Horn’s inequality for singular values via exterior algebra

\( \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Id}{Id} \)Horn’s inequality states that for any two compact operators \(\sigma,\tau\) on a Hilbert space \(E\), $$\prod_{k=1}^n s_k(\sigma\tau) \le \prod_{k=1}^n s_k(\sigma)s_k(\tau)$$ where \(s_1(\tau),s_2(\tau),\dots\) are the singular values of \(\tau\) arranged in descending order. Alfred Horn’s original 1950 paper provides a short proof that relies on the following result:

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Line integrals: 3. Applications to complex analysis

\( \newcommand{\bbc}{\mathbb{C}} \newcommand{\Int}{\operatorname{Int}} \)Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis Preliminaries Let \(F\) be a complex Banach space, let \(U\subseteq\bbc\) be an open set, and let \(f:U\to F\). Recall that the complex Fréchet derivative of \(f\) at \(z\in U\), if it […]

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