A Note On The Stahl’s Theorem

Let H be a Hilbert space and B(H) denotes the algebra of all bounded linear operators on H and A be a bounden operator in H. Let A; B 2 B(H) be positive operators and be a positive linear functional on B(H). We show that, if f is a non-negative operator decreasing function, then the function t ! ( f (A + tB)) can be written as a Laplace transform of a positive measure.

Keywords:

BMV conjecture , Laplace transform , Operator monotone functions..