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Norm inequalities involving convex and concave functions of operators. (English) Zbl 07089815
Summary: Let \(A_1,\dots,A_n\) be bounded linear operators on a complex separable Hilbert space \(\mathbb{H}\) and let \(\alpha_1,\dots,\alpha_n\) be positive real numbers such that \(\sum^n_{j=1}\alpha_j A_j=0\) and \(\sum^n_{j=1} \alpha_j=1\). Among other results, it is shown that
(a)
If \(f\) is a non-negative function on \([0,\infty)\) such that \(f(0)=0\) and \(g(t)=f(\sqrt{t})\) is convex, then for every unitarily invariant norm, \[ \left|\left|\left| \sum\limits^n_{j=1} \alpha_j f(|A_j|) \right|\right|\right| \geq \left|\left|\left| f\left(\sqrt{\frac{\alpha_\ell}{1-\alpha_\ell}}|A_\ell|\right) + \sum\limits_{j,k\in S_\ell} f\left(\sqrt{\frac{\alpha_j \alpha_k}{2(1-\alpha_\ell)}} |A_j - A_k|\right)\right|\right|\right| \] for \(\ell=1,\dots,n\).
(b)
If \(f\) is a non-negative function on \([0,\infty)\) such that \(g(t)=f(\sqrt{t})\) is concave, then for every unitarily invariant norm, \[ \left|\left|\left| \sum\limits^n_{j=1} \alpha_j f(|A_j|) \right|\right|\right| \leq \left|\left|\left| f\left(\sqrt{\frac{\alpha_\ell}{1-\alpha_\ell}}|A_\ell|\right) + \sum\limits_{j,k\in S_\ell} f\left(\sqrt{\frac{\alpha_j \alpha_k}{2(1-\alpha_\ell)}} |A_j - A_k|\right)\right|\right|\right| \] for \(\ell=1\). Here \(S_\ell =\{1,\dots,n\}\setminus \{\ell\}\).

MSC:
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
46B20 Geometry and structure of normed linear spaces
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