Orbits of operators commuting with the Volterra operator

Asymptotic estimates of the norms of orbits of certain operators that commute with the classical Volterra operator V acting on L-P[0,1], with 1 0, but also to operators of the form phi (V), where phi is a holomorphic function at zero. The method to obtain the estimates is based on the fact that the Riemann-Liouville operator as well as the Volterra operator can be related to the Levin-Pfluger theory of holomorphic functions of completely regular growth. Different methods, such as the Denjoy-Carleman theorem, are needed to analyze the behavior of the orbits of I - cV, where c > 0. The results are applied to the study of cyclic properties of phi (V), where phi is a holomorphic function at 0.

Non-weakly supercyclic operators

Several methods based on an easy geometric argument are provided to prove that a given operator is not weakly supercyclic. The methods apply to different kinds of operators like composition operators or bilateral weighted shifts. In particular, it is shown that the classical Volterra operator is not weakly supercyclic on any of the LP [0, 1] spaces, 1

Analyticity of smooth eigenfunctions and spectral analysis of the Gauss map

We provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form Uf(x) = integral a(x, y) f(b( x, y)) mu(dy) acting on functions f: [u, v] --> C ( evolution operators of one-dimensional dynamical systems and Markov processes have this form). We estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the Frobenius-Perron operator of the continuous fraction Gauss map. We prove that any infinitely differentiable eigenfunction f of this Frobenius-Perron operator, corresponding to a non-zero eigenvalue admits a (unique) analytic extension to the set C\(-infinity, 1]. Analyzing the spectrum of the Frobenius Perron operator in spaces of smooth functions, we extend significantly the domain of validity of the Mayer and Ropstorff asymptotic formula for the decay of correlations of the Gauss map.

Decay spectrum and decay subspace of normal operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Structure of spectra of linear operators in Banach spaces

Descriptive characterizations of the point, the continuous, and the residual spectra of operators in Banach spaces are put forward. In particular, necessary and sufficient conditions for three disjoint subsets of the complex plane to be the point spectrum, the continuous spectrum, and the residual spectrum of a linear continuous operator in a separable Banach space are obtained.