Two remarks about nilpotent operators of order two

Garcia, Stephan; Lutz, Bob; Timotin, Dan

doi:10.1090/S0002-9939-2014-11944-7

Two remarks about nilpotent operators of order two
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by Stephan Ramon Garcia, Bob Lutz and Dan Timotin PDF

Proc. Amer. Math. Soc. 142 (2014), 1749-1756 Request permission

Abstract:

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.

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