Invariant subspaces for certain finite-rank perturbations of diagonal operators (Q447907)

Suppose that \(D\) is a diagonal operator on a separable, infinite-dimensional Hilbert space \({\mathcal H},\) that is, there is an orthonormal basis \(\{e_k\}\) and there is a bounded sequence of complex scalars \(\{\lambda_k\}\) such that NEWLINE\[NEWLINE D= \sum_{k=1}^\infty \lambda_k e_k \otimes e_k. NEWLINE\]NEWLINE A long-standing open problem in operator theory is the question about the existence of invariant subspaces for a rank-one perturbation of a diagonal operator, that is, an operator of the form NEWLINE\[NEWLINE T=D + u \otimes v. NEWLINE\]NEWLINE This problem was partially solved by \textit{C. Foias} et al.\ [J. Funct. Anal. 253, No. 2, 628--646 (2007; Zbl 1134.47004)], who proved the following.NEWLINENEWLINETheorem 1. Suppose that \(u=\sum_{k=1}^\infty \alpha_k e_k\) and \(v=\sum_{k=1}^\infty \beta_k e_k\) satisfy the condition NEWLINE\[NEWLINE \sum_{k=1}^\infty (|\alpha_k|^{2/3}+|\beta_k|^{2/3}) < \infty. \tag{*} NEWLINE\]NEWLINE If the operator \(T=D + u \otimes v\) is not a scalar multiple of the identity operator, then \(T\) has a nontrivial hyperinvariant subspace.NEWLINENEWLINELet \(\ell_1(\{e_k\})\) denote the collection of vectors \(u = \sum_{k=1}^\infty \alpha_k e_k\) that satisfy the condition \(\sum_{k=1}^\infty |\alpha_k|<\infty\). The main result in the paper under review is the following.NEWLINENEWLINETheorem 2. Suppose that \(u_1, \dots, u_n\) and \(v_1, \dots , v_n\) are vectors in \(\ell_1(\{e_k\})\). If the operator NEWLINE\[NEWLINE T=D + u_1 \otimes v_1 + \dots + u_n \otimes v_n NEWLINE\]NEWLINE is not a scalar multiple of the identity operator, then \(T\) has a nontrivial hyperinvariant subspace.NEWLINENEWLINENotice that Theorem~2 represents a strong improvement over Theorem~1 in two aspects. First, condition \((*)\) is significantly weakened. Second, instead of rank-one perturbations, the authors consider perturbations of arbitrary finite rank.NEWLINENEWLINEThe strategy for the proof of Theorem~2 is sketched in the introduction, and the details are carried out in the subsequent sections. Although the paper focuses on the existence of hyperinvariant subspaces for finite rank perturbations of diagonal operators, the final section outlines the decomposability in the sense of [\textit{I. Colojoară} and \textit{C. Foias}, Theory of generalized spectral operators. New York-London-Paris: Gordon and Breach Science Publishers (1968; Zbl 0189.44201)] for the operator \(T\) considered in Theorem~2 under the additional assumption that \(T\) has at most countably many eigenvalues.