Let
H
H
be a Hilbert space with inner-product
(
x
,
y
)
(x,y)
, and let
R
R
be a bounded positive operator on
H
H
which determines an inner-product,
⟨
x
,
y
⟩
=
(
R
x
,
y
)
,
x
,
y
∈
H
\langle x,y\rangle =(Rx,y), x, y\in H
. Denote by
H
−
H^-
the completion of
H
H
with respect to the norm
‖
x
‖
=
⟨
x
,
x
⟩
1
/
2
\|x\|=\langle x,x\rangle ^{1/2}
. In this paper, operators having certain relationships with
R
R
are studied. In particular, if
T
=
S
R
1
/
2
T=SR^{1/2}
where
S
∈
B
(
H
)
S\in B(H)
, then
T
T
has an extension
T
−
∈
B
(
H
−
)
T^-\in B(H^-)
, and
T
T
and
T
−
T^-
have essentially the same spectral and Fredholm properties.