The essential norm, block sizes, and some Generalized Hilbert operators on lp

Abstract

We examine the generalized Hilbert (matrix) operators $$H_{g,\gamma }:(a_n)\mapsto \sum _{n=1}^{\infty }(\frac{k}{n}{)}^{\gamma }\frac{g_k{a}_n}{n+k}$$ on the ${\ell }^p$ spaces, $1<p<\infty$, where $g=(g_n)$ is a sequence and $-1/p<\gamma <1-1/p$. Given a partition of the natural numbers ${\bigcup }_j{I}_j=\mathbb{N}$, we encode $g$ to our investigation via its ${\ell }^p$-means $G_j$ over the sets $I_j$. We prove that if $I_j$ increases exponentially, the data $(G_j)$ characterizes boundedness, but is insufficient to determine the exact value of the essential norm. Nonetheless, if the growth rate of $(I_j)$ is subexponential, the corresponding data $(G_j)$, given that the tail converge, is sufficient to determine the exact value, in which case we also calculate it. Moreover, it is shown that the boundedness of $(G_j)$ is sufficient, but not necessary (and necessary, but not sufficient) to ensure $H_{g,\gamma }$ is bounded if the underlying partition $(I_j)$ increases subexponentially (and superexponentially, respectively). Using the dual operator, we also prove that the essential norm of $H_{g,\gamma }$ is comparable with ${\mathrm{limsup}}_{j}G(j)$ when $(I_j)$ grows exponentially.

10 pages