Subnormal generalized Hausdorff operators

Abstract

One type of generalized Hausdorff matrix is the lower triangular matrix with entries h n k = ( n − k n + α ) Δ n − k ν k {h_{nk}} = (_{n - k}^{n + \alpha }){\Delta ^{n - k}}{\nu _k} , where Δ ν n = ν n − ν n + 1 , ν n = ∫ 0 1 t n + α d β ( t ) \Delta {\nu _n} = {\nu _n} - {\nu _{n + 1}},{\nu _n} = \smallint _0^1{t^{n + \alpha }}d\beta (t) for some β ( t ) \beta (t) of bounded variation on [0, 1] and for some α ⩾ 0 \alpha \geqslant 0 . The matrix H α {H_\alpha } generated by ν n = ( n + α + 1 ) − 1 {\nu _n} = {(n + \alpha + 1)^{ - 1}} is shown to be a subnormal operator on l 2 {l^2} if α \alpha is a nonnegative integer.