Operator ideal explained
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
belongs to an operator ideal
, then for any operators
and
which can be composed with
as
, then
is class
as well. Additionally, in order for
to be an operator ideal, it must contain the class of all finite-rank Banach space operators.
Formal definition
Let
denote the class of continuous linear operators acting between arbitrary Banach spaces. For any subclass
of
and any two Banach spaces
and
over the same field
, denote by
the set of continuous linear operators of the form
such that
. In this case, we say that
is a
component of
. An operator ideal is a subclass
of
, containing every identity operator acting on a 1-dimensional Banach space, such that for any two Banach spaces
and
over the same field
, the following two conditions for
are satisfied:
(1) If
then
; and
(2) if
and
are Banach spaces over
with
and
, and if
, then
.
Properties and examples
Operator ideals enjoy the following nice properties.
of an operator ideal forms a linear subspace of
, 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
, every component of the form
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.
References
- Pietsch, Albrecht: Operator Ideals, Volume 16 of Mathematische Monographien, Deutscher Verlag d. Wiss., VEB, 1978.
