On Riesz bases of root functions for a class of functional-differential operators on a graph

For a functional-differential operator on a geometric graph consisting of two edges one of which is a cycle, we show that the system of root functions is a Riesz basis with parentheses.     

Riesz inequality for systems of root vector functions of the Dirac operator

We establish a criterion for the Riesz property of systems of root vector functions of the one-dimensional Dirac operator.     

On the existence of wave operators for Dirac operators

It is shown that Kuroda's criterion for the existence of wave operators in the Schrödinger case is also valid for Dirac operators if the mass m0. If m=0 a similar but stronger condition is sufficient.Part of the author's doctoral thesis at the University of Munich, Germany.     

Resonances for 1D massless Dirac operators

We consider the 1D massless Dirac operator on the real line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain, (3) the trace formula in terms of resonances.     

Dynamical lower bounds for 1D Dirac operators

Quantum dynamical lower bounds for continuous and discrete one-dimensional Dirac operators are established in terms of transfer matrices. Then such results are applied to various models, including the Bernoulli–Dirac one and, in contrast to the discrete case, critical energies are also found for the continuous Dirac case with positive mass. R. A. Prado was supported by FAPESP (Brazil). C. R. de Oliveira was partially supported by CNPq (Brazil).     

The Riesz bases consisting of eigen and associated functions for a functional differential operator with variable structure

We consider a functional differential operator with variable structure with an integral boundary condition. We prove that its eigen and associated functions form a Riesz basis with brackets in the space L ₂³[0, 1].     

Riesz bases for p-subordinate perturbations of normal operators

For a class of unbounded perturbations of unbounded normal operators, the change of the spectrum is studied and spectral criteria for the existence of a Riesz basis with parentheses of root vectors are established. A Riesz basis without parentheses is obtained under an additional a priori assumption on the spectrum of the perturbed operator. The results are applied to two classes of block operator matrices.     

Riesz bases formed by root functions of a functional-differential equation with a reflection operator

We find conditions under which the system of root functions of the operator

$$L_y = l[y] = ay'(x) + y'(1 - x) + p_1 (x)y(x) + p_2 (x)y(1 - x),x \in [0,1],U_1 (y) = \int\limits_0^1 {y(t)d\sigma (t) = 0,} $$

is a Riesz basis in L ₂[0, 1].

     

On the existence of wave operators for some Dirac operators with square summable potential

We study the Dirac operators on the half-line. If the potential is square summable, we prove existence of the wave operators.     

Riesz* homomorphisms on pre-Riesz spaces consisting of continuous functions

In the theory of operators on a Riesz space (vector lattice), an important result states that the Riesz homomorphisms (lattice homomorphisms) on C(X) are exactly the weighted composition operators. We extend this result to Riesz* homomorphisms on order dense subspaces of C(X). On those subspace we consider and compare various classes of operators that extend the notion of a Riesz homomorphism. Furthermore, using the weighted composition structure of Riesz* homomorphisms we obtain several results concerning bijective Riesz* homomorphisms. In particular, we characterize the automorphism group for order dense subspaces of C(X). Lastly, we develop a similar theory for Riesz* homomorphisms on subspace of \(C_0(X)\), for a locally compact Hausdorff space X, and apply it to smooth manifolds and Sobolev spaces.     

Sparse 1D discrete Dirac operators I: Quantum transport

Some dynamical lower bounds for one-dimensional discrete Dirac operators with different classes of sparse potentials are presented, and the particular role of the particle mass is emphasized.     

Two-sided estimates for root vector functions of the Dirac operator

We consider the one-dimensional Dirac operator. We derive a shift formula for its root vector functions and prove anti-a priori and two-sided estimates for various L _p -norms of these functions.     

Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

One dimensional Dirac operators $L_{bc}\left(v\right)y=i\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ?1\hfill \end{array}\right)\frac{dy}{dx}+v\left(x\right)y\text{,}y=\left(\begin{array}{c}\hfill y_1\hfill \\ \hfill y_2\hfill \end{array}\right)\text{,}x\in [0,\pi ]\text{,}$ considered with $L^2$-potentials $v\left(x\right)=\left(\begin{array}{cc}\hfill 0\hfill & \hfill P\left(x\right)\hfill \\ \hfill Q\left(x\right)\hfill & \hfill 0\hfill \end{array}\right)$ and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc$, the spectrum of the free operator $L_{bc}\left(0\right)$ is simple while the spectrum of $L_{bc}\left(v\right)$ is eventually simple, and the corresponding normalized root function systems are Riesz bases. For expansions of functions of bounded variation about these Riesz bases, we prove the uniform equiconvergence property and point-wise convergence on the closed interval $[0,\pi ]$. Analogous results are obtained for regular but not strictly regular $bc$.     

Maximal operators, Riesz transforms and Littlewood-Paley functions associated with Bessel operators on BMO

In this paper we study boundedness properties of certain harmonic analysis operators (maximal operators for heat and Poisson semigroups, Riesz transforms and Littlewood-Paley g-functions) associated with Bessel operators, on the space BMOo(R) that consists of the odd functions with bounded mean oscillation on R.     

Mean value formula for the root vector functions of the Dirac operator

We consider the one-dimensional Dirac operator on an arbitrary interval and obtain a mean value formula for the root vector functions of this operator.