- Source: Operator ideal
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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
{\displaystyle T}
belongs to an operator ideal
J
{\displaystyle {\mathcal {J}}}
, then for any operators
A
{\displaystyle A}
and
B
{\displaystyle B}
which can be composed with
T
{\displaystyle T}
as
B
T
A
{\displaystyle BTA}
, then
B
T
A
{\displaystyle BTA}
is class
J
{\displaystyle {\mathcal {J}}}
as well. Additionally, in order for
J
{\displaystyle {\mathcal {J}}}
to be an operator ideal, it must contain the class of all finite-rank Banach space operators.
Formal definition
Let
L
{\displaystyle {\mathcal {L}}}
denote the class of continuous linear operators acting between arbitrary Banach spaces. For any subclass
J
{\displaystyle {\mathcal {J}}}
of
L
{\displaystyle {\mathcal {L}}}
and any two Banach spaces
X
{\displaystyle X}
and
Y
{\displaystyle Y}
over the same field
K
∈
{
R
,
C
}
{\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}
, denote by
J
(
X
,
Y
)
{\displaystyle {\mathcal {J}}(X,Y)}
the set of continuous linear operators of the form
T
:
X
→
Y
{\displaystyle T:X\to Y}
such that
T
∈
J
{\displaystyle T\in {\mathcal {J}}}
. In this case, we say that
J
(
X
,
Y
)
{\displaystyle {\mathcal {J}}(X,Y)}
is a component of
J
{\displaystyle {\mathcal {J}}}
. An operator ideal is a subclass
J
{\displaystyle {\mathcal {J}}}
of
L
{\displaystyle {\mathcal {L}}}
, containing every identity operator acting on a 1-dimensional Banach space, such that for any two Banach spaces
X
{\displaystyle X}
and
Y
{\displaystyle Y}
over the same field
K
{\displaystyle \mathbb {K} }
, the following two conditions for
J
(
X
,
Y
)
{\displaystyle {\mathcal {J}}(X,Y)}
are satisfied:
(1) If
S
,
T
∈
J
(
X
,
Y
)
{\displaystyle S,T\in {\mathcal {J}}(X,Y)}
then
S
+
T
∈
J
(
X
,
Y
)
{\displaystyle S+T\in {\mathcal {J}}(X,Y)}
; and
(2) if
W
{\displaystyle W}
and
Z
{\displaystyle Z}
are Banach spaces over
K
{\displaystyle \mathbb {K} }
with
A
∈
L
(
W
,
X
)
{\displaystyle A\in {\mathcal {L}}(W,X)}
and
B
∈
L
(
Y
,
Z
)
{\displaystyle B\in {\mathcal {L}}(Y,Z)}
, and if
T
∈
J
(
X
,
Y
)
{\displaystyle T\in {\mathcal {J}}(X,Y)}
, then
B
T
A
∈
J
(
W
,
Z
)
{\displaystyle BTA\in {\mathcal {J}}(W,Z)}
.
Properties and examples
Operator ideals enjoy the following nice properties.
Every component
J
(
X
,
Y
)
{\displaystyle {\mathcal {J}}(X,Y)}
of an operator ideal forms a linear subspace of
L
(
X
,
Y
)
{\displaystyle {\mathcal {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
{\displaystyle {\mathcal {J}}}
, every component of the form
J
(
X
)
:=
J
(
X
,
X
)
{\displaystyle {\mathcal {J}}(X):={\mathcal {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
References
Pietsch, Albrecht: Operator Ideals, Volume 16 of Mathematische Monographien, Deutscher Verlag d. Wiss., VEB, 1978.