<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
branch	O
of	O
mathematics	O
,	O
the	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
is	O
a	O
functional	B-Algorithm
calculus	I-Algorithm
(	O
that	O
is	O
,	O
an	O
assignment	O
of	O
operators	O
from	O
commutative	O
algebras	O
to	O
functions	O
defined	O
on	O
their	O
spectra	O
)	O
,	O
which	O
has	O
particularly	O
broad	O
scope	O
.	O
</s>
<s>
The	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
is	O
more	O
general	O
than	O
the	O
continuous	O
functional	B-Algorithm
calculus	I-Algorithm
,	O
and	O
its	O
focus	O
is	O
different	O
than	O
the	O
holomorphic	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
allows	O
for	O
applying	O
an	O
arbitrary	O
Borel	O
function	O
to	O
a	O
self-adjoint	O
operator	O
,	O
in	O
a	O
way	O
that	O
generalizes	O
applying	O
a	O
polynomial	O
function	O
.	O
</s>
<s>
If	O
only	O
polynomials	O
in	O
T	O
are	O
considered	O
,	O
then	O
one	O
gets	O
the	O
holomorphic	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
is	O
desirable	O
to	O
formulate	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
in	O
a	O
way	O
that	O
does	O
not	O
depend	O
on	O
the	O
particular	O
representation	O
of	O
T	O
as	O
a	O
multiplication	O
operator	O
.	O
</s>
<s>
Formally	O
,	O
the	O
bounded	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
of	O
a	O
self	O
adjoint	O
operator	O
T	O
on	O
Hilbert	O
space	O
H	O
is	O
a	O
mapping	O
defined	O
on	O
the	O
space	O
of	O
bounded	O
complex-valued	O
Borel	O
functions	O
f	O
on	O
the	O
real	O
line	O
,	O
</s>
<s>
is	O
an	O
involution-preserving	O
and	O
unit-preserving	O
homomorphism	O
from	O
the	O
ring	O
of	O
complex-valued	O
bounded	O
measurable	O
functions	O
on	O
R	O
.	O
</s>
<s>
This	O
defines	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
for	O
bounded	O
functions	O
applied	O
to	O
possibly	O
unbounded	O
self-adjoint	O
operators	O
.	O
</s>
<s>
Using	O
the	O
bounded	O
functional	B-Algorithm
calculus	I-Algorithm
,	O
one	O
can	O
prove	O
part	O
of	O
the	O
Stone	O
's	O
theorem	O
on	O
one-parameter	O
unitary	O
groups	O
:	O
</s>
<s>
We	O
can	O
also	O
use	O
the	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
to	O
abstractly	O
solve	O
some	O
linear	O
initial	O
value	O
problems	O
such	O
as	O
the	O
heat	O
equation	O
,	O
or	O
Maxwell	O
's	O
equations	O
.	O
</s>
<s>
The	O
existence	O
of	O
a	O
mapping	O
with	O
the	O
properties	O
of	O
a	O
functional	B-Algorithm
calculus	I-Algorithm
requires	O
proof	O
.	O
</s>
<s>
For	O
the	O
case	O
of	O
a	O
bounded	O
self-adjoint	O
operator	O
T	O
,	O
the	O
existence	O
of	O
a	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
can	O
be	O
shown	O
in	O
an	O
elementary	O
way	O
as	O
follows	O
:	O
</s>
<s>
First	O
pass	O
from	O
polynomial	O
to	O
continuous	O
functional	B-Algorithm
calculus	I-Algorithm
by	O
using	O
the	O
Stone	O
–	O
Weierstrass	O
theorem	O
.	O
</s>
<s>
Extending	O
by	O
continuity	O
defines	O
f(T )	O
for	O
a	O
continuous	O
function	O
f	O
on	O
the	O
spectrum	O
of	O
T	O
.	O
The	O
Riesz-Markov	O
theorem	O
then	O
allows	O
us	O
to	O
pass	O
from	O
integration	O
on	O
continuous	O
functions	O
to	O
spectral	O
measures	O
,	O
and	O
this	O
is	O
the	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
Given	O
an	O
operator	O
T	O
,	O
the	O
range	O
of	O
the	O
continuous	O
functional	B-Algorithm
calculus	I-Algorithm
h	O
→	O
h(T )	O
is	O
the	O
(	O
abelian	O
)	O
C*	O
-algebra	O
C(T )	O
generated	O
by	O
T	O
.	O
The	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
has	O
a	O
larger	O
range	O
,	O
that	O
is	O
the	O
closure	O
of	O
C(T )	O
in	O
the	O
weak	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
,	O
a	O
(	O
still	O
abelian	O
)	O
von	O
Neumann	O
algebra	O
.	O
</s>
<s>
We	O
can	O
also	O
define	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
for	O
not	O
necessarily	O
bounded	O
Borel	O
functions	O
h	O
;	O
the	O
result	O
is	O
an	O
operator	O
which	O
in	O
general	O
fails	O
to	O
be	O
bounded	O
.	O
</s>
<s>
More	O
generally	O
,	O
a	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
also	O
exists	O
for	O
(	O
bounded	O
)	O
normal	O
operators	O
.	O
</s>
