Workshop

 

Interpolation, inequalities, invariants, multilinear forms and related topics

 

October 8, 2007

 

Centre for Mathematical Sciences

Lund university

Sweden

 

Organized by:

 

Michael Cwikel (Technion ľ Israel Institute of Technology, Haifa, Israel)

Jaak Peetre (Lund University, Lund, Sweden)

Lars-Erik Persson (Luleň University of Technology, Luleň, Sweden)

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

 

 

Supported by:

 

Crafoord FoundationSTINT Foundation, The Royal Swedish Academy of Sciences

 

Program

 

10.15-11.00 in MH:332A

Michael Cwikel, Technion ľ 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, Luleň University of Technology, Luleň, 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 Darp÷, 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 ľ 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, Luleň University of Technology, Luleň, 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 Darp÷, 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 (Y0,Y1) of a Banach couple (X0,X1) is K-closed if, for elements in Y0+Y1, 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 Hp-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.