<s>
where	O
denotes	O
function	B-Application
composition	I-Application
.	O
</s>
<s>
The	O
study	O
of	O
composition	B-Algorithm
operators	I-Algorithm
is	O
covered	O
by	O
.	O
</s>
<s>
In	O
physics	O
,	O
and	O
especially	O
the	O
area	O
of	O
dynamical	O
systems	O
,	O
the	O
composition	B-Algorithm
operator	I-Algorithm
is	O
usually	O
referred	O
to	O
as	O
the	O
Koopman	B-Algorithm
operator	I-Algorithm
(	O
and	O
its	O
wild	O
surge	O
in	O
popularity	O
is	O
sometimes	O
jokingly	O
called	O
"	O
Koopmania	O
"	O
)	O
,	O
named	O
after	O
Bernard	O
Koopman	O
.	O
</s>
<s>
Using	O
the	O
language	O
of	O
category	O
theory	O
,	O
the	O
composition	B-Algorithm
operator	I-Algorithm
is	O
a	O
pull-back	O
on	O
the	O
space	O
of	O
measurable	O
functions	O
;	O
it	O
is	O
adjoint	O
to	O
the	O
transfer	O
operator	O
in	O
the	O
same	O
way	O
that	O
the	O
pull-back	O
is	O
adjoint	O
to	O
the	O
push-forward	O
;	O
the	O
composition	B-Algorithm
operator	I-Algorithm
is	O
the	O
inverse	O
image	O
functor	O
.	O
</s>
<s>
Since	O
the	O
domain	B-Algorithm
considered	O
here	O
is	O
that	O
of	O
Borel	O
functions	O
,	O
the	O
above	O
describes	O
the	O
Koopman	B-Algorithm
operator	I-Algorithm
as	O
it	O
appears	O
in	O
Borel	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
The	O
domain	B-Algorithm
of	O
a	O
composition	B-Algorithm
operator	I-Algorithm
can	O
be	O
taken	O
more	O
narrowly	O
,	O
as	O
some	O
Banach	O
space	O
,	O
often	O
consisting	O
of	O
holomorphic	O
functions	O
:	O
for	O
example	O
,	O
some	O
Hardy	O
space	O
or	O
Bergman	O
space	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
composition	B-Algorithm
operator	I-Algorithm
lies	O
in	O
the	O
realm	O
of	O
some	O
functional	B-Algorithm
calculus	I-Algorithm
,	O
such	O
as	O
the	O
holomorphic	B-Algorithm
functional	I-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
Interesting	O
questions	O
posed	O
in	O
the	O
study	O
of	O
composition	B-Algorithm
operators	I-Algorithm
often	O
relate	O
to	O
how	O
the	O
spectral	O
properties	O
of	O
the	O
operator	O
depend	O
on	O
the	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Other	O
questions	O
include	O
whether	O
is	O
compact	O
or	O
trace-class	O
;	O
answers	O
typically	O
depend	O
on	O
how	O
the	O
function	O
behaves	O
on	O
the	O
boundary	O
of	O
some	O
domain	B-Algorithm
.	O
</s>
<s>
When	O
the	O
transfer	O
operator	O
is	O
a	O
left-shift	O
operator	O
,	O
the	O
Koopman	B-Algorithm
operator	I-Algorithm
,	O
as	O
its	O
adjoint	O
,	O
can	O
be	O
taken	O
to	O
be	O
the	O
right-shift	O
operator	O
.	O
</s>
<s>
When	O
the	O
polynomials	O
are	O
orthogonal	O
on	O
some	O
region	O
of	O
the	O
complex	O
plane	O
(	O
viz	O
,	O
in	O
Bergman	O
space	O
)	O
,	O
the	O
Jacobi	O
operator	O
is	O
replaced	O
by	O
a	O
Hessenberg	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
In	O
mathematics	O
,	O
composition	B-Algorithm
operators	I-Algorithm
commonly	O
occur	O
in	O
the	O
study	O
of	O
shift	B-Algorithm
operators	I-Algorithm
,	O
for	O
example	O
,	O
in	O
the	O
Beurling	O
–	O
Lax	O
theorem	O
and	O
the	O
Wold	O
decomposition	O
.	O
</s>
<s>
Shift	B-Algorithm
operators	I-Algorithm
can	O
be	O
studied	O
as	O
one-dimensional	O
spin	O
lattices	O
.	O
</s>
<s>
Composition	B-Algorithm
operators	I-Algorithm
appear	O
in	O
the	O
theory	O
of	O
Aleksandrov	O
–	O
Clark	O
measures	O
.	O
</s>
<s>
The	O
eigenvalue	O
equation	O
of	O
the	O
composition	B-Algorithm
operator	I-Algorithm
is	O
Schröder	O
's	O
equation	O
,	O
and	O
the	O
principal	O
eigenfunction	B-Algorithm
is	O
often	O
called	O
Schröder	O
's	O
function	O
or	O
Koenigs	O
function	O
.	O
</s>
<s>
Composition	B-Algorithm
operator	I-Algorithm
has	O
been	O
used	O
in	O
data-driven	O
techniques	O
for	O
dynamical	O
systems	O
by	O
the	O
use	O
of	O
dynamic	O
mode	O
decomposition	O
algorithms	O
,	O
which	O
approximate	O
the	O
modes	O
and	O
eigenvalues	O
of	O
the	O
composition	B-Algorithm
operator	I-Algorithm
.	O
</s>
