A determinantal inequality for projectors in a unitary space

Abstract:

Let $V$ be a finite dimensional unitary space and $V = {V_1} + \cdots + {V_k}$ a direct decomposition. Let ${P_i}$ be the orthogonal projector in $V$ with range ${V_i}$. If $A = {P_1} + \cdots + {P_k}$ we prove that $0 < \det (A) \leqq 1$ and $\det (A) = 1$ if and only if $V = {V_1} + \cdots + {V_k}$ is an orthogonal decomposition. Let ${N_i}(1 \leqq i \leqq k)$ be a normal operator in $V$, of rank ${r_i}$. Assume that $N = \sum _{v = 1}^k{N_i}$ has rank $r = {r_1} + \cdots + {r_k} \leqq n = \dim V$. If the nonzero eigenvalues of $N$ (counting multiplicities) are the same as the nonzero eigenvalues of all ${N_i}(1 \leqq i \leqq k)$ together, then ${N_i}{N_j} = 0$ for $i \ne j$. This generalizes a recent result of L. Brand.