Past Functional Analysis Seminar

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

$f(A) \leq f(B)$.

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.

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.

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.