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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A determinantal inequality for projectors in a unitary space


Author: D. Ž. Djoković
Journal: Proc. Amer. Math. Soc. 27 (1971), 19-23
MSC: Primary 15.58
DOI: https://doi.org/10.1090/S0002-9939-1971-0266944-0
MathSciNet review: 0266944
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0266944-0
Keywords: Unitary space, orthogonal decomposition, eigenvalues, eigenvectors, projector, normal operator, hermitian positive definite operator
Article copyright: © Copyright 1971 American Mathematical Society