Reducible representations of abelian groups
by
Aharon Atzmon
Tel Aviv University

It is not known if every representation of an abelian group in an infinite dimensional complex Hilbert space, or a reflexive Banach space, is reducible. For the integer group, this problem is equivalent to the problem whether every invertible operator on such a space, has a common nontrivial invariant subspace with its inverse. The known sufficient conditions for reducibility involve growth conditions on certain orbits of the representation, which yield reducibility by spectral splitting. However, these conditions do not hold for representations with rigid spectral properties, such as completely indecomposable representation. In the talk, we present a reducibility criterion of different type. We show that if G is an abelian group and T is a representation which yield reducibility by spectral splitting. However, these conditions do not hold for representations with rigid spectral properties, such as completely indecomposable representation. In the talk, we present a reducibility criterion of different type. We show that if G is an abelian group and T is a representation of G in an infinite dimensional complex Banach space X, and there exists a conjugate linear involution J on X such that JT(g) = T(-g) J, for allg in G, and nonzero vectors u in X and v in X^* such that Ju = u, and the function on G defined by g --> <T(g) u, v> is positive definite, then T is reducible. The proof is achieved by a variational argument which involves the Bishop-Phelps Theorem. This result implies that Harmonic analysis on Hermitian translation invariant Banach spaces of functions on a locally compact abelian group is non-void, and yields, in particular, a positive solution to the translation invariant subspace problem for weighted L^\rho spaces on such groups, for even weights. The result for reflexive spaces appears in [1] and its extension to the general case in [2].

1. A. Atzmon, Reducible representations of abelian groups, Ann. Inst. Fourier, Grenoble 51 (2001), 1407-1418.
2. A. Atzmon and G. Godefroy, An application of the smooth variational principle to the existence of non-trivial invariant subspaces, C.R. Acad. Sci., Paris, t. 332(1) (2001), 151-156.

Date received: May 30, 2002