Let
T
T
be an operator on a Hilbert space
H
H
with numerical radius
w
(
T
)
≤
1
w(T)\le 1
. According to a theorem of Berger and Stampfli, if
f
f
is a function in the disk algebra such that
f
(
0
)
=
0
f(0)=0
, then
w
(
f
(
T
)
)
≤
‖
f
‖
∞
w(f(T))\le \|f\|_\infty
. We give a new and elementary proof of this result using finite Blaschke products.
A well-known result relating numerical radius and norm says
‖
T
‖
≤
2
w
(
T
)
\|T\| \leq 2w(T)
. We obtain a local improvement of this estimate, namely, if
w
(
T
)
≤
1
w(T)\le 1
, then
\[
‖
T
x
‖
2
≤
2
+
2
1
−
|
⟨
T
x
,
x
⟩
|
2
(
x
∈
H
,
‖
x
‖
≤
1
)
.
\|Tx\|^2\le 2+2\sqrt {1-|\langle Tx,x\rangle |^2} \qquad (x\in H,~\|x\|\le 1).
\]
Using this refinement, we give a simplified proof of Drury’s teardrop theorem, which extends the Berger–Stampfli theorem to the case
f
(
0
)
≠
0
f(0)\ne 0
.