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Barbour path functions and related operator means. (English) Zbl 1297.47019

Let \(a,b\) be positive real numbers. It is rather well known that the fractional linear function known as the Barbour function \(F_x(a,b)=a\frac{\frac{b}{a}x+\sqrt{\frac{b}{a}}(1-x)}{x+\sqrt{\frac{b}{a}}(1-x)},~ x\in [0,1]\), interpolates the function \(G_x(a,b) = a^{1-x}b^x,~x \in [0,1]\), at the points \(x=0,\frac{1}{2}, 1\). A bounded linear operator \(A\) on a Hilbert space \(\mathcal H\) is called positive and denoted by \(A \geq 0\) if \(\langle Ax,x \rangle \geq 0\) for all \(x \in \mathcal H\). This induces an order relation between two bounded linear operators \(A\) and \(B\), namely, \(A \geq B\) for self-adjoint operators \(A\) and \(B\) on \(\mathcal H\). A real valued function \(f\) on \((0, \infty)\) is called operator monotone if \(f(A) \leq f(B)\) whenever \(0 \leq A \leq B\). The author introduces the Barbour path functions as operator monotone functions and studies the integral mean for this function. He also considers the order relation between the integral mean for the Barbour function and a certain other mean.

MSC:

47A64 Operator means involving linear operators, shorted linear operators, etc.
47A63 Linear operator inequalities
26A48 Monotonic functions, generalizations
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References:

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