When is Hyperlat\(T\) = Hyperlat\(f(T)\) in finite dimension? (English) Zbl 1045.47008
Let \(T\) be an operator on \(\mathbb{C}^n\) and let \(f\) be an analytic function in a neighbourhood of \(\sigma(T)\) (the spectrum of \(T\)). Let \(\operatorname{Hyperlat}T\) denote the hyperinvariant subspace lattice of \(T\). The main result of the paper gives the following necessary and sufficient condition under which \(\operatorname{Hyperlat}T =\operatorname{Hyperlat}f(T)\):
(a) \(f\) is one-to-one on \(\sigma(T)\) and, if there is \(\lambda\in\sigma(T)\) such that \(f'(\lambda)=0\), then either
(b1) \(f''(\lambda)=0\) and all Jordan blocks corresponding to \(\lambda\) have dimension \(1\), or
(b2) \(f''(\lambda)\neq 0\) and all Jordan blocks corresponding to \(\lambda\) have the same odd dimension.
(a) \(f\) is one-to-one on \(\sigma(T)\) and, if there is \(\lambda\in\sigma(T)\) such that \(f'(\lambda)=0\), then either
(b1) \(f''(\lambda)=0\) and all Jordan blocks corresponding to \(\lambda\) have dimension \(1\), or
(b2) \(f''(\lambda)\neq 0\) and all Jordan blocks corresponding to \(\lambda\) have the same odd dimension.
Reviewer: Michal Zajac (Bratislava)
MSC:
| 47A15 | Invariant subspaces of linear operators |
| 15A04 | Linear transformations, semilinear transformations |
References:
| [1] | Conway, J. B.; Wu, P. Y., The splitting of \(A(T1⊕T2)\) and related questions, Indiana Univ. Math. J., 26, 41-56 (1976) · Zbl 0352.46041 |
| [2] | Gohberg, I.; Lancaster, P.; Rodman, L., Invariant Subspaces of Matrices with Applications (1986), Wiley: Wiley New York · Zbl 0608.15004 |
| [3] | Brickman, L.; Fillmore, P. A., The invariant subspace lattice of a linear transformation, Canad. J. Math., 19, 810-822 (1967) · Zbl 0153.04801 |
| [4] | Fillmore, P. A.; Herrero, D. A.; Longstaff, W. E., The hyperinvariant subspace lattice of a linear transformation, Linear Algebra Appl., 17, 125-132 (1977) · Zbl 0359.47005 |
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