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Publications and source records attributed to H. C. Rhaly Jr.
The procedure that was used in an earlier paper on the generalized Cesàro matrices of order two is adapted here to show that generalized Cesàro matrices of order three are supraposinormal on $\ell^2$. This leads to information about their posinormality, coposinormality, and hyponormality. The reader is then invited to formulate a conjecture regarding generalized Cesàro matrices of other orders.
A weighted mean matrix whose weight sequence is linear with positive coefficients is shown to be a posinormal operator on $\ell^2$. This operator is also shown to be coposinormal, so it and its adjoint have the same null space and the same range. The posinormality result leads to a proof that the weighted mean matrix associated with the sequence of odd positive integers is hyponormal, as well as a conjecture regarding a more general linear case.
This note calls attention to an alternative version of the main result from [4], which can be used together with Maclaurin series expansions and trigonometric identities to show that the terraced matrices generated by the sequences $\{\ln (1+1/(n+1)) : n \geq 0\}$, $\{\tan (1/(n+2)) : n \geq 0\}$, and $\{\sinh (1/(n+2)) : n \geq 0\}$ are hyponormal operators on $\ell^2$. Along the way it is also shown that the matrix generated by $\{\tan (1/(n+1)) : n \geq 0\}$ is not a hyponormal operator on $\ell^2$.
It is already known that the Cesàro matrices of orders one and two are coposinormal, hyponormal operators on $\ell^2$. Here it is shown that the Cesàro matrices of order three and four are also coposinormal, hyponormal; the proofs employ posinormality, achieved by means of a diagonal interrupter, and elementary computational techniques from calculus. A conjecture is then propounded for the Cesàro matrix of positive integer order greater than four.
It is well known that the generalized Cesàro matrices of order one are hyponormal operators on $\ell^2$, and it has recently been shown that the Cesàro matrix of order two is also hyponormal. Here the relatively new concept of supraposinormality is used to show that the generalized Cesàro matrices of order two are both posinormal and coposinormal, and that "most" of them are also hyponormal. A conjecture is propounded that would extend the hyponormality result.