Complex Analysis and Operator Theory

, Volume 13, Issue 3, pp 583–613 | Cite as

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

  • Danko R. JocićEmail author


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.


Unitarily invariant norm Circulant block operator matrix Convex function Concave function Non-commutative Clarkson inequalities Finite Fourier transform 

Mathematics Subject Classification

Primary 47A30 47A60 Secondary 47A65 47B10 47B15 46B20 15A57 15A60 



The author is grateful to the referees and to the editor for their valuable suggestions as to how to improve the presentation, and to professor Srdjan Petrovic for his overall support during the preparation of the manuscript.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BelgradeBelgradeSerbia

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