On the matrix Cauchy-Schwarz inequality. (English) Zbl 07812134
Summary: The main goal of this work is to present new matrix inequalities of Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz inequality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that
\[
\| T\| \leqslant \frac{1}{4} (\| |T| + |T^*| +2\mathfrak{R} T \|+\| |T| + |T^*| -2\mathfrak{R}T\|),
\]
where \(\mathfrak{R}T\) is the real part of the matrix \(T\).
MSC:
| 47A63 | Linear operator inequalities |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 46L05 | General theory of \(C^*\)-algebras |
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