<s>
In	O
functional	B-Application
analysis	I-Application
,	O
one	O
is	O
interested	O
in	O
extensions	B-Algorithm
of	I-Algorithm
symmetric	I-Algorithm
operators	I-Algorithm
acting	O
on	O
a	O
Hilbert	O
space	O
.	O
</s>
<s>
When	O
dealing	O
with	O
unbounded	B-Algorithm
operators	I-Algorithm
,	O
it	O
is	O
often	O
desirable	O
to	O
be	O
able	O
to	O
assume	O
that	O
the	O
operator	O
in	O
question	O
is	O
closed	O
.	O
</s>
<s>
closable	B-Algorithm
.	O
</s>
<s>
In	O
the	O
sequel	O
,	O
a	O
symmetric	O
operator	O
will	O
be	O
assumed	O
to	O
be	O
densely	B-Algorithm
defined	I-Algorithm
and	O
closed	O
.	O
</s>
<s>
Problem	O
Given	O
a	O
densely	B-Algorithm
defined	I-Algorithm
closed	O
symmetric	O
operator	O
A	O
,	O
find	O
its	O
self-adjoint	O
extensions	O
.	O
</s>
<s>
is	O
also	O
called	O
the	O
Cayley	B-Algorithm
transform	I-Algorithm
of	O
the	O
symmetric	O
operator	O
A	O
.	O
</s>
<s>
Theorem	O
The	O
symmetric	O
extensions	O
of	O
a	O
closed	O
symmetric	O
operator	O
A	O
is	O
in	O
one-to-one	O
correspondence	O
with	O
the	O
isometric	O
extensions	O
of	O
its	O
Cayley	B-Algorithm
transform	I-Algorithm
UA	O
.	O
</s>
<s>
when	O
its	O
Cayley	B-Algorithm
transform	I-Algorithm
UA	O
is	O
a	O
unitary	O
operator	O
on	O
H	O
.	O
</s>
<s>
Corollary	O
The	O
self-adjoint	O
extensions	O
of	O
a	O
closed	O
symmetric	O
operator	O
A	O
is	O
in	O
one-to-one	O
correspondence	O
with	O
the	O
unitary	O
extensions	O
of	O
its	O
Cayley	B-Algorithm
transform	I-Algorithm
UA	O
.	O
</s>
<s>
In	O
this	O
language	O
,	O
the	O
description	O
of	O
the	O
self-adjoint	O
extension	O
problem	O
given	O
by	O
the	O
corollary	O
can	O
be	O
restated	O
as	O
follows	O
:	O
a	O
symmetric	O
operator	O
A	O
has	O
self-adjoint	O
extensions	O
if	O
and	O
only	O
if	O
its	O
Cayley	B-Algorithm
transform	I-Algorithm
UA	O
has	O
unitary	O
extensions	O
to	O
H	O
,	O
i.e.	O
</s>
<s>
The	O
Cayley	B-Algorithm
transform	I-Algorithm
for	O
general	O
symmetric	O
operators	O
can	O
be	O
adapted	O
to	O
this	O
special	O
case	O
.	O
</s>
<s>
The	O
converse	O
is	O
also	O
true	O
:	O
given	O
a	O
positive	O
symmetric	O
extension	O
of	O
A	O
,	O
its	O
Cayley	B-Algorithm
transform	I-Algorithm
is	O
a	O
contraction	O
satisfying	O
the	O
stated	O
"	O
partial	O
"	O
self-adjoint	O
property	O
.	O
</s>
<s>
Theorem	O
The	O
positive	O
symmetric	O
extensions	O
of	O
A	O
are	O
in	O
one-to-one	O
correspondence	O
with	O
the	O
extensions	O
of	O
its	O
Cayley	B-Algorithm
transform	I-Algorithm
where	O
if	O
C	O
is	O
such	O
an	O
extension	O
,	O
we	O
require	O
C	O
projected	O
onto	O
Dom(C )	O
be	O
self-adjoint	O
.	O
</s>
<s>
The	O
unitarity	O
criterion	O
of	O
the	O
Cayley	B-Algorithm
transform	I-Algorithm
is	O
replaced	O
by	O
self-adjointness	O
for	O
positive	O
operators	O
.	O
</s>
<s>
Theorem	O
A	O
symmetric	O
positive	O
operator	O
A	O
is	O
self-adjoint	O
if	O
and	O
only	O
if	O
its	O
Cayley	B-Algorithm
transform	I-Algorithm
is	O
a	O
self-adjoint	O
contraction	O
defined	O
on	O
all	O
of	O
H	O
,	O
i.e.	O
</s>
