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functional analysis seminar - spring 2013

time

We will meet on Fridays from 14.00-17.00 in the Snellius building, Leiden University. This is the (still tentative) program:

February 15, room 312.
Introduction, begin of Section 1.2.
Speaker: Onno van Gaans

February 22.
Rest of 1.2, 2.3.
Speaker: Onno van Gaans

March 8.
3.1: AL and AM-spaces
3.2: Complex Banach lattices
Speaker: Hent van Imhoff

March 15, room 412.
4.2: Multiplication operators
4.3: Lattice homomorphisms and algebra homomorphisms (through p. 148)
Speaker: Onno van Gaans

March 22, room 176.
4.3: from paragraph above 4.31
5.1: Integral operators
Speaker: Anna Rusinek

April 12, room Gorlaeus C7
5.3: Conditional expectations and positive projections
Speaker: Fengnan Gao

April 19, room Gorlaeus C6
5.2: Abstract integral operators
Speaker: Berry Bakker

April 26, room 176.
6.3: Resolvents of positive operators
6.4: Functional calculus
Speaker: Ozkan Tan

May 3, room 312.
7.1: Spectrum of compact operators
7.2: Approximate eigenvalues
7.3: Spectrum of lattice homomorphisms
Speaker: Nick Lindemulder

May 17, room Gorlaeus C7
8.3: Positive matrices
8.4: Irreducible matrices
8.5: Perron-Frobenius
Speaker: Anco Moritz

May 31, room 312.
9.1: Irreducible operators
9.2: Ideal irreducible operators
Speaker: Floris Claassens

subject

The functional analysis seminar in Spring 2013 will be on positive operators. We will consider Banach spaces that have in addition to their linear structure and norm structure also a partial order, which is in a suitable way compatible with the vector space operations and the norm. A typical example of such an order is a the natural order in a space of real valued functions, where f less than or equal to g means that at every t the number f(t) is less than or equal to g(t). A linear map between two ordered vector spaces is called positive if it preserves the order relation between any two elements.

The main aim of the seminar will be to study spectral theory of positive operators. Classically, the Perron-Frobenius theorem says that the spectral radius r of a positive matrix A (that is, a matrix with all entries greater than or equal to zero) is an eigenvalue and that there is a positive eigenvector for this eigenvalue. There is an infinite dimensional version of this theorem, composed of two parts. The first part is a theorem by Krein and Rutman, which says that every compact positive operator on a Banach lattice with strictly positive spectral radius has its spectral radius as an eigenvalue with a positive eigenvector. The second part deals with the question when the spectral radius is strictly positive. It turns out that a certain `irreducibility' condition is sufficient. The key result here is a theorem by de Pagter: every ideal irreducible compact positive operator on a Banach lattice has a strictly positive spectral radius.

On our way to the spectral theory for positive operators on Banach lattices, we will give generous attention to the general theory of partially ordered vector spaces, vector lattices, Banach lattices and operators on them.

book

We will follow a selection from the first nine chapters of the following book:

Y.A. Abramovich and C.D. Aliprantis, An invitation to Operator Theory, Graduate Studies in Mathematics Vol. 50, AMS, 2002.

participation

By active participation it is possible to obtain 6 EC for this seminar. Active participation includes attending (almost) all sessions and giving a one afternoon lecture on an assigned part of the book. Please inquire at your own university if it is allowed to include this seminar in your master program. Some more formal details can be found in the study guide.

If you have further questions or if you like to participate, please contact Onno van Gaans.