A Characterisation of Anti-Löwner Functions
نویسنده
چکیده
According to a celebrated result by Löwner, a real-valued function f is operator monotone if and only if its Löwner matrix, which is the matrix of divided differences Lf = ( f(xi)−f(xj) xi−xj )N i,j=1 , is positive semidefinite for every integer N > 0 and any choice of x1, x2, . . . , xN . In this paper we answer a question of R. Bhatia, who asked for a characterisation of real-valued functions g defined on (0,+∞) for which the matrix of divided sums Kg = ( g(xi)+g(xj) xi+xj )N i,j=1 , which we call its anti-Löwner matrix, is positive semidefinite for every integer N > 0 and any choice of x1, x2, . . . , xN ∈ (0,+∞). Such functions, which we call anti-Löwner functions, have applications in the theory of Lyapunov-type equations.
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