On Fractional Functional Calculus of Positive Operators

Let N 2 B(H) be a normal operator acting on a real or complex Hilbert space H. Define Ny := N1 1 0 : R(N) K(N) ! H, where N1 = NjR(N). Let the fractional semigroup Fr(W) denote the collection of all words of the form f 1 f 2 f k in which f j 2 L1(W) and f is either f or f y, where f y = f f,0g=( f + f f=0g) and L1(W) is a certain normed functional algebra of functions defined on F(W), besides that, W = W 2 B(H) and F = R or C indicates the underlying scalar field. The fractional calculus ( f 1 f 2 f k )(W) on Fr(W) is defined as f 1 (W) f 2 (W) f k (W), where f y j (W) = ( f j(W))y. The present paper studies sucient conditions on f j to ensure such fractional calculus are unbounded normal operators. The results will be extended beyond continuous functions.