<s>
In	O
mathematics	O
,	O
more	O
specifically	O
functional	B-Application
analysis	I-Application
and	O
operator	O
theory	O
,	O
the	O
notion	O
of	O
unbounded	B-Algorithm
operator	I-Algorithm
provides	O
an	O
abstract	O
framework	O
for	O
dealing	O
with	O
differential	O
operators	O
,	O
unbounded	O
observables	O
in	O
quantum	O
mechanics	O
,	O
and	O
other	O
cases	O
.	O
</s>
<s>
"	O
operator	O
"	O
should	O
be	O
understood	O
as	O
"	O
linear	B-Architecture
operator	I-Architecture
"	O
(	O
as	O
in	O
the	O
case	O
of	O
"	O
bounded	O
operator	O
"	O
)	O
;	O
</s>
<s>
In	O
contrast	O
to	O
bounded	O
operators	O
,	O
unbounded	B-Algorithm
operators	I-Algorithm
on	O
a	O
given	O
space	O
do	O
not	O
form	O
an	O
algebra	O
,	O
nor	O
even	O
a	O
linear	O
space	O
,	O
because	O
each	O
one	O
is	O
defined	O
on	O
its	O
own	O
domain	O
.	O
</s>
<s>
The	O
term	O
"	O
operator	O
"	O
often	O
means	O
"	O
bounded	O
linear	B-Architecture
operator	I-Architecture
"	O
,	O
but	O
in	O
the	O
context	O
of	O
this	O
article	O
it	O
means	O
"	O
unbounded	B-Algorithm
operator	I-Algorithm
"	O
,	O
with	O
the	O
reservations	O
made	O
above	O
.	O
</s>
<s>
Some	O
generalizations	O
to	O
Banach	O
spaces	O
and	O
more	O
general	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
are	O
possible	O
.	O
</s>
<s>
The	O
theory	O
of	O
unbounded	B-Algorithm
operators	I-Algorithm
developed	O
in	O
the	O
late	O
1920s	O
and	O
early	O
1930s	O
as	O
part	O
of	O
developing	O
a	O
rigorous	O
mathematical	O
framework	O
for	O
quantum	O
mechanics	O
.	O
</s>
<s>
Von	O
Neumann	O
introduced	O
using	O
graphs	B-Application
to	O
analyze	O
unbounded	B-Algorithm
operators	I-Algorithm
in	O
1932	O
.	O
</s>
<s>
An	O
unbounded	B-Algorithm
operator	I-Algorithm
(	O
or	O
simply	O
operator	O
)	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
a	O
linear	O
subspace	O
—	O
the	O
domain	O
of	O
—	O
to	O
the	O
space	O
.	O
</s>
<s>
An	O
operator	O
is	O
said	O
to	O
be	O
closed	O
if	O
its	O
graph	B-Application
is	O
a	O
closed	O
set	O
.	O
</s>
<s>
(	O
Here	O
,	O
the	O
graph	B-Application
is	O
a	O
linear	O
subspace	O
of	O
the	O
direct	O
sum	O
,	O
defined	O
as	O
the	O
set	O
of	O
all	O
pairs	O
,	O
where	O
runs	O
over	O
the	O
domain	O
of	O
.	O
)	O
</s>
<s>
The	O
closedness	O
can	O
also	O
be	O
formulated	O
in	O
terms	O
of	O
the	O
graph	B-Application
norm	O
:	O
an	O
operator	O
is	O
closed	O
if	O
and	O
only	O
if	O
its	O
domain	O
is	O
a	O
complete	O
space	O
with	O
respect	O
to	O
the	O
norm	O
:	O
</s>
<s>
An	O
operator	O
is	O
said	O
to	O
be	O
densely	B-Algorithm
defined	I-Algorithm
if	O
its	O
domain	O
is	O
dense	O
in	O
.	O
</s>
<s>
If	O
is	O
closed	O
,	O
densely	B-Algorithm
defined	I-Algorithm
and	O
continuous	O
on	O
its	O
domain	O
,	O
then	O
its	O
domain	O
is	O
all	O
of	O
.	O
</s>
<s>
A	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
on	O
a	O
Hilbert	O
space	O
is	O
called	O
bounded	O
from	O
below	O
if	O
is	O
a	O
positive	O
operator	O
for	O
some	O
real	O
number	O
.	O
</s>
<s>
We	O
claim	O
that	O
is	O
a	O
well-defined	O
unbounded	B-Algorithm
operator	I-Algorithm
,	O
with	O
domain	O
.	O
</s>
<s>
The	O
operator	O
is	O
densely	B-Algorithm
defined	I-Algorithm
,	O
and	O
closed	O
.	O
</s>
<s>
At	O
the	O
same	O
time	O
,	O
it	O
can	O
be	O
bounded	O
as	O
an	O
operator	O
for	O
other	O
pairs	O
of	O
Banach	O
spaces	O
,	O
and	O
also	O
as	O
operator	O
for	O
some	O
topological	B-Architecture
vector	I-Architecture
spaces	I-Architecture
.	O
</s>
<s>
The	O
adjoint	O
of	O
an	O
unbounded	B-Algorithm
operator	I-Algorithm
can	O
be	O
defined	O
in	O
two	O
equivalent	O
ways	O
.	O
</s>
<s>
Let	O
be	O
an	O
unbounded	B-Algorithm
operator	I-Algorithm
between	O
Hilbert	O
spaces	O
.	O
</s>
<s>
This	O
vector	O
is	O
uniquely	O
determined	O
by	O
if	O
and	O
only	O
if	O
the	O
linear	O
functional	O
is	O
densely	B-Algorithm
defined	I-Algorithm
;	O
or	O
equivalently	O
,	O
if	O
is	O
densely	B-Algorithm
defined	I-Algorithm
.	O
</s>
<s>
Finally	O
,	O
letting	O
completes	O
the	O
construction	O
of	O
which	O
is	O
necessarily	O
a	O
linear	B-Architecture
map	I-Architecture
.	O
</s>
<s>
The	O
adjoint	O
exists	O
if	O
and	O
only	O
if	O
is	O
densely	B-Algorithm
defined	I-Algorithm
.	O
</s>
<s>
If	O
the	O
domain	O
of	O
is	O
dense	O
,	O
then	O
it	O
has	O
its	O
adjoint	O
A	O
closed	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
is	O
bounded	O
if	O
and	O
only	O
if	O
is	O
bounded	O
.	O
</s>
<s>
Define	O
a	O
linear	B-Architecture
operator	I-Architecture
as	O
follows	O
:	O
</s>
<s>
Hence	O
:	O
is	O
the	O
graph	B-Application
of	O
some	O
operator	O
if	O
and	O
only	O
if	O
is	O
densely	B-Algorithm
defined	I-Algorithm
.	O
</s>
<s>
Some	O
well-known	O
properties	O
for	O
bounded	O
operators	O
generalize	O
to	O
closed	O
densely	B-Algorithm
defined	I-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
the	O
kernel	O
of	O
a	O
closed	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
coincides	O
with	O
the	O
orthogonal	O
complement	O
of	O
the	O
range	O
of	O
the	O
adjoint	O
.	O
</s>
<s>
A	O
densely	B-Algorithm
defined	I-Algorithm
,	O
closed	O
operator	O
is	O
called	O
normal	O
if	O
it	O
satisfies	O
the	O
following	O
equivalent	O
conditions	O
:	O
</s>
<s>
Then	O
the	O
transpose	O
(	O
or	O
dual	O
)	O
of	O
is	O
the	O
linear	B-Architecture
operator	I-Architecture
satisfying	O
:	O
</s>
<s>
The	O
necessary	O
and	O
sufficient	O
condition	O
for	O
the	O
transpose	O
of	O
to	O
exist	O
is	O
that	O
is	O
densely	B-Algorithm
defined	I-Algorithm
(	O
for	O
essentially	O
the	O
same	O
reason	O
as	O
to	O
adjoints	O
,	O
as	O
discussed	O
above	O
.	O
)	O
</s>
<s>
Closed	O
linear	B-Architecture
operators	I-Architecture
are	O
a	O
class	O
of	O
linear	B-Architecture
operators	I-Architecture
on	O
Banach	O
spaces	O
.	O
</s>
<s>
They	O
are	O
more	O
general	O
than	O
bounded	O
operators	O
,	O
and	O
therefore	O
not	O
necessarily	O
continuous	O
,	O
but	O
they	O
still	O
retain	O
nice	O
enough	O
properties	O
that	O
one	O
can	O
define	O
the	O
spectrum	O
and	O
(	O
with	O
certain	O
assumptions	O
)	O
functional	B-Algorithm
calculus	I-Algorithm
for	O
such	O
operators	O
.	O
</s>
<s>
Many	O
important	O
linear	B-Architecture
operators	I-Architecture
which	O
fail	O
to	O
be	O
bounded	O
turn	O
out	O
to	O
be	O
closed	O
,	O
such	O
as	O
the	O
derivative	B-Algorithm
and	O
a	O
large	O
class	O
of	O
differential	O
operators	O
.	O
</s>
<s>
A	O
linear	B-Architecture
operator	I-Architecture
is	O
closed	O
if	O
for	O
every	O
sequence	O
in	O
converging	B-Algorithm
to	O
in	O
such	O
that	O
as	O
one	O
has	O
and	O
.	O
</s>
<s>
Equivalently	O
,	O
is	O
closed	O
if	O
its	O
graph	B-Application
is	O
closed	O
in	O
the	O
direct	O
sum	O
.	O
</s>
<s>
Given	O
a	O
linear	B-Architecture
operator	I-Architecture
,	O
not	O
necessarily	O
closed	O
,	O
if	O
the	O
closure	O
of	O
its	O
graph	B-Application
in	O
happens	O
to	O
be	O
the	O
graph	B-Application
of	O
some	O
operator	O
,	O
that	O
operator	O
is	O
called	O
the	O
closure	O
of	O
,	O
and	O
we	O
say	O
that	O
is	O
closable	O
.	O
</s>
<s>
A	O
core	O
(	O
or	O
essential	O
domain	O
)	O
of	O
a	O
closable	B-Algorithm
operator	I-Algorithm
is	O
a	O
subset	O
of	O
such	O
that	O
the	O
closure	O
of	O
the	O
restriction	O
of	O
to	O
is	O
.	O
</s>
<s>
Consider	O
the	O
derivative	B-Algorithm
operator	O
where	O
is	O
the	O
Banach	O
space	O
of	O
all	O
continuous	O
functions	O
on	O
an	O
interval	O
.	O
</s>
<s>
A	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
is	O
symmetric	O
if	O
and	O
only	O
if	O
it	O
agrees	O
with	O
its	O
adjoint	O
T∗	O
restricted	O
to	O
the	O
domain	O
of	O
T	O
,	O
in	O
other	O
words	O
when	O
T∗	O
is	O
an	O
extension	O
of	O
.	O
</s>
<s>
In	O
general	O
,	O
if	O
T	O
is	O
densely	B-Algorithm
defined	I-Algorithm
and	O
symmetric	O
,	O
the	O
domain	O
of	O
the	O
adjoint	O
T∗	O
need	O
not	O
equal	O
the	O
domain	O
of	O
T	O
.	O
If	O
T	O
is	O
symmetric	O
and	O
the	O
domain	O
of	O
T	O
and	O
the	O
domain	O
of	O
the	O
adjoint	O
coincide	O
,	O
then	O
we	O
say	O
that	O
T	O
is	O
self-adjoint	O
.	O
</s>
<s>
Note	O
that	O
,	O
when	O
T	O
is	O
self-adjoint	O
,	O
the	O
existence	O
of	O
the	O
adjoint	O
implies	O
that	O
T	O
is	O
densely	B-Algorithm
defined	I-Algorithm
and	O
since	O
T∗	O
is	O
necessarily	O
closed	O
,	O
T	O
is	O
closed	O
.	O
</s>
<s>
A	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
T	O
is	O
symmetric	O
,	O
if	O
the	O
subspace	O
(	O
defined	O
in	O
a	O
previous	O
section	O
)	O
is	O
orthogonal	O
to	O
its	O
image	O
under	O
J	O
(	O
where	O
J(x,y )	O
:=(	O
y	O
,	O
-x	O
)	O
)	O
.	O
</s>
<s>
Equivalently	O
,	O
an	O
operator	O
T	O
is	O
self-adjoint	O
if	O
it	O
is	O
densely	B-Algorithm
defined	I-Algorithm
,	O
closed	O
,	O
symmetric	O
,	O
and	O
satisfies	O
the	O
fourth	O
condition	O
:	O
both	O
operators	O
,	O
are	O
surjective	O
,	O
that	O
is	O
,	O
map	O
the	O
domain	O
of	O
T	O
onto	O
the	O
whole	O
space	O
H	O
.	O
In	O
other	O
words	O
:	O
for	O
every	O
x	O
in	O
H	O
there	O
exist	O
y	O
and	O
z	O
in	O
the	O
domain	O
of	O
T	O
such	O
that	O
and	O
.	O
</s>
<s>
Non-densely	O
defined	O
symmetric	O
operators	O
can	O
be	O
defined	O
directly	O
or	O
via	O
graphs	B-Application
,	O
but	O
not	O
via	O
adjoint	O
operators	O
.	O
</s>
<s>
A	O
symmetric	O
operator	O
is	O
often	O
studied	O
via	O
its	O
Cayley	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
A	O
densely	B-Algorithm
defined	I-Algorithm
closed	O
symmetric	O
operator	O
T	O
is	O
self-adjoint	O
if	O
and	O
only	O
if	O
T∗	O
is	O
symmetric	O
.	O
</s>
<s>
A	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
T	O
is	O
called	O
positive	O
(	O
or	O
nonnegative	O
)	O
if	O
its	O
quadratic	O
form	O
is	O
nonnegative	O
,	O
that	O
is	O
,	O
for	O
all	O
x	O
in	O
the	O
domain	O
of	O
T	O
.	O
Such	O
operator	O
is	O
necessarily	O
symmetric	O
.	O
</s>
<s>
The	O
operator	O
T∗T	O
is	O
self-adjoint	O
and	O
positive	O
for	O
every	O
densely	B-Algorithm
defined	I-Algorithm
,	O
closed	O
T	O
.	O
</s>
<s>
The	O
spectral	O
theorem	O
applies	O
to	O
self-adjoint	O
operators	O
and	O
moreover	O
,	O
to	O
normal	O
operators	O
,	O
but	O
not	O
to	O
densely	B-Algorithm
defined	I-Algorithm
,	O
closed	O
operators	O
in	O
general	O
,	O
since	O
in	O
this	O
case	O
the	O
spectrum	O
can	O
be	O
empty	O
.	O
</s>
<s>
Note	O
that	O
an	O
everywhere	O
defined	O
extension	O
exists	O
for	O
every	O
operator	O
,	O
which	O
is	O
a	O
purely	O
algebraic	O
fact	O
explained	O
at	O
Discontinuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
#General	O
existence	O
theorem	O
and	O
based	O
on	O
the	O
axiom	O
of	O
choice	O
.	O
</s>
<s>
If	O
the	O
given	O
operator	O
is	O
not	O
bounded	O
then	O
the	O
extension	O
is	O
a	O
discontinuous	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
.	O
</s>
<s>
the	O
closure	O
of	O
the	O
graph	B-Application
of	O
T	O
is	O
the	O
graph	B-Application
of	O
some	O
operator	O
;	O
</s>
<s>
A	O
closable	B-Algorithm
operator	I-Algorithm
T	O
has	O
the	O
least	O
closed	O
extension	O
called	O
the	O
closure	O
of	O
T	O
.	O
The	O
closure	O
of	O
the	O
graph	B-Application
of	O
T	O
is	O
equal	O
to	O
the	O
graph	B-Application
of	O
Other	O
,	O
non-minimal	O
closed	O
extensions	O
may	O
exist	O
.	O
</s>
<s>
A	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
T	O
is	O
closable	O
if	O
and	O
only	O
if	O
T∗	O
is	O
densely	B-Algorithm
defined	I-Algorithm
.	O
</s>
<s>
If	O
S	O
is	O
densely	B-Algorithm
defined	I-Algorithm
and	O
T	O
is	O
an	O
extension	O
of	O
S	O
then	O
S∗	O
is	O
an	O
extension	O
of	O
T∗	O
.	O
</s>
<s>
A	O
densely	B-Algorithm
defined	I-Algorithm
,	O
symmetric	O
operator	O
T	O
is	O
essentially	O
self-adjoint	O
if	O
and	O
only	O
if	O
both	O
operators	O
,	O
have	O
dense	O
range	O
.	O
</s>
<s>
Let	O
T	O
be	O
a	O
densely	B-Algorithm
defined	I-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
Every	O
self-adjoint	O
operator	O
is	O
densely	B-Algorithm
defined	I-Algorithm
,	O
closed	O
and	O
symmetric	O
.	O
</s>
