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- W4298174950 abstract "By a result of Lundquist-Barrett, it follows that the rank of a positive semi-definite matrix is less than or equal to the sum of the ranks of its principal diagonal submatrices when written in block form. In this article, we take a general operator algebraic approach which provides insight as to why the above rank inequality resembles the Hadamard-Fischer determinant inequality in form, with multiplication replaced by addition. It also helps in identifying the necessary and sufficient conditions under which equality holds. Let $mathscr{R}$ be a von Neumann algebra, and $Phi$ be a normal conditional expectation from $mathscr{R}$ onto a von Neumann subalgebra $mathscr{S}$ of $mathscr{R}$. Let $mathfrak{R}[T]$ denote the range projection of an operator $T$. For a positive operator $A$ in $mathscr{R}$, we prove that $Phi(mathfrak{R}[A]) le mathfrak{R}[Phi(A)]$ with equality if and only if $mathfrak{R}[A] in mathscr{S}$." @default.
- W4298174950 created "2022-10-01" @default.
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- W4298174950 date "2018-04-25" @default.
- W4298174950 modified "2023-10-17" @default.
- W4298174950 title "An operator inequality for range projections" @default.
- W4298174950 doi "https://doi.org/10.48550/arxiv.1804.09683" @default.
- W4298174950 hasPublicationYear "2018" @default.
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