<s>
In	O
mathematics	O
,	O
holomorphic	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
is	O
functional	B-Algorithm
calculus	I-Algorithm
with	O
holomorphic	O
functions	O
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
defines	O
a	O
continuous	O
algebra	O
homomorphism	O
from	O
the	O
holomorphic	O
functions	O
on	O
a	O
neighbourhood	O
of	O
the	O
spectrum	O
of	O
T	O
to	O
the	O
bounded	O
operators	O
.	O
</s>
<s>
In	O
particular	O
,	O
T	O
can	O
be	O
a	O
square	B-Algorithm
matrix	I-Algorithm
with	O
complex	O
entries	O
,	O
a	O
case	O
which	O
will	O
be	O
used	O
to	O
illustrate	O
functional	B-Algorithm
calculus	I-Algorithm
and	O
provide	O
some	O
heuristic	O
insights	O
for	O
the	O
assumptions	O
involved	O
in	O
the	O
general	O
construction	O
.	O
</s>
<s>
where	O
T0	O
=	O
I	O
,	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
This	O
is	O
the	O
polynomial	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
Thus	O
a	O
more	O
general	O
functional	B-Algorithm
calculus	I-Algorithm
is	O
needed	O
.	O
</s>
<s>
This	O
assumption	O
turns	O
out	O
to	O
be	O
an	O
enabling	O
condition	O
such	O
that	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
map	O
,	O
f	O
→	O
f(T )	O
,	O
has	O
certain	O
desirable	O
properties	O
.	O
</s>
<s>
Assuming	O
this	O
Banach	O
space-valued	O
integral	O
is	O
appropriately	O
defined	O
,	O
this	O
proposed	O
functional	B-Algorithm
calculus	I-Algorithm
implies	O
the	O
following	O
necessary	O
conditions	O
:	O
</s>
<s>
So	O
to	O
obtain	O
a	O
non-trivial	O
functional	B-Algorithm
calculus	I-Algorithm
,	O
Γ	O
must	O
enclose	O
(	O
at	O
least	O
part	O
of	O
)	O
σ(T )	O
.	O
</s>
<s>
The	O
functional	B-Algorithm
calculus	I-Algorithm
should	O
be	O
well-defined	O
in	O
the	O
sense	O
that	O
f(T )	O
has	O
to	O
be	O
independent	O
of	O
Γ	O
.	O
</s>
<s>
The	O
Riemann	O
sums	O
converge	O
in	O
the	O
uniform	B-Algorithm
operator	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
In	O
the	O
definition	O
of	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
,	O
f	O
is	O
assumed	O
to	O
be	O
holomorphic	O
in	O
an	O
open	O
neighborhood	O
of	O
Γ	O
.	O
</s>
<s>
The	O
holomorphic	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
is	O
similar	O
in	O
that	O
the	O
resolvent	O
mapping	O
plays	O
a	O
crucial	O
role	O
in	O
obtaining	O
properties	O
one	O
requires	O
from	O
a	O
nice	O
functional	B-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
The	O
formula	O
shows	O
(	O
z1−T	O
)	O
−1	O
and	O
(	O
z2−T	O
)	O
−1	O
commute	O
,	O
which	O
hints	O
at	O
the	O
fact	O
that	O
the	O
image	O
of	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
will	O
be	O
a	O
commutative	O
algebra	O
.	O
</s>
<s>
Letting	O
z2	O
→	O
z1	O
shows	O
the	O
resolvent	O
map	O
is	O
(	O
complex	O
-	O
)	O
differentiable	O
at	O
each	O
z1	O
∈	O
ρ(T )	O
;	O
so	O
the	O
integral	O
in	O
the	O
expression	O
of	O
functional	B-Algorithm
calculus	I-Algorithm
converges	O
in	O
L(X )	O
.	O
</s>
<s>
This	O
property	O
will	O
be	O
used	O
in	O
subsequent	O
arguments	O
for	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
Hence	O
,	O
for	O
the	O
definition	O
of	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
,	O
indeed	O
a	O
suitable	O
family	O
of	O
Jordan	O
curves	O
can	O
be	O
found	O
for	O
each	O
f	O
that	O
is	O
holomorphic	O
on	O
some	O
D	O
.	O
</s>
<s>
What	O
has	O
not	O
been	O
shown	O
is	O
that	O
the	O
definition	O
of	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
is	O
unambiguous	O
,	O
i.e.	O
</s>
<s>
The	O
well-definedness	O
of	O
functional	B-Algorithm
calculus	I-Algorithm
now	O
follows	O
as	O
an	O
easy	O
consequence	O
.	O
</s>
<s>
Suppose	O
Γ	O
=	O
 { γi } 	O
and	O
Ω	O
=	O
 { ωj } 	O
are	O
two	O
(	O
finite	O
)	O
collections	O
of	O
Jordan	O
curves	O
satisfying	O
the	O
assumption	O
given	O
for	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
Hence	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
is	O
well-defined	O
.	O
</s>
<s>
The	O
assumption	O
will	O
be	O
applied	O
in	O
its	O
entirety	O
in	O
showing	O
the	O
homomorphism	O
property	O
of	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
We	O
recover	O
the	O
polynomial	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
when	O
f(z )	O
=	O
Σ0	O
≤	O
i	O
≤	O
m	O
ai	O
zi	O
is	O
a	O
polynomial	O
.	O
</s>
<s>
To	O
summarize	O
,	O
we	O
have	O
shown	O
the	O
holomorphic	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
,	O
f	O
→	O
f(T )	O
,	O
has	O
the	O
following	O
properties	O
:	O
</s>
<s>
It	O
extends	O
the	O
polynomial	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
The	O
functional	B-Algorithm
calculus	I-Algorithm
can	O
be	O
defined	O
in	O
exactly	O
the	O
same	O
way	O
for	O
an	O
element	O
in	O
A	O
.	O
</s>
<s>
It	O
is	O
known	O
that	O
the	O
spectral	O
mapping	O
theorem	O
holds	O
for	O
the	O
polynomial	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
:	O
for	O
any	O
polynomial	O
p	O
,	O
σ(p(T )	O
)	O
=	O
p(σ(T )	O
)	O
.	O
</s>
<s>
If	O
the	O
spectrum	O
σ(T )	O
is	O
not	O
connected	O
,	O
X	O
can	O
be	O
decomposed	O
into	O
invariant	O
subspaces	O
of	O
T	O
using	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
With	O
stronger	O
assumptions	O
,	O
when	O
T	O
is	O
a	O
normal	O
operator	O
acting	O
on	O
a	O
Hilbert	O
space	O
,	O
the	O
domain	O
of	O
the	O
functional	B-Algorithm
calculus	I-Algorithm
can	O
be	O
broadened	O
.	O
</s>
<s>
When	O
T	O
is	O
a	O
normal	O
operator	O
,	O
a	O
continuous	O
functional	B-Algorithm
calculus	I-Algorithm
can	O
be	O
obtained	O
,	O
that	O
is	O
,	O
one	O
can	O
evaluate	O
f(T )	O
with	O
f	O
being	O
a	O
continuous	O
function	O
defined	O
on	O
σ(T )	O
.	O
</s>
<s>
Using	O
the	O
machinery	O
of	O
measure	O
theory	O
,	O
this	O
can	O
be	O
extended	O
to	O
functions	O
which	O
are	O
only	O
measurable	O
(	O
see	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
)	O
.	O
</s>
<s>
The	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
extends	O
to	O
unbounded	O
self-adjoint	O
operators	O
on	O
a	O
Hilbert	O
space	O
.	O
</s>
<s>
In	O
slightly	O
more	O
abstract	O
language	O
,	O
the	O
holomorphic	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
can	O
be	O
extended	O
to	O
any	O
element	O
of	O
a	O
Banach	O
algebra	O
,	O
using	O
essentially	O
the	O
same	O
arguments	O
as	O
above	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
continuous	O
functional	B-Algorithm
calculus	I-Algorithm
holds	O
for	O
normal	O
elements	O
in	O
any	O
C*	B-Algorithm
-algebra	I-Algorithm
and	O
the	O
measurable	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
for	O
normal	O
elements	O
in	O
any	O
von	O
Neumann	O
algebra	O
.	O
</s>
<s>
A	O
holomorphic	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
can	O
be	O
defined	O
in	O
a	O
similar	O
fashion	O
for	O
unbounded	O
closed	O
operators	O
with	O
non-empty	O
resolvent	O
set	O
.	O
</s>
