Characterization of diagonality for operators (Q2839853)

Let \(A\) be a square matrix of order \(n\) with complex entries. Let the \(k\)-fold Hadamard or the Schur product of \(A\) with itself be denoted by \(A^{\circ k}\). This matrix has as its \((i,j)\)th entry the number \(a_{i,j}^k\). For \(k \geq 1\), we use \(A^k\) to denote the usual \(k\)th power of \(A\). The author shows that, if \(A^{\circ k}=A^k\), for \(k=1,2,\dots,n+1\), then \(A^{\circ k}=A^k\) holds for any positive integer \(k\). In addition, if \(A\) is invertible, then they show that \(A\) must be a diagonal matrix. It is also shown that, for an invertible operator \(A\) over a Hilbert space with a countable orthonormal basis, if \(A^{\circ k}=A^k\) for all positive integers \(k\), then \(A\) is a diagonal operator, with respect to the given countable orthonormal basis.