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Matrix subadditivity inequalities and block-matrices. (English) Zbl 1181.15030

This paper is a sequel to the work by J.-C. Bourin [Linear Algebra Appl. 413, 212–217 (2006; Zbl 1092.47009)], J.-C. Bourin and E.-Y. Lee [J. Oper. Theory, to appear] and J.-C. Bourin and M. Uchiyama [Linear Algebra Appl. 423, 512–518 (2007; Zbl 1123.15013)].
Let \(f:[0,\infty)\rightarrow [0,\infty)\) be concave, let \(A_1,\dots,A_m\in\mathbb{C}^{n\times n}\) be normal, let \(Z_1,\dots,Z_m\in\mathbb{C}^{n\times n}\) be expansive (i.e. \(Z^*Z\geq I\) where \(\geq\) denotes the Löwner partial ordering), and let \(\|.\|\) be a symmetric (or unitarily invariant) norm (i.e. \(\|A\|=\|UAV\|\) for all \(A\in\mathbb{C}^{n\times n}\) and unitary \(U,V\in\mathbb{C}^{n\times n}\)). The author conjectures that \[ \|f(|\sum_{i=1}^mZ_i^*A_iZ_i|)\|\leq \|\sum_{i=1}^mZ_i^*f(|A_i|)Z_i\| \] (where \(|A|=\sqrt{A^*A})\) and proves it in some special cases.
As an application, consider a partitioned matrix \(A=(A_{ij})\in\mathbb{C}^{n\times n}\) where all the blocks \(A_{ij}\) are of same size. Assume that \(A\) is normal or all its blocks are normal. The author conjectures that
\[ \|f(|A|)\|\leq\|\sum_{i,j}f(|A_{ij}|)\| \]
and proves it in some special cases. (In Conjecture 3.10, “for all symmetric norms” should read “for all nonnegative concave functions on \([0,\infty)\)”. A similar remark concerns Conjecture 3.11.)
Recently the author [Proc. Am. Math. Soc. 138, 495–504 (2010)] proved the first conjecture. He also proved the second, weakened as follows: “Assume that \(A\) is Hermitian or all its blocks are normal”.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A45 Miscellaneous inequalities involving matrices
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A60 Functional calculus for linear operators
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