Non-commutative Clarkson inequalities for \(n\)-tuples of operators. (English) Zbl 1155.47013

Hirzallah, Omar; Kittaneh, Fuad

Integral Equations Oper. Theory 60, No. 3, 369-379 (2008).

Clarkson 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)].

Reviewer: Janko Bračič (Ljubljana)

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.)
47A13	Several-variable operator theory (spectral, Fredholm, etc.)

Keywords:

Clarkson inequalities; Clarkson-McCarthy inequalities; \(n\)-tuple of operators