Forthcoming Seminars › Seminar Series

Functional Analysis Seminar

Tue, 03/02/2009
17:00 		Pierre Portal (Lille) Functional Analysis Seminar Add to calendar L3
Hardy spaces associated with operators
Tue, 17/02/2009
17:00 		John Weir (King's College, London) Functional Analysis Seminar Add to calendar L3
A non-self-adjoint operator with real spectrum
Fri, 27/02/2009
15:15 		Alexandru Aleman (NBFAS Meeting) (Lund) Functional Analysis Seminar Add to calendar L3
Nearly Invariant Spaces of Analytic Functions
We consider Hilbert spaces $ H $ which consist of analytic functions in a domain $ \Omega\subset\mathbb{C} $ and have the property that any zero of an element of $ H $ which is not a common zero of the whole space, can be divided out without leaving $ H $. This property is called near invariance and is related to a number of interesting problems that connect complex analysis and operator theory. The concept probably appeared first in L. de Branges' work on Hilbert spaces of entire functions and played later a decisive role in the description of invariant subspaces of the shift operator on Hardy spaces over multiply connected domains. There are a number of structure theorems for nearly invariant spaces obtained by de Branges, Hitt and Sarason, and more recently by Feldman, Ross and myself, but the emphasis of my talk will be on some applications; the study of differentiation invariant subspaces of $ C^\infty(\mathbb{R}) $, or invariant subspaces of Volterra operators on spaces of power series on the unit disc. Finally, we discuss near invariance in the vector-valued case and show how it can be related to kernels of products of Toeplitz operators. More precisely, I will present in more detail the solution of the following problem: If a finite product of Toeplitz operators is the zero operator then one of the factors is zero.
Fri, 27/02/2009
16:45 		Alexandru Aleman (NBFAS Meeting) Functional Analysis Seminar Add to calendar L3
Nearly Invariant Spaces of Analytic Functions (cont)
Sat, 28/02/2009
10:00 		Ralph Chill (NBFAS Meeting) (Metz) Functional Analysis Seminar Add to calendar L3
Regularity of functions, and the norm-continuity problem for semigroups
It is a fundamental problem in harmonic analysis to deduce regularity or asymptotic properties of a bounded, vector-valued function, defined on a half-line, from properties of its Laplace transform. In the first part of this talk, we will study how the analytic extendability of the Laplace transform to certain large domains, and the boundedness therein, (almost) characterizes regularity properties like analyticity and differentiability of the original function. We will also see that it is not clear how to characterize continuity in this way; naive counterparts/generalizations of the results which hold for analyticity and differentiability admit easy counterexamples. Characterizing continuity becomes not easier, if one considers bounded, strongly continuous semigroups: it was a longstanding open problem whether the decay to zero of the resolvent of the generator along vertical lines characterizes immediate norm-continuity of the semigroup with respect to the operator-norm. After several affirmative results in Hilbert space and for positive semigroups on $ L^p $ spaces, a negative answer to this question was recently given by Tamas Matrai. In the second part of this talk, we will give some counterexamples which are conceptually different to the one given by Matrai. In fact, we will present a new method of constructing semigroups, by considering operators and algebra homomorphisms on $ L^1 $ with specific properties. Our examples rule out the possibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent-norm decay on vertical lines.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

Syndicate content