<s>
In	O
mathematics	O
,	O
a	O
functional	B-Algorithm
calculus	I-Algorithm
is	O
a	O
theory	O
allowing	O
one	O
to	O
apply	O
mathematical	O
functions	O
to	O
mathematical	O
operators	O
.	O
</s>
<s>
It	O
is	O
now	O
a	O
branch	O
(	O
more	O
accurately	O
,	O
several	O
related	O
areas	O
)	O
of	O
the	O
field	O
of	O
functional	B-Application
analysis	I-Application
,	O
connected	O
with	O
spectral	O
theory	O
.	O
</s>
<s>
(	O
Historically	O
,	O
the	O
term	O
was	O
also	O
used	O
synonymously	O
with	O
calculus	B-Algorithm
of	I-Algorithm
variations	I-Algorithm
;	O
this	O
usage	O
is	O
obsolete	O
,	O
except	O
for	O
functional	O
derivative	O
.	O
</s>
<s>
Sometimes	O
it	O
is	O
used	O
in	O
relation	O
to	O
types	O
of	O
functional	O
equations	O
,	O
or	O
in	O
logic	O
for	O
systems	O
of	O
predicate	B-Application
calculus	I-Application
.	O
)	O
</s>
<s>
If	O
it	O
does	O
,	O
then	O
we	O
are	O
no	O
longer	O
using	O
on	O
its	O
original	O
function	B-Algorithm
domain	I-Algorithm
.	O
</s>
<s>
This	O
passes	O
nearly	O
unnoticed	O
if	O
we	O
talk	O
about	O
'	O
squaring	O
a	O
matrix	B-Architecture
 '	O
,	O
though	O
,	O
which	O
is	O
the	O
case	O
of	O
and	O
an	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
idea	O
of	O
a	O
functional	B-Algorithm
calculus	I-Algorithm
is	O
to	O
create	O
a	O
principled	O
approach	O
to	O
this	O
kind	O
of	O
overloading	B-Language
of	O
the	O
notation	O
.	O
</s>
<s>
The	O
most	O
immediate	O
case	O
is	O
to	O
apply	O
polynomial	O
functions	O
to	O
a	O
square	B-Algorithm
matrix	I-Algorithm
,	O
extending	O
what	O
has	O
just	O
been	O
discussed	O
.	O
</s>
<s>
In	O
the	O
finite-dimensional	O
case	O
,	O
the	O
polynomial	O
functional	B-Algorithm
calculus	I-Algorithm
yields	O
quite	O
a	O
bit	O
of	O
information	O
about	O
the	O
operator	O
.	O
</s>
<s>
Consider	O
the	O
unilateral	B-Algorithm
shift	I-Algorithm
with	O
the	O
polynomials	O
calculus	O
;	O
the	O
ideal	O
defined	O
above	O
is	O
now	O
trivial	O
.	O
</s>
<s>
The	O
subject	O
is	O
closely	O
linked	O
to	O
spectral	O
theory	O
,	O
since	O
for	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
or	O
multiplication	B-Algorithm
operator	I-Algorithm
,	O
it	O
is	O
rather	O
clear	O
what	O
the	O
definitions	O
should	O
be	O
.	O
</s>
