Upper triangular Toeplitz matrices and real parts of quasinilpotent operators
Ken Dykema, Junsheng Fang, Anna Skripka · Indiana University Mathematics Journal · 2014
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 = XU n , where X is an n × n permutation matrix and U n 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.