Non-commutative Clarkson inequalities for \(n\)-tuples of operators.

*(English)*Zbl 1155.47013Clarkson inequalities give bounds for the \(L_p\)-norms of the sum and difference of two measurable functions in \(L_p\) in terms of the \(L_p\)-norms of those functions individually. The non-commutative version of these inequalities has been obtained first by C. A. McCarthy [Isr. J. Math. 5, 249–271 (1967; Zbl 0156.37902)] for Schatten \(p\)-norms and reads as follows. Let \(A\) and \(B\) be bounded linear operators on a separable complex Hilbert space. Then
\[
2\bigl( \| A\| _{p}^{p}+\| B\| _{p}^{p}\bigr) \leq \| A+B\| _{p}^{p}+\| A-B\| _{p}^{p}\leq 2^{p-1}\bigl( \| A\| _{p}^{p}+\| B\| _{p}^{p}\bigr),
\]
for \(2\leq p<\infty\), and
\[
2\bigl( \| A\| _{p}^{p}+\| B\| _{p}^{p}\bigr) \geq \| A+B\| _{p}^{p}+\| A-B\| _{p}^{p}\geq 2^{p-1}\bigl( \| A\| _{p}^{p}+\| B\| _{p}^{p}\bigr)
\]
for \(0<p\leq 2\). There are many generalizations of these inequalities, see, for instance, R. Bhatia and J. Holbrook [Math. Ann. 281, 7–12 (1988; Zbl 0618.47008)], R. Bhatia and F. Kittaneh [Bull. Lond. Math. Soc. 36, 820–832 (2004; Zbl 1071.47011)], O. Hirzallah and F. Kittaneh [Pac. J. Math. 202, 363–369 (2002; Zbl 1054.47011)], and E. Kissin [Proc. Am. Math. Soc. 135, 2483–2495 (2007; Zbl 1140.47005)]

In the paper under review, some new Clarkson type inequalities for \(n\)-tuples of operators are obtained. Also, there are some refinements of known Clarkson inequalities. The results are obtained by using the recent convexity and concavity inequalities for positive operators due to T. Kosem [Linear Algebra Appl. 418, 153–160 (2006; Zbl 1105.15016)].

In the paper under review, some new Clarkson type inequalities for \(n\)-tuples of operators are obtained. Also, there are some refinements of known Clarkson inequalities. The results are obtained by using the recent convexity and concavity inequalities for positive operators due to T. Kosem [Linear Algebra Appl. 418, 153–160 (2006; Zbl 1105.15016)].

Reviewer: Janko Bračič (Ljubljana)