Hyperinvariant subspaces and extended eigenvalues (Q1774329)

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 \(\lambda\) for which there is an operator \(T\neq 0\) satisfying \(AT=\lambda TA\). For a given \(\lambda\in EE(A)\), define \({\mathcal E}={\mathcal E}(A,\lambda)\) as the set of all \(\lambda\) eigen-operators for \(A\). In this paper, several scenarios are investigated for which the existence of non-unimodular extended eigenvalues leads to invariant or hyperinvariant subspaces. Some of the results are as follows: (1) Let \(\lambda\in EE(A)\) and let \({\mathcal E}={\mathcal E}(A,A)\). Define \(S= \{x:{\mathcal E}x= \{0\}\}\). Then \(S\) in hyperinvariant for \(A\). (2) Suppose that \(EE(A)\) is not closed under multiplication. Then \(A\) has a proper hyperinvariant subspace. (3) Suppose that \(A\) is invertible and \(T\in{\mathcal E}(A,\lambda)\), and \(\lambda\) is not a root of unity. Then the spectrum of \(T\) must be circularly symmetric. (4) Let \(\gamma\) be a nonzero scalar. Then \(EE(\gamma+ V)= \{1\}\), where \(V\) is the classical Volterra operator on \(L^2(0, 1)\).