Square Functions for Commuting Families of Ritt Operators

Differential Operators Commuting with Invariant Functions

Common Lyapunov functions for families of commuting nonlinear systems

We present constructions of a local and global common Lyapunov function for a finite family of pairwise commuting globally asymptotically stable nonlinear systems. The constructions are based on an iterative procedure, which at each step invokes a converse Lyapunov theorem for one of the individual systems. Our results extend a previously available one which relies on exponential stability of t...

Square Functions for Bi-lipschitz Maps and Directional Operators

First we prove a Littlewood-Paley diagonalization result for bi-Lipschitz perturbations of the identity map on the real line. This result entails a number of corollaries for the Hilbert transform along lines and monomial curves in the plane. Second, we prove a square function bound for a single scale directional operator. As a corollary we give a new proof of part of a theorem of Katz on direct...

On Commuting Differential Operators

The theory of commuting linear differential expressions has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P,L). Gesztesy and the present author have established a relationship of this circle of ideas with the property that all solutions of the differential equations Ly = zy, z ∈ C, are meromorphic. In this paper this relationship is explored ...

Essentially Commuting Toeplitz Operators

For f in L∞, the space of essentially bounded Lebesgue measurable functions on the unit circle, ∂D, the Toeplitz operator with symbol f is the operator Tf on the Hardy space H2 of the unit circle defined by Tfh = P (fh). Here P denotes the orthogonal projection in L2 with range H2. There are many fascinating problems about Toeplitz operators ([3], [6], [7] and [20]). In this paper we shall conc...