In functional analysis, a branch of mathematics, an operator ideal is a special kind of class of continuous linear operators between Banach spaces. If an operator
T
belongs to an operator ideal
J
, then for any operators
A
and
B
which can be composed with
T
as
B
T
A
, then
B
T
A
is class
J
as well. Additionally, in order for
J
to be an operator ideal, it must contain the class of all finite-rank Banach space operators.
Let
L
denote the class of continuous linear operators acting between two Banach spaces. For any subclass
J
of
L
and any two Banach spaces
X
and
Y
over the same field
K
, denote by
J
(
X
,
Y
)
the set of continuous linear operators of the form
T
:
X
→
Y
. In this case, we say that
J
(
X
,
Y
)
is a component of
J
. An operator ideal is a subclass
J
of
L
, containing every identity operator acting on a 1-dimensional Banach space, such that for any two Banach spaces
X
and
Y
over the same field
K
, the following two conditions for
J
(
X
,
Y
)
are satisfied: (1) If
S
,
T
∈
J
(
X
,
Y
)
then
S
+
T
∈
J
(
X
,
Y
)
; and (2) if
W
and
Z
are Banach spaces over
K
with
A
∈
L
(
W
,
X
)
and
B
∈
L
(
Y
,
Z
)
, and if
T
∈
J
(
X
,
Y
)
, then
B
T
A
∈
J
(
W
,
Z
)
.
Properties and examples
Operator ideals enjoy the following nice properties.
Every component
J
(
X
,
Y
)
of an operator ideal forms a linear subspace of
L
(
X
,
Y
)
, although in general this need not be norm-closed.
Every operator ideal contains all finite-rank operators. In particular, the finite-rank operators form the smallest operator ideal.
For each operator ideal
J
, every component of the form
J
(
X
)
:=
J
(
X
,
X
)
forms an ideal in the algebraic sense.
Furthermore, some very well-known classes are norm-closed operator ideals, i.e., operator ideals whose components are always norm-closed. These include but are not limited to the following.
Compact operators
Weakly compact operators
Finitely strictly singular operators
Strictly singular operators
Completely continuous operators
