Upper Triangular Toeplitz Matrices and Real Parts of Quasinilpotent Operators

We show that every self-adjoint matrix B of trace 0 can be realized as B = T + T* for a nilpotent matrix T with ∥T∥ ≤ K∥B∥, for a constant K that is independent of matrix size. More particularly, if D is a diagonal, self-adjoint n × n matrix of trace 0, then there is a unitary matrix V = XUn, where...

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Bibliographic Details
Published in:Indiana University mathematics journal 2014-01, Vol.63 (1), p.53-75
Main Authors: Dykema, Ken, Fang, Junsheng, Skripka, Anna
Format: Article
Language:English
Subjects:
Algebra
Automorphisms
College mathematics
Mathematical theorems
Mathematics
Matrices
Partial sums
Subalgebras
Von Neumann algebra
Zero
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Summary:We show that every self-adjoint matrix B of trace 0 can be realized as B = T + T* for a nilpotent matrix T with ∥T∥ ≤ K∥B∥, for a constant K that is independent of matrix size. More particularly, if D is a diagonal, self-adjoint n × n matrix of trace 0, then there is a unitary matrix V = XUn, where X is an n × n permutation matrix and Un is the n × n Fourier matrix, such that the upper triangular part, T, of the conjugate V* DV of D satisfies ∥T∥ ≤ K∥D∥. This matrix T is a strictly upper triangular Toeplitz matrix such that T + T* = V* DV. We apply this and related results to give partial answers to questions about real parts of quasinilpotent elements in finite von Neumann algebras.
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.2014.63.5193