A generalized reverse Cauchy inequality for matrices. (English) Zbl 1347.15028
Let \(A,B\geq 0\) (i.e., \(A\) and \(B\) are nonnegative definite matrices of the same dimension) satisfy \(AB+BA\geq 0\), and let \(f\) be an operator monotone function on \([0,\infty)\). Denote
\[
A\,\sigma_f\,B=A^\frac{1}{2}f(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})A^\frac{1}{2}
\]
if \(A,B>0\), and
\[
A\,\sigma_f\,B=\lim_{\varepsilon\to 0}(A+\varepsilon I)\sigma_f(B+\varepsilon I)
\]
otherwise. The authors prove that if \(f\) is positive and \(f(1)=1\), then
\[
A+B-|A-B|\leq 2\,A\,\sigma_f\,B.
\]
The reverse Cauchy inequality for positive definite matrices
\[
\frac{A+B}{2}\leq A^\frac{1}{2}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^\frac{1}{2}A^\frac{1}{2}+ \frac{|A-B|}{2}
\]
is so extended. Define \(g(t)=t/f(t)\) for \(t>0\), and \(g(0)=0\). The authors also prove that if \(f(0)=0\) and \(f(t)>0\) for all \(t>0\), then
\[
\|A+B-|A-B|\|\leq 2\,\|f(A)^\frac{1}{2}g(B)f(A)^\frac{1}{2}\|
\]
for any unitarily invariant norm \(\|\cdot\|\). The generalized Powers-Størmer inequality
\[
\mathrm{tr}(A+B-|A-B|)\leq 2\,\mathrm{tr}\,(f(A)^\frac{1}{2}g(B)f(A)^\frac{1}{2}),
\]
proved by the first author et al. [Linear Algebra Appl. 438, 242–249 (2013; Zbl 1270.46057)], is so extended.
Reviewer: Jorma K. Merikoski (Tampere)
MSC:
15A45 | Miscellaneous inequalities involving matrices |
15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
15B48 | Positive matrices and their generalizations; cones of matrices |
Keywords:
Cauchy inequality; Powers-Størmer inequality; operator monotone functions; unitarily invariant norms; reverse Cauchy inequality; positive definite matricesCitations:
Zbl 1270.46057References:
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