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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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