## 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

### Keywords:

normal matrices; norms; concave functions

### Citations:

Zbl 1092.47009; Zbl 1123.15013
Full Text:

### References:

 [1] DOI: 10.1007/s002080050335 · Zbl 0941.47004 [2] DOI: 10.1016/j.laa.2005.08.017 · Zbl 1092.15017 [3] DOI: 10.1007/978-1-4612-0653-8 [4] DOI: 10.1007/BF01446925 · Zbl 0688.47005 [5] DOI: 10.1023/A:1007432816893 · Zbl 0912.47005 [6] Bourin J.-C., Math. Ineq. Appl. 7 pp 607– [7] DOI: 10.1016/j.laa.2005.09.007 · Zbl 1092.47009 [8] DOI: 10.1016/j.laa.2007.02.019 · Zbl 1123.15013 [9] Gohberg I. C., English Trans. Amer. Math. Soc. Trans 20 pp 201– [10] Hiai F., Means for Hilbert Space Operators · Zbl 1048.47001 [11] DOI: 10.1007/s00220-003-0955-9 · Zbl 1049.15013 [12] DOI: 10.1016/j.laa.2004.03.006 · Zbl 1064.15022 [13] DOI: 10.1016/j.laa.2006.01.028 · Zbl 1105.15016 [14] S. Ju. Rotfel’d, Topics in Mathematical Physics, Consultants Bureau 3 (1969) pp. 73–78.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.