Operator monotone functions, positive definite kernels and majorization. (English) Zbl 1225.47016
Let \(f(t)\) be a real-valued continuous function defined on an interval \(I\) in the reals and consider the operator-valued function \(f(X)\) defined on Hermitian operators \(X\) on a Hilbert space. It is shown that, if \(f(X)\) is an increasing function with respect to the operator order, then for \(F(t)=\int f(t)\,dt\) the function \(F(X)\) is convex. Moreover, let \(h(t)\) and \(g(t)\) be \(C^1\) functions on \(I\) with \(h'(t)\geq 0\) and \(g'(t)> 0\). The continuous kernel on \(I\times I\) satisfying
\[
K_{h,g}(s, t)= (h(t)- h(s))/(g(t)- g(s))
\]
for \(s\neq t\) is a generalization of the Löwner kernel. It is shown that \(K_{h,g}\) is positive definite on \(I\) if and only if \(h\) is majorized by \(g\) on \(I\), meaning that \(h(A)\leq h(B)\) whenever \(g(A)\leq g(B)\) for Hermitian operators \(A, B\) with spectra in \(I\). In this case, if also \(h(I)\subset g(I)= (0,\infty)\), then the kernel \(K_{g,hg}\) is infinitely divisible.
Reviewer: Alan J. Pryde (Clayton)
MSC:
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 15A39 | Linear inequalities of matrices |
| 47B34 | Kernel operators |
Keywords:
positive definite kernel; pick function; matrix order; Löwner theorem; operator monotone function; majorization; infinitely divisible kernelReferences:
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