In this set of lecture notes we focus on the point-value representation obtained by looking at a particular set of points, the nth roots of unity. In the field of complex numbers C, there are exactly n different solutions to the equation x = 1. We call these solutions the n-th roots of unity. One of these roots of unity is ωn = cos(2π/n)+ i sin(2π/n), and this is called the principal nth root o...

If T is an operator on a Hubert space %, this paper gives necessary and sufficient conditions on 7" such that C*(T), the C*-algebra generated by T, is generated by a unilateral shift of some multiplicity. This result is then specialized to the cases in which T is a hyponormal or subnormal operator. In particular, it is shown how to prove a recent conjecture of C. R. Putnam as a consequence of o...

In this set of lecture notes we focus on the point-value representation obtained by looking at a particular set of points, the nth roots of unity. In the field of complex numbers C, there are exactly n different solutions to the equation x = 1. We call these solutions the n-th roots of unity. One of these roots of unity is ωn = cos(2π/n)+ i sin(2π/n), and this is called the principal nth root o...

We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1)-hyponormality for each k ≥ 1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals. 1. Notation and Preliminaries The Lifting Problem for Commuti...

We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros of the nth eigenfunction of the Schrödinger operator on a quantum graph is related to the stability of the nth eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the nth eigenvalue as a function of the magnetic perturbation and show that its Morse index at zer...