We study infinite-dimensional well-posed linear systems with output feedback such that the closed-loop system is well-posed. The generator A of the open-loop system is assumed to be diagonal, i.e., the state space X (a Hilbert space) has a Riesz basis consisting of eigenvectors of A. We investigate when the closed-loop generator A is Riesz spectral, i.e, its generalized eigenvectors form a Ries...

Assuming that a Nemytskii operator maps a subset of the space of bounded variation functions in the sense of Riesz into another space of the same type, and is uniformly continuous, we prove that the generator of the operator is an affine function.

For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL on L for some α > 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave e...

The continuity for some multilinear operators generated by certain integral operators and Lipschitz functions on some Hardy and Herz-type spaces are obtained. The operators include Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.

It is well known that short-time expansions of heat kernels correlate to formal high-frequency expansions of spectral densities. It is also well known that the latter expansions are generally not literally true beyond the first term. However, the terms in the heat-kernel expansion correspond rigorously to quantities called Riesz means of the spectral expansion, which damp out oscillations in th...

We use Fourier multipliers of the Dirac operator and Cauchy transform to obtain composition theorems and integral representations. In particular we calculate the multiplier of the Π-operator. This operator is the hypercomplex version of the Beurling Ahlfors transform in the plane. The hypercomplex Beuling Ahlfors transform is a direct generalization of the Beurling Ahlfors transform and reduces...

We utilize the Bellman function technique to prove a bilinear dimension-free inequality for the Hermite operator. The Bellman technique is applied here to a non-local operator, which at first did not seem to be feasible. As a consequence of our bilinear inequality one proves dimension-free boundedness for the Riesz-Hermite transforms on L with linear growth in terms of p. A feature of the proof...