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Positive definiteness of functions with applications to operator norm inequalities. (English) Zbl 1227.47005

Mem. Am. Math. Soc. 997, v, 80 p. (2011).
Let \(H\geq 0\), \(K \geq 0\) and \(X\) denote operators acting on a separable Hilbert space. F. Hiai and H. Kosaki in [Indiana Univ. Math. J. 48, No. 3, 899–936 (1999; Zbl 0934.15023)] and [Means of Hilbert space operators. Lecture Notes in Mathematics 1820. Berlin: Springer (2003; Zbl 1048.47001)] introduced a certain class \(\mathfrak{M}\) of scalar means \(M(s,t)\), associated a certain operator mean \(M(H,K)X\) to each \(M(s,t)\in \mathfrak{M}\), and showed that the positive definiteness of the ratio \(M(e^x,1)/N(e^x,1)\) is equivalent to the validity of \(\||M(H,K)X|\| \leq \||N(H,K)X|\|\) for each unitarily invariant norm \(\||\cdot|\|\). In the present work, the author computes the Fourier transforms of certain ratios \(M(e^x,1)/N(e^x,1)\) between various typical scalar means \(M(s,t), N(s,t)\) and then uses Bochner’s theorem to determine whether these ratios are positive definite. He then presents the following Heinz-type norm inequality
\[ \||H^{1/2+\beta}XK^{1/2-\beta}-H^{1/2-\beta}XK^{1/2+\beta}|\|\;\frac{1}{\pi} \leq \log\left(\frac{1+\sin(\pi\beta)}{1-\sin(\pi\beta)}\right)\||HX+XK|\| \]
for \(\beta \in [0,1/2)\) as well as the commutator estimate
\[ \||AX-XB|\| \leq \||e^{A/2}Xe^{-B/2}+e^{-A/2}Xe^{B/2}|\|, \]
where \(A, B\) are selfadjoint operators. He also provides several interesting unitarily invariant norm inequalities involving the “Heinz means” \(H^{1/2+\beta}XK^{1/2-\beta}+H^{1/2-\beta}XK^{1/2+\beta}\), the operators \(H^{1/2+\beta}XK^{1/2-\beta}+H^{1/2-\beta}XK^{1/2+\beta}\pm H^{1/2}XK^{1/2}\), and the operators \(H^{1/2+\beta}XK^{1/2-\beta}+H^{1/2-\beta}XK^{1/2+\beta}+2\beta H^{1/2}XK^{1/2}\) related to the Heron means \((s+t)/2+\beta\sqrt{st}\), in which \(\beta \in (-1,1]\).

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A63 Linear operator inequalities
47A64 Operator means involving linear operators, shorted linear operators, etc.
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
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