Abstract
Let
A
be a positive bounded operator on an infinite dimensional complex and separable Hilbert space
(
H
,
,
)
and
A
:
=
B
(
H
)
/
K
(
H
)
be the Calkin algebra and
A
′
its dual; here
K
(
H
)
denotes the closed ideal of
B
(
H
)
consisting of all compact operators on
H
. For an operator
T
on
H
, let
‖
T
‖
A
and
V
A
(
T
)
denote the
A
-operator semi-norm and the
A
-numerical range induced by
A
. The
A
-essential numerical range and the
A
-essential norm of
T
are defined by
V
A
e
(
T
)
=
V
A
^
(
T
^
)
and
‖
T
‖
A
e
:
=
sup
S
A
^
(
A
)
f
T
∗
^
A
^
T
^
.
where
S
A
^
(
A
)
=
f
∈
A
′
:
f
≥
0
,
f
(
A
^
)
=
1
and
T
^
:
=
T
+
K
(
H
)
is the coset containing
T
. In this paper, we establish some properties of the
A
-essential numerical range. In particular we show that if
T
has an
A
-adjoint then
V
A
e
(
T
)
=
⋂
K
∈
K
A
(
H
)
V
A
(
T
+
K
)
.
Further, we prove that
λ
∈
V
A
e
(
T
)
if and only if there exists a sequence
(
h
n
)
∈
H
of
A
-unit vectors such that
h
n
→
0
weakly and
A
T
h
n
,
h
n
→
λ
. Other auxiliary results are established.