Eva Gallardo Gutierrez, Universidad Complutense de Madrid. Compact perturbations of normal operators: invariant subspaces, spectral idempotents and decomposability

Giovedì 2 Maggio ore 16 aula Dal Passo

Abstract:

After addressing the problem regarding the existence of non-trivial closed invariant subspaces of finite-rank perturbations of diagonalizable normal operators acting boundedly on separable, infinite-dimensional complex Hilbert spaces, we will show that a large class of such operators are decomposable, extending in particular recent results of Foias, Jung, Ko and Pearcy. Decomposable operators were introduced by Foias in the sixties and many operators in Hilbert spaces are decomposable as unitary operators, self-adjoint operators or more generally normal operators. In a broad sense, decomposable operators have the most general kind of spectral decomposition possible.  Consequently, every operator in the aforementioned class has a rich spectral structure and plenty of non-trivial closed hyperinvariant subspaces. Based on joint works with F. J. González-Doña.

Anders Karlsson, Genéve. A fixed point theorem for isometries of metric spaces

Mercoledì 10 aprile ore 16 aula A0

Abstract:

Any isometry of a metric spaces must have a fixed point in a certain canonical compactification of the space. This result is more concrete if the space admits a convex bicombing, such as the lines in a Banach space or geodesics in a space of nonpositive curvature. It applies to invertible bounded operators on a Hilbert space H acting on Pos, the space of positive operator on H, giving a non-trivial invariant metric functional. Another application is to mean ergodic theorems when the usual formulation fails. There are also other examples of diffeomorphims acting by isometry on certain spaces of Riemannian or Kähler metrics. Biholomorphisms provide yet another source of isometries, but in that setting the conclusion is somewhat less clear in general. For proper metric spaces, the compactification in question is the horofunction compactification. All terms will be defined and explained.

Luna Lomonaco, IMPA. The Mandelbrot set and its Satellite copies

Giovedì 5 Ottobre ore 16.00 , Aula D'Antoni

Abstract:

For a polynomial on the Riemann sphere, infinity is a (super) attracting fixed point, and the filled Julia set is the set of points with bounded orbit. Consider the quadratic family P_c(z)=z^2+c. The Mandelbrot set M  is the set of parameters c such that the filled Julia set of P_c is connected. Douady and Hubbard proved the existence of homeomorphic copies of M inside of M, which can be primitive (roughly speaking the ones with a cusp) or a satellite (without a cusp). Lyubich proved that the primitive copies of M are quasiconformally homeomorphic to M, and that the satellite ones are quasiconformally homeomorphic to M outside any small neighbourhood of the root. The satellite copies are not quasiconformally homeomorphic to M (as we cannot straighten a cusp quasiconformally), but are they mutually quasiconformally homeomorphic? In a joint work with C. Petersen we prove that this question has in general a negative answer, but positive in the case the satellite copies have rotation number with same denominator (this last part is work in progress). 

Jesús Oliva Maza, Universidad de La Rioja. Spectra of invertible weighted composition operators (on D) of hyperbolic symbol

Giovedì 14 Settembre ore 14.00 , Aula D'Antoni

Abstract:

Weighted composition operators, i.e., operators of the form (uCψf)(z) = u(z)f(ψ(z)) for holomorphic u : D → C, ψ : D → D, play an important role in the study of Banach spaces of holomorphic functions on the unit disc D. For instance, all surjective isometries on Hardy spaces and weighted Bergman spaces are given by {(φ′)γCφ | φ : D → D automorphism}, for a parameter γ > 0 depending on the space.

Properties of such operators, such as norm, compactness, spectra, etc., have been thoroughly studied through the last decades. In particular, the spectrum of invertible weighted composition operators uCφ is known (under mild assumptions on u) in the case the automorphism φ of D is either elliptic or parabolic. If φ is hyperbolic, there are only partial results about the spectrum of uCφ.

In this talk, we present an ongoing work where we obtain the spectrum of an invertible uCφ with φ hyperbolic. Therefore, such a result completes the spectral description of invertible weighted composition operators (again, under mild assump- tions on u). The key of our work is the following remarkable finding: for each in- vertible uCφ with φ hyperbolic, there exists a one-parameter group (utCφt)t∈R such that u1Cφ1 = uCφ.