Hyperinvariant subspaces and extended eigenvalues

An extended eigenvalue for an operator A is a scalar λ for which the operator equation AX = λXA has a nonzero solution. Several scenarios are investigated where the existence of non-unimodular extended eigenvalues leads to invariant or hyperinvariant subspaces. For a bounded operator A on a complex Hilbert space H, the set EE(A) of extended eigenvalues for A is defined to be the set of those complex numbers λ for which there is an operator T = 0 satisfying AT = λTA. T is then referred to as a λ eigen-operator for A. The eigenvalue terminology, although not perfectly accurate, seems useful on two levels. The first was described in [2]; briefly, if A has dense range, then the equation AX = φ(X)A; φ(X) ∈ L(H) has a unital algebra as its solution set, and φ is a unital homomorphism. Our extended eigenvalues are precisely the eigenvalues for φ. The second point of view is that one can easily show that for an operator on a finite dimensional space, the set of extended eigenvalues for that operator is the set of ratios of eigenvalues, with the obvious restriction on the use of 0. This is shown explicitly in [3]. In other works this concept of extended eigenvalue has appeared as α commuting or λ commuting, but we choose to use a term which is parameter free. For A ∈ L(H) (the set of bounded operators on H), a (closed, linear) subspace of H is a nontrivial invariant subspace (n.i.s.) for A if it is neither H nor {0} and is invariant under A. This space is hyperinvariant for A if it is invariant for every operator in (A)′, the commutant of A. More generally, a subspace is defined to be invariant for a set of operators if it is invariant for each member of that set. Extended eigenvalues and invariant subspaces. For a given λ ∈ EE(A) we define E = E(A, λ) as the set of all λ eigen-operators for A. This is a (weakly) closed linear space of operators, and E(A, 1) is (A)′, the commutant of A; that is, the set of all operators commuting with A. Direct multiplication leads to the next result: Received June 15, 2003. Mathematics Subject Classification. 47A15, 47A62.