INVARIANT SUBSPACES FOR SOME FAMILIES OF UNBOUNDED SUBNORMAL OPERATORS This work has been prepared while the second named author was a guest of the University of the Saarland as a Humboldt Fellow. He would like to express his gratitude to the Alexander von Humboldt-Foundation for its support and to the University of the Saarland for its hospitality.

Abstract

Extending results and methods of Thomson and Trent, we prove the existence of non trivial quasi-invariant subspaces for subnormal families of unbounded operators having sufficiently rich domains. In some special cases, proper subspaces are obtained. 2000 Mathematics Subject Classification. 47A15, 47B20. 1. Preliminaries. The problem of existence of proper subspaces for arbitrary bounded linear operators has been and still is an obsessive question for operator theorists. While the counterexamples of Enflo and Read have completely settled this problem in the frame of general Banach spaces, the Hilbert space case, as well as that of some particular Banach spaces, offer hopes for a positive answer to some optimistic scholars. Although the case of bounded operators remains a permanent temptation, we shall try in the following to turn the discussion to some classes of unbounded operators. Also, because the Scott Brown technique (3 )s eems to be very much related to bounded operators, we shall exploit the resources of Thomson's and Trent's techniques (10), (12 )t og et some information concerning the existence of subspaces for the unbounded ones. This starting point will force us to restrict ourselves to some families of (unbounded) subnormal operators in Hilbert spaces. Even for the concept of invariant subspace it turns out that one has to formulate (at least) two possible definitions. Let H be a complex Hilbert space and let T : D(T ) ⊂ H → H be a closed, densely defined linear operator. Also, let L ⊂ H be a closed linear subspace. DEFINITION 1. We say that L is under T if D0(T; L ): = D(T) ∩ L is dense in L and TD0(T; L) ⊂ L.