Clarkson–McCarthy Inequalities for Several Operators and Related Norm Inequalities for p-Modified Unitarily Invariant Norms

Let \(\left| {\left| {\cdot }\right| }\right| _\Phi \) be a unitarily invariant norm related to a symmetrically norming (s.n.) function \(\Phi \), defined on the associated ideal \({ {\mathcal C}_{\Phi }({\mathcal H})}\) of compact Hilbert space operators, let \(\left| {\left| {\cdot }\right| }\right| _{\Phi ^{^(\!\,^{q}\!\,^)}}\) be its degree q-modification, let \(\left| {\left| {\cdot }\right| }\right| _{\Phi ^{{^(\!\,^{q}\!\,^)}^{_*}}}\) be a dual norm to \(\left| {\left| {\cdot }\right| }\right| _{\Phi ^{^(\!\,^{q}\!\,^)}}\) and let \(\bigl [{A_{m,n}^{\phantom {}}}\bigr ]_{m,n\in {\mathbb {Z}}}\) be a block operator matrix. We show that, if \(0< p \le 2 \) and \(q\ge p,\) then $$\begin{aligned} \left\| {\bigl [{A_{m,n}^{\phantom {}}}\bigr ]_{m,n\in {\mathbb {Z}}}}\right\| _{\Phi ^{{^(\!\,^{q}\!\,^)}}}^p \le \sum _{m\in {\mathbb {Z}}}\left\| {\bigl [{A_{m,n}^{\phantom {}}}\bigr ]_{n\in {\mathbb {Z}}}}\right\| _{\Phi ^{{^(\!\,^{q}\!\,^)}}}^p \le \sum _{m,n\in {\mathbb {Z}}} \left\| {A_{m,n}^{\phantom {}}}\right\| _{\Phi ^{{^(\!\,^{q}\!\,^)}}}^p. \end{aligned}$$ If \(2\le p <+\infty \) and \(q\ge {p}/(p-1)\), then $$\begin{aligned} \left\| {\bigl [{A_{m,n}^{\phantom {}}}\bigr ]_{m,n\in {\mathbb {Z}}}}\right\| _{\Phi ^{{^(\!\,^{q}\!\,^{)*}}}}^p \ge \sum _{m\in {\mathbb {Z}}}\left\| {\bigl [{A_{m,n}^{\phantom {}}}\bigr ]_{n\in {\mathbb {Z}}}}\right\| _{\Phi ^{{^(\!\,^{q}\!\,^{)*}}}}^p \ge \sum _{m,n\in {\mathbb {Z}}} \left\| {A_{m,n}^{\phantom {}}}\right\| _{\Phi ^{{^(\!\,^{q}\!\,^{)*}}}}^p. \end{aligned}$$ If \(2\le p<+\infty , q\ge {p}/(p-1)\) and \({\Phi ^{{^(\!\,^{q}\!\,^)}^{_*}}}=\Psi ^{{^(\!\,^{r}\!\,^)}},\) for some \(1\le r\le p\) and for some s.n. function \(\Psi ,\) we extend Clarkson–McCarthy inequalities to an n-tuple of operators \((A_1^{\phantom {}},A_2^{\phantom {}},\dots ,A_{_N})\) as $$\begin{aligned}&{\scriptstyle N}\sum _{n=1}^{\scriptscriptstyle N}\left\| {A_n}\right\| _{{\Phi ^{{^(\!\,^{q}\!\,^)}^{_*}}}}^p\le \biggl ({\sum _{n=1}^{\scriptscriptstyle N}\left\| {\sum _{k=1}^{\scriptscriptstyle N}\omega _{\scriptscriptstyle N}^{nk}A_k^{\phantom {}}}\right\| _{{\Phi ^{{^(\!\,^{q}\!\,^)}^{_*}}}}^r}\biggr )^{\,\frac{p}{r}} \le {\scriptstyle N}^{\frac{p}{r}-1}\sum _{n=1}^{\scriptscriptstyle N}\left\| {\sum _{k=1}^{\scriptscriptstyle N}\omega _{\scriptscriptstyle N}^{nk}A_k^{\phantom {}}}\right\| _{{\Phi ^{{^(\!\,^{q}\!\,^)}^{_*}}}}^p\\&\quad \le {\scriptstyle N}^{\frac{p}{r}+p-2}\biggl ({\sum _{n=1}^{\scriptscriptstyle N}\left\| {A_n}\right\| _{{\Phi ^{{^(\!\,^{q}\!\,^)}^{_*}}}}^r}\biggr )^{\,\frac{p}{r}} \le {\scriptstyle N}^{\frac{2p}{r}+p-3}\sum _{n=1}^{\scriptscriptstyle N}\left\| {A_n}\right\| _{{\Phi ^{{^(\!\,^{q}\!\,^)}^{_*}}}}^p. \end{aligned}$$ In addition, we provide some refinements of the above inequalities, as well as some new norm inequalities.