<s>
In	O
functional	B-Application
analysis	I-Application
,	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
is	O
a	O
surjective	B-Algorithm
bounded	O
operator	O
on	O
a	O
Hilbert	O
space	O
that	O
preserves	O
the	O
inner	O
product	O
.	O
</s>
<s>
Unitary	B-Algorithm
operators	I-Algorithm
are	O
usually	O
taken	O
as	O
operating	O
on	O
a	O
Hilbert	O
space	O
,	O
but	O
the	O
same	O
notion	O
serves	O
to	O
define	O
the	O
concept	O
of	O
isomorphism	O
between	O
Hilbert	O
spaces	O
.	O
</s>
<s>
A	O
unitary	O
element	O
is	O
a	O
generalization	O
of	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
A	O
unitary	B-Algorithm
operator	I-Algorithm
is	O
a	O
bounded	O
linear	O
operator	O
on	O
a	O
Hilbert	O
space	O
that	O
satisfies	O
,	O
where	O
is	O
the	O
adjoint	O
of	O
,	O
and	O
is	O
the	O
identity	O
operator	O
.	O
</s>
<s>
The	O
other	O
condition	O
,	O
,	O
defines	O
a	O
coisometry	B-Algorithm
.	O
</s>
<s>
Thus	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
is	O
a	O
bounded	O
linear	O
operator	O
which	O
is	O
both	O
an	O
isometry	O
and	O
a	O
coisometry	B-Algorithm
,	O
or	O
,	O
equivalently	O
,	O
a	O
surjective	B-Algorithm
isometry	O
.	O
</s>
<s>
A	O
unitary	B-Algorithm
operator	I-Algorithm
is	O
a	O
bounded	O
linear	O
operator	O
on	O
a	O
Hilbert	O
space	O
for	O
which	O
the	O
following	O
hold	O
:	O
</s>
<s>
A	O
unitary	B-Algorithm
operator	I-Algorithm
is	O
a	O
bounded	O
linear	O
operator	O
on	O
a	O
Hilbert	O
space	O
for	O
which	O
the	O
following	O
hold	O
:	O
</s>
<s>
Thus	O
,	O
unitary	B-Algorithm
operators	I-Algorithm
are	O
just	O
automorphisms	O
of	O
Hilbert	O
spaces	O
,	O
i.e.	O
,	O
they	O
preserve	O
the	O
structure	O
(	O
the	O
linear	O
space	O
structure	O
,	O
the	O
inner	O
product	O
,	O
and	O
hence	O
the	O
topology	B-Architecture
)	O
of	O
the	O
space	O
on	O
which	O
they	O
act	O
.	O
</s>
<s>
The	O
group	O
of	O
all	O
unitary	B-Algorithm
operators	I-Algorithm
from	O
a	O
given	O
Hilbert	O
space	O
to	O
itself	O
is	O
sometimes	O
referred	O
to	O
as	O
the	O
Hilbert	B-Algorithm
group	I-Algorithm
of	O
,	O
denoted	O
or	O
.	O
</s>
<s>
The	O
identity	O
function	O
is	O
trivially	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
Rotations	O
in	O
are	O
the	O
simplest	O
nontrivial	O
example	O
of	O
unitary	B-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
On	O
the	O
vector	O
space	O
of	O
complex	O
numbers	O
,	O
multiplication	O
by	O
a	O
number	O
of	O
absolute	O
value	O
,	O
that	O
is	O
,	O
a	O
number	O
of	O
the	O
form	O
for	O
,	O
is	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
Notice	O
that	O
the	O
value	O
of	O
modulo	O
does	O
not	O
affect	O
the	O
result	O
of	O
the	O
multiplication	O
,	O
and	O
so	O
the	O
independent	O
unitary	B-Algorithm
operators	I-Algorithm
on	O
are	O
parametrized	O
by	O
a	O
circle	O
.	O
</s>
<s>
More	O
generally	O
,	O
unitary	B-Algorithm
matrices	I-Algorithm
are	O
precisely	O
the	O
unitary	B-Algorithm
operators	I-Algorithm
on	O
finite-dimensional	O
Hilbert	O
spaces	O
,	O
so	O
the	O
notion	O
of	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
is	O
a	O
generalization	O
of	O
the	O
notion	O
of	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Orthogonal	B-Algorithm
matrices	I-Algorithm
are	O
the	O
special	O
case	O
of	O
unitary	B-Algorithm
matrices	I-Algorithm
in	O
which	O
all	O
entries	O
are	O
real	O
.	O
</s>
<s>
They	O
are	O
the	O
unitary	B-Algorithm
operators	I-Algorithm
on	O
.	O
</s>
<s>
The	O
bilateral	B-Algorithm
shift	I-Algorithm
on	O
the	O
sequence	O
space	O
indexed	O
by	O
the	O
integers	O
is	O
unitary	O
.	O
</s>
<s>
In	O
the	O
finite	O
dimensional	O
case	O
,	O
such	O
operators	O
are	O
the	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
unilateral	B-Algorithm
shift	I-Algorithm
(	O
right	O
shift	O
)	O
is	O
an	O
isometry	O
;	O
its	O
conjugate	O
(	O
left	O
shift	O
)	O
is	O
a	O
coisometry	B-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
operator	I-Algorithm
is	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
,	O
i.e.	O
</s>
<s>
the	O
operator	O
which	O
performs	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
with	O
proper	O
normalization	O
)	O
.	O
</s>
<s>
Unitary	B-Algorithm
operators	I-Algorithm
are	O
used	O
in	O
unitary	O
representations	O
.	O
</s>
<s>
Quantum	O
logic	O
gates	O
are	O
unitary	B-Algorithm
operators	I-Algorithm
.	O
</s>
<s>
Not	O
all	O
gates	O
are	O
Hermitian	B-Algorithm
.	O
</s>
<s>
The	O
linearity	O
requirement	O
in	O
the	O
definition	O
of	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
can	O
be	O
dropped	O
without	O
changing	O
the	O
meaning	O
because	O
it	O
can	O
be	O
derived	O
from	O
linearity	O
and	O
positive-definiteness	O
of	O
the	O
scalar	O
product	O
:	O
</s>
<s>
The	O
spectrum	O
of	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
lies	O
on	O
the	O
unit	O
circle	O
.	O
</s>
<s>
A	O
linear	O
map	O
is	O
unitary	O
if	O
it	O
is	O
surjective	B-Algorithm
and	O
isometric	O
.	O
</s>
