{C}:|z|< 1\}\), and let \(A^p\) denote the Bergman space on \(\mathbb{D}\).
An infinite lower triangular Hausdorff matrix \(H= \{c_{n,k}\}\) has entries \(c_{n,k}\) defined by \(c_{n,k}= {^nC_k}\int^1_0 t^k(1- t)^{n-k} d\mu(t)\), with \({^nC_k}= n!/((n- k)!k!)\), \(k= 0,1,2,\dots, n\). If \((X,\|.\|_X)\) is a Banach space of analytic functions on \(\mathbb{D}\) with norm \(\|.\|_X\) and if \(f\in X\) has the representation \(f(z)= \sum^\infty_{n=0} a_n z^n\), then the transform \({\mathcal H}_\mu(f)\) and transpose operator \({\mathcal A}_\mu\) are defined formally as \[ {\mathcal H}_\mu(f)(z)= \sum^\infty_{n= 0}\,\Biggl(\sum^n_{k=0} c_{n,k}a_k\Biggr) z^n;\;{\mathcal A}_\mu(f)(z)= \sum^\infty_{k= 0}\,\Biggl(\sum^\infty_{n=k} c_{n,k} a_n\Biggr)z^n. \] In addition, the operators \({\mathcal S}_\mu\), \({\mathcal T}_\mu\) are defined by \[ {\mathcal S}_\mu(f)(z)= \int^1_0 w_t(z) f(\varphi_t(z))\,d\mu(t);\;{\mathcal T}_\mu(f)(z)= \int^1_0 f(\psi_t(z))\,d\mu(t), \] where \(\varphi_t(z)= tz/((t- 1)z+ 1)\), \(\psi_t(z)= zt+ 1-t\); \(w_t(z)= ((t- 1)z+ 1)- 1\), \(z\in\mathbb{D}\).
The main results of this paper provide necessary and sufficient conditions so that
(i) \({\mathcal H}_\mu={\mathcal S}_\mu: H^p\to H^p\) or (ii) \({\mathcal H}_\mu={\mathcal S}_\mu: A^p\to A^p\) is a bounded operator;
(iii) \({\mathcal A}_\mu: H^p\to H^p\) or (iv) \({\mathcal A}_\mu: A^p\to A^p\) is a bounded operator.
In particular, in case (i), the applicable conditions include \(\int^1_0 t^{(1/p)-1}d\mu(t)< \infty\) and in the cases (ii), (iii), (iv), the applicable conditions include \(\int^1_0 t^{-\delta/p} d\mu(t)< \infty\), \(\delta= 1\) or \(2\). It is also shown that \({\mathcal A}_\mu\) in \(H^{p'}\) is the adjoint of \({\mathcal H}_\mu\) in \(H^p\), where \((1/p)+ (1/p')= 1\).
The introduction of the paper contains statements that earlier papers, including [P. Galanopoulos and A. Siskakis, Ill.J.Math.45, No. 3, 757–773 (2001; Zbl 0994.47026)] and O. Rudolf [“Hausdorff–Operatoren auf BK-Räumen und Halbgruppen linearer Operatoren” (Mitteillungen aus dem Mathematischen Seminar Gießen 241) (Diss. Univ. Gießen) (2000; Zbl 1104.47302)] contain results which indicate sufficient conditions for (i) in the case \(2\leq p<\infty\), and state different proofs for the results (i) and (iv).