A study of Rhaly operators on Dirichlet series with real frequencies
Koh, Ronald Joon Wei
Date of Issue2019-05-09
School of Physical and Mathematical Sciences
In this thesis, we recall several important properties of the general Dirichlet series and characterize the entire Dirichlet series in the ﬁrst two chapters. We also recall the Cesaro operator, an operator which was deﬁned based on the Cesaro sum, and is heavily related to the Cesaro limits. A more general version of the Cesaro operator; Rhaly operators, will also be deﬁned and the study of its properties on the Hilbert space of entire Dirichlet series is the bedrock of this thesis. Criterions will be provided for properties of Rhaly operators on the Hilbert space of entire Dirichlet series like boundedness and invariance. Compactness and Schatten p-class inclusions will also be studied on a smaller class of entire Dirichlet series; the small weighted Hilbert space of Dirichlet series. An additional section will also study injectivity and the closed range property of bounded Rhaly operators.