New unitary equivalences for some operator matrices with applications
Ozra Abdollahi, Saeed Karami, Jamal Rooin · Linear and Multilinear Algebra · 2022
In this paper, first we consider cross-diagonal, in particular off-diagonal, operator matrices; those operator matrices such that all entries except those on the main and off diagonals are zero. We show that this operator matrix is unitarily equivalent to a block diagonal operator matrix whose diagonal blocks are all two-by-two, except at most one of them which is one-by-one. Then, using this unitary equivalence, we show that any left circulant operator matrix is unitarily equivalent to a direct sum of a one-by-one operator and an off-diagonal operator matrix. As an application, we give equalities for the numerical radius of some important operator matrices. In particular, for the following left circulant operator matrix, we show that w([A1A2…AnA2…AnA1⋮⋱⋱⋮AnA1…An−1])=max1≤k≤nw([0∑m=1nωk(m−1)Am∑m=1nω(n−k)(m−1)Am0]),where ω=e2π in is the nth root of unity. Meanwhile, some inequalities for general operator matrices are obtained.