Cesaro-Like Operators

Abstract

In previous work it was shown that the lower triangular generalized Hausdorff matrix $H_{\alpha}$ with nonzero entries $h_{nk} = (n+\alpha+1)^{-1}$, for $\alpha \geq 0$, is subnormal on $\ell^2$ if and only if $\alpha = 0, 1, 2, \ldots$. For $0 < h \leq 1$, the weighted Cesaro operator $C^{\prime}_h : \{a_n\} \to \{b_n\}$ on $\ell^2$, when $b_n = (a_0 + d_1a_1 + \cdots + d_na_n)/(n + 1)d_n$, is subnormal when $d_j^2 = \Gamma(j + 1)\Gamma(h)/\Gamma(j + h)$. In this paper we show that, when $d_j = \Gamma(j + 1)\Gamma(h)/\Gamma(j + h)$, the square of the weights chosen above, then the corresponding operator $C_h$ is bounded on $\ell^2$ for $0 < h < 3/2$, that $H_{\alpha}$ is bounded on $\ell^2$ for all non-integer $\alpha < 0$, and that $C_h$ is closely related to $H_{h-1}$. This relationship leads to our main result that $C_h$ is only subnormal when $h = 1$, when it corresponds to the original Cesaro operator with $\alpha = 0$ and each $d_j = 1$.