**Workshop **

Interpolation, inequalities, invariants, multilinear forms and related topics

**
October 8, 2007**** **

**
Centre
for Mathematical Sciences**

**Lund**** university**

**Sweden**** **

**Organized
by: **

**Michael Cwikel** (Technion
– Israel Institute of Technology, Haifa, Israel)

**
Jaak Peetre**
(Lund University, Lund, Sweden)

**
Lars-Erik
Persson** (Luleå University of
Technology, Luleå, Sweden)

**
Sergei Silvestrov
(**Lund University, Sweden, Lund, Sweden)

**Supported by: **

Crafoord Foundation, STINT Foundation, The Royal Swedish Academy of Sciences

**Program**

10.15-11.00 in MH:332A

**Michael Cwikel**,
Technion – Israel Institute of Technology, Haifa, Israel:

Complex interpolation of compact operators. Creeping towards an answer?

**11.10-11.55 i
MH:332A**

**Lars-Erik
Persson**, Luleå University of
Technology, Luleå, Sweden:

Hardy-type inequalities and convexity- historical remarks and some

recent results and questions

**12.00 -- 13.00 Lunch (Organized, follow
the organizers)**

13.15-13:45 i MH:332A

**Jaak Peetre**,
Lund University, Lund, Sweden:

Neoclassical Invariant theory

**13.50-14.10 in
MH:332A**

**Christian
Svensson**, Lund University,
Lund, Sweden:

Crossed product algebras and Banach algebras associated to dynamical systems.

14.10-14.30 in MH:332A

**Johan Oinert**,
Lund University, Lund, Sweden:

Crystalline graded rings and generalized crossed products

14.35-14.55 in MH:332A

**Erik Darpö**,
Uppsala University, Lund, Sweden:

On the classification of finite-dimensional division algebras

15.30-15.50 in MH:332A

**Olivier Verdier**,
Lund University, Lund, Sweden:

Application of interpolation between anisotropic Sobolev spaces to PDEs

15.50-16.35 in MH:332A

**Svante Janson**,
Uppsala University, Sweden:

Interpolation of subcouples and quotient couples

16.40-17.10: in MH:332A

**Sten Kaijser**,
Uppsala University, Uppsala, Sweden:

Pseudodeterminants

**17.15-18.00 in
MH:332A**

**Frank Hansen**,
Copenhagen University, Copenhagen, Denmark:

The application of operator monotone functions in economics

**Abstracts**

10.15-11.00 in MH:332A

**Michael Cwikel**,
Technion – Israel Institute of Technology, Haifa, Israel:

Complex interpolation of compact operators. Creeping towards an answer?

**Abstract: **

The
problem that we are struggling with goes back to Alberto Calderon's celebrated
and beautiful work on complex interpolation spaces. Thus it has been open since
August 15, 1963.

Suppose that T is a linear operator which acts compactly on both of the Banach
spaces X and Y. We still do not know whether, in general, T is also compact on
Calderon's space [X,Y]_{θ}.

Partial answers to this question have been given by Calderon himself, by Lions
and Peetre, Arne Persson, and many others. I
will report on some recent partial answers which at least give the impression
(illusion?) that we might be creeping closer to a full answer.

Unless the audience prefers otherwise, I will begin by recalling some of the
history and applications of interpolation theory and the definitions and some of
the basic relevant facts about Calderon's spaces. I will also indicate various
connections with Fourier series. Indeed Fourier series are apparently a
significant part of the arsenal we have for attacking this problem. (As
some of you may recall, Svante Janson has characterised Calderon's spaces via
sequence spaces of Fourier coefficients, and Fedor Nazarov has used Fourier
series to give an "almost counterexample" to a closely related question.)

Some background about these things can be found at

http://www.math.technion.ac.il/~mcwikel/compact

**11.10-11.55 i
MH:332A**

**Lars-Erik
Persson**, Luleå University of
Technology, Luleå, Sweden:

Hardy-type inequalities and convexity- historical remarks and some recent results and questions

**Abstract: **

`
I will first shortly present some
remarkably facts from the period 1915-1925, which finally led Hardy to present
and prove his inequality in his famous 1925 paper (in particular, the Swedish
mathematician Riesz was important for this development !), see [1]. After that I
will present the easiest (and, in my opinion, most natural) proof of this
inequality I know. This proof depends only on a convexity argument and this way
of thinking is also related to interpolation theory. Finally, I present some
facts from the books [2] and [3] but also some even more recent results and open
questions.
[1]A. Kufner, L. Maligranda and L.E. Persson, The prehistory of the Hardy
inequality, Amer. Math. Monthly 113 (2006), No. 8, 715-732.
[2]C.P. Niculescu and L.E. Persson, CONVEX FUNCTIONS AND THEIR APPLICATIONS-A
CONTEMPORARY APPROACH, Canad. Math. Series Books in Mathematics, Springer, 2006
(252 pages).
[3] A. Kufner, L. Maligranda and L.E. Persson, THE HARDY INEQUALITY. ABOUT ITS
HISTORY AND SOME RELATED RESULTS, Vydavatelsky Servis Publishing House, Pilsen,
2007 (161 pages).`

13.15-13:45 i MH:332A

**Jaak Peetre**,
Lund University, Lund, Sweden:

Neoclassical Invariant theory

**Abstract: **

Some topics and ideas related to this interesting neoclassical subject will be presented.

**13.50-14.10 in
MH:332A**

**Christian
Svensson**, Lund University,
Lund, Sweden:

Crossed product algebras and Banach algebras associated to dynamical systems.

**Abstract: **

Given a discrete dynamical system, one may construct an associative (non-commutative) complex algebra with multiplication determined via the action defining the system, being an example of a crossed product. It turns out that for large classes of systems, one obtains striking equivalences between, in particular, topological properties of the system and algebraic properties of the crossed product. In this talk we will discuss the ideal structure of crossed products and ideal intersection properties of (commutative) subalgebras between the canonical "coefficient algebra" and its commutant.

14.10-14.30 in MH:332A

**Johan Oinert**,
Lund University, Lund, Sweden:

Crystalline graded rings and generalized crossed products

**Abstract: **

In this talk we will focus on a certain 'intersection property' of ideals in certain kinds of rings and algebras. We start by looking at C*-crossed product algebras associated with dynamical systems, which is our main motivating example. We will then show that the 'intersection property' also holds in a purely algebraic framework, namely for algebraic crossed products. Thereafter we generalize this result to a more general class of rings, the so called 'crystalline graded rings'. Both of these classes of rings are group graded, but we will define a new class of rings (with motivation coming from irreversible dynamical systems) which is only assumed to be monoid graded. After giving the definition, we shall give some concrete examples of such rings and show that in general the 'intersection property' need not hold for these rings.

14.35-14.55 in MH:332A

**Erik Darpö**,
Uppsala University, Lund, Sweden:

On the classification of finite-dimensional division algebras

**Abstract: **

Let *k* be a field. A division algebra
over *k* is a vector space A over *k* endowed with a bilinear
multiplication map AxA --> A; (x,y) --> xy such that left and right
multiplication with any non-zero element in A are invertible linear operators on
A. In general, a division algebra is not assumed to be associative or
commutative, nor to have an identity element.

We shall give a survey of the classification problem for finite-dimensional
division algebras, with focus on algebras over the real ground field. This
includes the construction of the four classical real

division algebras, theorems by Frobenius and Zorn on associative and alternative
division algebras respectively, and the celebrated (1,2,4,8)-theorem, which
asserts that every finite-dimensional division algebra over **R** has
dimension either 1, 2, 4 or 8.

We will also touch upon some recent development in the field, originating mainly
from Osborns work on quadratic division algebras in the sixties.

15.30-15.50 in MH:332A

**Olivier Verdier**,
Lund University, Lund, Sweden:

Application of interpolation between anisotropic Sobolev spaces to PDEs

**Abstract: **

The interpolation between anisotropic Sobolev spaces with p=2 is elementary but turns out to be an effective tool, coupled with the vectorial Sobolev inequalities, to obtain estimates of nonlinear terms. I will show some applications to the Burgers equation with highly irregular source terms.

15.50-16.35 in MH:332A

**Svante Janson**,
Uppsala University, Sweden:

Interpolation of subcouples and quotient couples

**Abstract: **

A subcouple (Y_{0},Y_{1})
of a Banach couple (X_{0},X_{1}) is K-closed if, for elements in
Y_{0}+Y_{1}, the K-functional evaluated in the smaller couple is
equivalent to the K-functional evaluated in the larger couple.

This is not true in general, but it is a useful property when it holds. This
property was used and investigated by Pisier, in connection with interpolation
of H^{p}-spaces. I will discuss this and general results concerning
interpolation by the real method of subcouples and quotient couples. It seems
that very little is known about corresponding results for the complex method.

This is a survey based on my paper with the same title in Ark. Mat. 31 (1993),
no. 2, 307--338.

16.40-17.10: in MH:332A

**Sten Kaijser**,
Uppsala University, Uppsala, Sweden:

Pseudodeterminants

**Abstract: **

This talk will be devoted to Pseudodeterminants.

**17.15-18.00 in
MH:332A**

**Frank Hansen**,
Copenhagen University, Copenhagen, Denmark:

The application of operator monotone functions in economics

Two notions in microeconomics, decreasing
relative risk premium and risk vulnerability, are connected to matrix or
operator monotonicity. We show that a decision maker with an increasing utility
function has preferences representing decreasing relative risk premium, if and
only if his utility function is matrix monotone of order two. We then show that
this property may be equivalently formulated in terms of preferences on binary
lotteries. We finally explain the notion of risk vulnerability and show that an

operator monotone function necessarily is risk vulnerable.

`
`