Application on a Factor Derived from Hilbert and Carleson Measure on Hardy Spaces

Abstract

For \(\mu_j\) to a positive Borel measure on the interval \([0,1)\). The Hankel matrix \(\mathcal{H}_{\mu_j}=\left(\left(\mu_j\right)_{n, k}\right)_{j, n, k \geq 0}\) with entries \(\left(\mu_j\right)_{n, k}=\left(\mu_j\right)_{n+k}\), where \(\left(\mu_j\right)_n=\int_{[0,1)} t^n d \mu_j(t)\), the operator is formally induced\[\sum_i \mathcal{D} \mathcal{H}_{\mu_j}\left(f_j\right)(z)=\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \sum_i\left(\left(\mu_j\right)_{n, k} a_k\right)(n+1) z^n\]in the space of each analytical function \(f_j(z)=\sum_{k=0}^{\infty} a_k z_n\) in the unit disc \(\mathbb{D}\). We classify positive Borel measures on [0,1) as such \(\mathcal{D} \mathcal{H}_{\mu_j}\left(f_j\right)(z)=\int_{[0,1)} \frac{f_j(t)}{(1-t z)^2} d \mu_j(t)\) for all in Hardy spaces \(H^{1+\epsilon}(0 \leq \epsilon<\infty)\), and we describe those for which \(\mathcal{D} \mathcal{H}_{\mu_j}\) is a bounded* operator from \(H^{1+\epsilon}(0 \leq \epsilon<\infty)\) into \(H^{1+2 \epsilon}(\epsilon \geq 0)\).