On the Cesáro operator in weighted‎ ‎$\ell^2$-sequence spaces and the generalized concept of normality

Abstract

‎The weighted Cesáro operator $C_h$ in $\ell^2(h)$-spaces is‎ ‎investigated in terms of several concepts of normality‎, ‎where $h$‎ ‎denotes a positive discrete measure on $\mathcal{N}$‎. ‎We classify exactly‎ ‎those $h$ for which $C_h$ is hyponormal‎. ‎Two examples related to the‎ ‎Haar measures of orthogonal polynomials are discussed‎. ‎We show that‎ ‎the Cesáro operator is not always paranormal‎. ‎Furthermore‎, ‎we‎ ‎prove that the Cesáro operator is not quasinormal for any choice‎ ‎of $h$.