Generalized Hilbert operators on Hardy spaces

Let $g\in H(\mathbb D)$, the generalized Hilbert operator $\mathcal H_g$ is defined by $$\mathcal H_g(f)(z)=\int_0^1 f(t)g'(tz)dt,\ \ z\in \mathbb D\, \ \ f \in H(\mathbb D).$$ Let $\mathcal R_p=\mathcal H(H^p)$ be the range of the classical Hilbert operator on Hardy space, equipped with the pullback norm, and let $(\mathcal R_p,H^p)$ denote the Hadamard multiplier space. For $1<p<\infty$, we prove the exact multiplier characterization $$\mathcal H_g:H^{p}\longrightarrow H^{p} \ \ \text{is bounded} \quad\Longleftrightarrow\quad g'\in(\mathcal R_p,H^p),$$ and an equivalent Hilbert-matrix bilinear criterion $\mathfrak B_p(g)<\infty$. We identify the multiplier space completely when $1<p\le2$: $$(\mathcal R_p,H^p)=H\left(p,\infty,\frac1{p'}\right).$$ For $p>2$, we prove that the multiplier space $(\mathcal R_p,H^p)$ is strictly contained in $H\left(p,\infty,\frac1{p'}\right)$. This shows that $g\in \Lambda(p,1/p)$ does not imply that $\mathcal H_g$ is bounded on $H^p$, giving a negative answer to the conjecture posed by Galanopoulos, Girela, Pel\'aez and Siskakis. In addition, we locate two previously known sufficient classes inside the multiplier space. This allow us obtain a complete coefficient characterization of $\mathcal H_g$ on $H^{p}$ for $g \in H(\mathbb D)$ with nonnegative decreasing Taylor coefficients. We then study the structure of $(\mathcal R_p,H^p)$. % It turns out that $(\mathcal{R}_p,H^p)$ contains all polynomials as well as Cauchy transforms. We show that the multiplier spaces $(\mathcal{R}_p,H^p)$ form a strictly increasing family with respect to the exponent $p$.