From nonlinear eigenvalue problems to fast transforms, number theoretic operators, and special quantum states

I will discuss the properties of a special class of Hilbert and Banach space operators, called D-matrix operators, that arise naturally in the analysis of differential-algebraic eigenvalue problems. Onone hand, these objects furnish a matrix representation of the classical Dirichlet series and through that endow a new analytic perspective at some problems of number theory. On the other hand, D-matrix operators allow the construction of a large class of novel fast transforms for signal processing applications. They also suggest a new point of view at the morphology of nonselfadjoint operators. Time permitting, I will briey outline the related concept of E-matrices which represent specialized quantum states. I will show how such states can be implemented using a quantum circuit and, as an upshot, give a quantum algorithm for evaluation of the Dirichlet product.