<s>
In	O
operator	O
theory	O
,	O
a	O
dilation	B-Algorithm
of	O
an	O
operator	O
T	O
on	O
a	O
Hilbert	O
space	O
H	O
is	O
an	O
operator	O
on	O
a	O
larger	O
Hilbert	O
space	O
K	O
,	O
whose	O
restriction	O
to	O
H	O
composed	O
with	O
the	O
orthogonal	O
projection	O
onto	O
H	O
is	O
T	O
.	O
</s>
<s>
V	O
is	O
said	O
to	O
be	O
a	O
unitary	B-Algorithm
dilation	I-Algorithm
(	O
respectively	O
,	O
normal	O
,	O
isometric	O
,	O
etc	O
.	O
)	O
</s>
<s>
T	O
is	O
said	O
to	O
be	O
a	O
compression	O
of	O
V	O
.	O
If	O
an	O
operator	O
T	O
has	O
a	O
spectral	B-Algorithm
set	I-Algorithm
,	O
we	O
say	O
that	O
V	O
is	O
a	O
normal	O
boundary	O
dilation	B-Algorithm
or	O
a	O
normal	O
dilation	B-Algorithm
if	O
V	O
is	O
a	O
normal	O
dilation	B-Algorithm
of	O
T	O
and	O
.	O
</s>
<s>
Namely	O
,	O
that	O
a	O
dilation	B-Algorithm
satisfy	O
the	O
following	O
(	O
calculus	O
)	O
property	O
:	O
</s>
<s>
where	O
f(T )	O
is	O
some	O
specified	O
functional	B-Algorithm
calculus	I-Algorithm
(	O
for	O
example	O
,	O
the	O
polynomial	O
or	O
H∞	O
calculus	O
)	O
.	O
</s>
<s>
The	O
utility	O
of	O
a	O
dilation	B-Algorithm
is	O
that	O
it	O
allows	O
the	O
"	O
lifting	O
"	O
of	O
objects	O
associated	O
to	O
T	O
to	O
the	O
level	O
of	O
V	O
,	O
where	O
the	O
lifted	O
objects	O
may	O
have	O
nicer	O
properties	O
.	O
</s>
<s>
We	O
can	O
show	O
that	O
every	O
contraction	O
on	O
Hilbert	O
spaces	O
has	O
a	O
unitary	B-Algorithm
dilation	I-Algorithm
.	O
</s>
<s>
A	O
possible	O
construction	O
of	O
this	O
dilation	B-Algorithm
is	O
as	O
follows	O
.	O
</s>
<s>
is	O
positive	O
,	O
where	O
the	O
continuous	O
functional	B-Algorithm
calculus	I-Algorithm
is	O
used	O
to	O
define	O
the	O
square	O
root	O
.	O
</s>
<s>
Using	O
this	O
one	O
can	O
show	O
,	O
by	O
calculating	O
directly	O
,	O
that	O
V	O
is	O
unitary	O
,	O
therefore	O
a	O
unitary	B-Algorithm
dilation	I-Algorithm
of	O
T	O
.	O
This	O
operator	O
V	O
is	O
sometimes	O
called	O
the	O
Julia	O
operator	O
of	O
T	O
.	O
</s>
<s>
We	O
note	O
here	O
that	O
in	O
the	O
above	O
discussion	O
we	O
have	O
not	O
required	O
the	O
calculus	O
property	O
for	O
a	O
dilation	B-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
direct	O
calculation	O
shows	O
the	O
Julia	O
operator	O
fails	O
to	O
be	O
a	O
"	O
degree-2	O
"	O
dilation	B-Algorithm
in	O
general	O
,	O
i.e.	O
</s>
<s>
However	O
,	O
it	O
can	O
also	O
be	O
shown	O
that	O
any	O
contraction	O
has	O
a	O
unitary	B-Algorithm
dilation	I-Algorithm
which	O
does	O
have	O
the	O
calculus	O
property	O
above	O
.	O
</s>
<s>
This	O
is	O
Sz.-Nagy	O
'	O
s	O
dilation	B-Algorithm
theorem	O
.	O
</s>
<s>
More	O
generally	O
,	O
if	O
is	O
a	O
Dirichlet	O
algebra	O
,	O
any	O
operator	O
T	O
with	O
as	O
a	O
spectral	B-Algorithm
set	I-Algorithm
will	O
have	O
a	O
normal	O
dilation	B-Algorithm
with	O
this	O
property	O
.	O
</s>
<s>
This	O
generalises	O
Sz.-Nagy	O
'	O
s	O
dilation	B-Algorithm
theorem	O
as	O
all	O
contractions	O
have	O
the	O
unit	O
disc	O
as	O
a	O
spectral	B-Algorithm
set	I-Algorithm
.	O
</s>
