Upper triangular Toeplitz matrices and real parts of quasinilpotent operators
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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 KB, for a constant K that is independent of matrix size. More particularly, if D is a diagonal, self-adjoint nn 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 KD. 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. Indiana University Mathematics Journal.
Indiana University Mathematics Journal
author list (cited authors)
Dykema, K., Fang, J., & Skripka, A.
complete list of authors
Dykema, Ken||Fang, Junsheng||Skripka, Anna