Mathematics Department

Dr JA Erdos Department of Mathematics King's College London Strand, London WC2R 2LS United Kingdom Room 407a, Strand Building Tel: +44-(0)20-7848 2225 (direct) Tel: +44-(0)20-7848 2217 (general office) Fax: +44-(0)20-1848 2017 E-mail: john.erdos@kcl.ac.uk

Member of the Analysis Group

Research Interests

Operator theory and non-selfadjoint operator algebras

In the study of linear maps, an important and useful technique is to represent them in some sort of standard form. The analysis frequently focuses on the action of the maps on subspaces. Thus, for maps on finite-dimensional vector spaces, the presence of enough eigenvectors leads to matrices in diagonal form; more general invariant subspaces give rise to triangular forms. My research follows this theme with operators on Hilbert space. There is a large body of established theory for the self-adjoint (Hermitian) case but more general results are sketchy. The question whether every Hilbert space operator has a non-trivial invariant subspace is a famous long-standing open problem. While this remains unsolved, there are two ways to sidestep this question : to consider special classes of operators (e.g. compact operators) or to specify the action first and study the set of all operators that perform this action.

In my research, I mainly follow the second course. For example, one might consider the set of all operators that leave invariant a given a family of subspaces. This set forms an algebra whose properties are determined by the initial choice of subspaces; for example, if the subspaces are ordered by inclusion we obtain what are called "nest algebras" - the analogue of the triangular matrices. More general families produce more complicated spaces of operators. There are many structural questions about these spaces and also problems concerning their concrete representation (eg as integral operators) and the approximation of operators by simpler ones within the same space. The subject has links to lattice theory (the set of invariant subspaces of a family of operators always forms a complete lattice) and also to basis theory (e.g. if {xi} is a basis or a basis-like family of vectors, the operators having {xi} as eigenvectors provide a source of interesting cases).

Of the large literature on this topic here is a small selection that gives some idea of the kind of work that might be done in this area at King's:

1. K.R. Davidson, Nest algebras, Pitman Research Notes in Mathematics No. 191, Longman 1988.
2. J.A. Erdos, Reflexivity for subspace maps and linear spaces of operators, Proc. London Math. Soc. (3) 52 (1986), 582 - 600.
3. J.A. Erdos, Basis theory and operator algebras, in "Operator Algebras and Applications (Ed. A. Katavolos), NATO ASI Series, Kluwer, (1997) 209-223.
4. J.A. Erdos, A. Katavolos and V.S. Shulman, Rank one subspaces of bimodules over maximal abelian selfadjoint algebras, Journal of Functional Analysis, 157 (1998) 554 - 587.
5. P.R. Halmos, Reflexive lattices of subspaces, J. London Math. Soc., (2) 4 (1971), 257 - 263.

[5] is an easy, pioneering paper initiating part of this subject. Most parts of [2] and [3] are fairly accessible. The paper [4] is much more technical and gives an idea of some recent developments. [1] is an advanced, rather comprehensive textbook.

Recent Publications

• JA Erdos, MS Lambrou and NK Spanoudakis "Spectral synthesis and reflexive operators", Comptes Rendus Math. Rep. Acad. Sci. Canada 17 193-196, 1995.
• JA Erdos "A completely distributive CSL algebra with no complement in C_p", Proc. Amer. Math. Soc. 124 1127-1131, 1996.
• JA Erdos, MS Lambrou and NK Spanoudakis "Block strong M-bases and spectral synthesis", J. London Math. Soc. (2) 57, 183-195, 1998.