<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
branch	O
of	O
mathematics	O
,	O
an	O
operator	B-Algorithm
ideal	I-Algorithm
is	O
a	O
special	O
kind	O
of	O
class	O
of	O
continuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
between	O
Banach	O
spaces	O
.	O
</s>
<s>
If	O
an	O
operator	O
belongs	O
to	O
an	O
operator	B-Algorithm
ideal	I-Algorithm
,	O
then	O
for	O
any	O
operators	O
and	O
which	O
can	O
be	O
composed	O
with	O
as	O
,	O
then	O
is	O
class	O
as	O
well	O
.	O
</s>
<s>
Additionally	O
,	O
in	O
order	O
for	O
to	O
be	O
an	O
operator	B-Algorithm
ideal	I-Algorithm
,	O
it	O
must	O
contain	O
the	O
class	O
of	O
all	O
finite-rank	O
Banach	O
space	O
operators	O
.	O
</s>
<s>
Let	O
denote	O
the	O
class	O
of	O
continuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
acting	O
between	O
arbitrary	O
Banach	O
spaces	O
.	O
</s>
<s>
For	O
any	O
subclass	O
of	O
and	O
any	O
two	O
Banach	O
spaces	O
and	O
over	O
the	O
same	O
field	O
,	O
denote	O
by	O
the	O
set	O
of	O
continuous	B-Algorithm
linear	I-Algorithm
operators	I-Algorithm
of	O
the	O
form	O
such	O
that	O
.	O
</s>
<s>
An	O
operator	B-Algorithm
ideal	I-Algorithm
is	O
a	O
subclass	O
of	O
,	O
containing	O
every	O
identity	O
operator	O
acting	O
on	O
a	O
1-dimensional	O
Banach	O
space	O
,	O
such	O
that	O
for	O
any	O
two	O
Banach	O
spaces	O
and	O
over	O
the	O
same	O
field	O
,	O
the	O
following	O
two	O
conditions	O
for	O
are	O
satisfied	O
:	O
</s>
<s>
Operator	B-Algorithm
ideals	I-Algorithm
enjoy	O
the	O
following	O
nice	O
properties	O
.	O
</s>
<s>
Every	O
component	O
of	O
an	O
operator	B-Algorithm
ideal	I-Algorithm
forms	O
a	O
linear	O
subspace	O
of	O
,	O
although	O
in	O
general	O
this	O
need	O
not	O
be	O
norm-closed	O
.	O
</s>
<s>
Every	O
operator	B-Algorithm
ideal	I-Algorithm
contains	O
all	O
finite-rank	O
operators	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
finite-rank	O
operators	O
form	O
the	O
smallest	O
operator	B-Algorithm
ideal	I-Algorithm
.	O
</s>
<s>
For	O
each	O
operator	B-Algorithm
ideal	I-Algorithm
,	O
every	O
component	O
of	O
the	O
form	O
forms	O
an	O
ideal	O
in	O
the	O
algebraic	O
sense	O
.	O
</s>
<s>
Furthermore	O
,	O
some	O
very	O
well-known	O
classes	O
are	O
norm-closed	O
operator	B-Algorithm
ideals	I-Algorithm
,	O
i.e.	O
,	O
operator	B-Algorithm
ideals	I-Algorithm
whose	O
components	O
are	O
always	O
norm-closed	O
.	O
</s>
