Caratheodory inequality for analytic operator function

Suppose H is a complex Hilbert space, AH (∆) denotes the set of all analytic operator functions on ∆, and the set NH(∆)= {f (z) ǀ f(z) is an analytic operator function on the open uint disk ∆, f (z) f (w) = f (w) f (z), f* (z) f (z) = f (z) f* (z),∀ z, w ∈ ∆}. The note proves that if f(z) ∈ NH(∆), (or AH(∆))ǁf(z)ǁ ≤ l, ∀ z ∈ ∆ then (eqution found), where T ∈ ℒ (H) (orT*T = TT*,respectively), ǁTǁ < 1, Tf = fT. © 2001, Springer Verlag. All rights reserved.