Past Functional Analysis Seminar

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.

We study a number of regularity properties of C*-algebras which are

intimately related in the case of nuclear C*-algebras.

These properties can be expressed topologically (as dimension type

conditions), C*-algebraically (as stability under tensoring with suitable

strongly self-absorbing C*-algebras), and at the level of homological

invariants (in terms of comparison properties of projections, or positive

elements, respectively).

We explain these concepts and some known relations between them,

and outline their relevance for the classification program. (As a particularly

satisfying application, one obtains a classification result for C*-algebras

associated to compact, finite-dimensional, minimal, uniquely ergodic,

dynamical systems.)

Furthermore, we investigate potential applications of these technologies

to other areas, such as coarse geometry.

The talk will give two entirely different answers to the question asked in the title of the talk. A topological answer will be based on the classical notion of Baire category. A measure theoretical answer will be based on the much newer notion of prevalence/shyness.

A number of results are known establishing exponential/polynomial/logarithmic decay of energy for (damped) wave equations. Typically the results have been obtained by estimating the resolvent of the generator of a certain bounded $C_0$-semigroup, and then showing that the estimates imply certain rates of decay for the smooth orbits of the semigroup. We shall present a result of this type, which is both general and sharp, and which has a simple proof thanks to a device of Newman and Korevaar.

The Schur product is the commutative operation of entrywise

multiplication of two (possibly infinite) matrices. If we fix a matrix

A and require that the Schur product of A with the matrix of any

bounded operator is again the matrix of a bounded operator, then A is

said to be a Schur multiplier; Schur multiplication by A then turns

out to be a completely bounded map. The Schur multipliers were

characterised by Grothendieck in the 1950s. In a 2006 paper, Kissin

and Shulman study a noncommutative generalisation which they call

"operator multipliers", in which the theory of operator spaces plays

an important role. We will present joint work with Katja Juschenko,

Ivan Todorov and Ludmilla Turowska in which we determine the operator

multipliers which are completely compact (that is, they satisfy a

strengthening of the usual notion of compactness which is appropriate

for completely bounded maps).

A problem that has been open since (at least) 1960 is whether there exists an infinite-dimensional Banach space on which every bounded linear operator is a compact perturbation of a scalar multiple of the identity. The HI spaces constructed by Gowers and Maurey have "few operators" in a slightly weaker sense than this. Combining HI methods with a technique due to Bourgain, Spiros Argyros and the speaker have recently constructed a space which solves the original problem. The seminar talk will attempt to convey some of the underlying ideas.