Forthcoming events in this series
Monotone Matrix functions in several variables
Abstract
In 1934, K. Loewner characterized functions that preserve
matrix inequalities,
i.e.\ those f with the property that whenever A and B are self-adjoint
matrices of the same dimension,
with $A \leq B$, then $f(A) \leq f(B)$.
In this talk, I shall discuss how to characterize monotone matrix
functions of several variables,
namely functions f with the property that if $A = (A_1, \dots , A_n) $
is an n-tuple of commuting self-adjoint matrices,
and $B = (B_1, \dots, B_n)$ is another, with each $A_i \leq B_i$, then
$f(A) \leq f(B)$.
Brascamp-Lieb inequalities and some applications
Abstract
I will give an overview of the classical Brascamp-Lieb inequality
from its birth to recent developments. I will discuss certain nonlinear
generalisations of the Brascamp-Lieb inequality and applications of
such inequalities in harmonic and geometric analysis.
Current state of the almost periodic factorization problem: a survey
Regularity of functions, and the norm-continuity problem for semigroups (cont)
Regularity of functions, and the norm-continuity problem for semigroups
Abstract
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.
Nearly Invariant Spaces of Analytic Functions
Abstract
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 {\it 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.