| Abstract: | Suppose H is a complex Hilbert space, A
H
(Δ) denotes the set of all analytic operator function on Δ, and the set N
H
(Δ)={f(z)/f(z) is an analytic operator function on the open unit disk Δ, f(z)f(w)=f(w)f(z), f
*
(z)f(z)=f(z)f
*
(z), ∀ z, w ≡ δ }. The note proves that if f(z)∈N
H
(Δ), (or A
H
(Δ)| f(z)|⩽1, ∀ z∈Δ then |f′(T)|⩽(1-|T|
2
)
−1
|I−f
*
(T)f(T)|
1/2
|I−f(T)f
*
(T)|
1/2
, where T∃−(H) (or T
*
T=TT
*, respectively), |T|<1, Tf=fT.
Supported by Education Foundation of Henan Province (98110012). |
