# Universal operators and invariant subspaces

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## Abstract

Suppose that each of H and K is a Hilbert space, 0 is a class of operators from H to and S is an operator from K to K. The operator S is said to be O-universal provided that for each member A of 0 there is a linear homeomorphism [sigma] from H to K such that [sigma]A[sigma]=S|sigma. This dissertation presents a study of the above universality concept. Its prime motivation is the following, to be referred to as the main conjecture: If G is a Hilbert space and T is an operator from G to G, then there is a non-trivial invariant subspace for T. The property of being O-universal, for a given class 0, is shown to be invariant under linear homeomorphism. Specific choices of a class 0 of operators and of an O-universal operator, together with related results, constitute the main development. In addition, four new versions of the main conjecture are presented.