On the functions of one selfadjoint integral operator.
We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed...
Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.
A new version of the maximum principle is presented. The classical Kantorovich-Rubinstein principle gives necessary conditions for the maxima of a linear functional acting on the space of Lipschitzian functions. The maximum value of this functional defines the Hutchinson metric on the space of probability measures. We show an analogous result for the Fortet-Mourier metric. This principle is then applied in the stability theory of Markov-Feller semigroups.
It is proved that Riesz elements in the intersection of the kernel and the closure of the image of a family of derivations on a Banach algebra are quasinilpotent. Some related results are obtained.
In a recent paper by H. X. Cao, J. H. Zhang and Z. B. Xu an -Lipschitz operator from a compact metric space into a Banach space is defined and characterized in a natural way in the sence that is a -Lipschitz operator if and only if for each the mapping is a -Lipschitz function. The Lipschitz operators algebras and are developed here further, and we study their amenability and weak amenability of these algebras. Moreover, we prove an interesting result that and are isometrically...
We study the local spectral properties of both unilateral and bilateral weighted shift operators.
In the present work, using a formula describing all scalar spectral functions of a Carleman operator of defect indices in the Hilbert space that we obtained in a previous paper, we derive certain results concerning the localization of the spectrum of quasi-selfadjoint extensions of the operator .