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Let  $  T _ {1} $
be a [[Contraction(2)|contraction]] on a [[Hilbert space|Hilbert space]]  $  {\mathcal H} _ {1} $, 
that is,  $  \| {T _ {1} } \| \leq  1 $. 
Recall that  $  U $
is an isometric dilation of  $  T _ {1} $
if  $  U $
is an isometry (cf. [[Isometric operator|Isometric operator]]) on a Hilbert space  $  {\mathcal K} \supseteq {\mathcal H} _ {1} $
and  $  {\mathcal H} _ {1} $
is an invariant subspace for  $  U  ^ {*} $
satisfying  $  U  ^ {*} \mid  {\mathcal H} _ {1} = T _ {1}  ^ {*} $. 
The Sz.-Nagy–Shäffer construction shows that all contractions admit an isometric dilation [[#References|[a1]]], [[#References|[a5]]]. This sets the stage for the following result, known as the Sz.-Nagy–Foias commutant lifting theorem [[#References|[a1]]], [[#References|[a4]]], [[#References|[a5]]].

Let  $  U $
on  $  {\mathcal K} $
be an isometric dilation for a contraction  $  T _ {1} $
on  $  {\mathcal H} _ {1} $. 
Let  $  A $
be an operator from the Hilbert space  $  {\mathcal H} $
into  $  {\mathcal H} _ {1} $
and  $  T $
an isometry on  $  {\mathcal H} $
satisfying  $  T _ {1} A = AT $. 
Then there exists an operator  $  B $
from  $  {\mathcal H} $
into  $  {\mathcal K} $
satisfying the following three conditions:  $  UB = BT $, 
$  \| B \| = \| A \| $
and  $  PB = A $, 
where  $  P $
is the orthogonal projection onto  $  {\mathcal H} _ {1} $.

The commutant lifting theorem was inspired by seminal work of D. Sarason [[#References|[a3]]] on  $  H  ^  \infty  $
interpolation. It can be used to solve many classical and modern  $  H  ^  \infty  $
interpolation problems, including the Carathéodory, Nevanlinna–Pick, Hermite–Féjer, Nudelman, Nehari, and Löwner interpolation problems in both their classical and tangential form (see [[#References|[a1]]] and also [[Carathéodory interpolation|Carathéodory interpolation]]; [[Nevanlinna–Pick interpolation|Nevanlinna–Pick interpolation]]). The commutant lifting theorem can also be used to solve problems in [[H^infinity-control-theory| $  H  ^  \infty  $
control theory]] and inverse scattering [[#References|[a1]]], [[#References|[a2]]].

There is a one-to-one correspondence between the set of all solutions in the commutant lifting theorem and a certain choice sequence of contractions. This choice sequence is a generalization of the Schur numbers used to solve the Carathéodory interpolation problem or the reflection coefficients appearing in inverse scattering problems for layered media in geophysics. There is also a one-to-one correspondence between the sets of all solutions for the commutant lifting theorem and a certain contractive analytic function in the open unit disc. This characterization of all solutions has several different network interpretations [[#References|[a1]]].

As an illustration of the commutant lifting theorem, consider the Nehari interpolation problem

$$ 
d _  \infty  =  \inf  \left \{ {\left \| {f - h } \right \| _  \infty  } : {h \in H  ^  \infty  } \right \} ,
$$

where  $  f $
is a given function in  $  L  ^  \infty  $. 
Here,  $  L  ^  \infty  $
is the Banach space of all Lebesgue-measurable functions  $  g $
on the unit circle whose norm  $  \| g \| _  \infty  = { \mathop{\rm ess}  \sup } \{ {| {g ( e ^ {it } ) } | } : {0 \leq  t < 2 \pi } \} $
is finite, and  $  H  ^  \infty  $
is the subspace of  $  L  ^  \infty  $
consisting of all functions  $  g $
in  $  L  ^  \infty  $
whose Fourier coefficients at  $  e ^ {int } $
are zero for all  $  n < 0 $. 
Likewise,  $  L  ^ {2} $
is the Hilbert space of all Lebesgue-measurable, square-integrable functions on the unit circle, and  $  H  ^ {2} $
is the subspace of  $  L  ^ {2} $
consisting of all functions in  $  L  ^ {2} $
whose Fourier coefficients at  $  e ^ {int } $
vanish for all  $  n < 0 $. 
Now, let  $  A $
be the Hankel operator from  $  {\mathcal H} = H  ^ {2} $
into  $  {\mathcal H} _ {1} = L  ^ {2} \omn H  ^ {2} $
defined by  $  Ax = Pfx $
for  $  x $
in  $  H  ^ {2} $. 
Let  $  T $
be the isometry on  $  H  ^ {2} $
and  $  U $
the unitary operator on  $  {\mathcal K} = L  ^ {2} $
defined by  $  Tx = e ^ {it } x $
and  $  Uy = e ^ {it } y $, 
respectively. Let  $  T _ {1} $
be the contraction on  $  {\mathcal H} _ {1} $
defined by  $  T _ {1} h _ {1} = PUh _ {1} $
for  $  h _ {1} $
in  $  {\mathcal H} _ {1} $. 
Since  $  T _ {1}  ^ {*} = U  ^ {*} \mid  {\mathcal H} _ {1} $, 
it follows that  $  U $
is an isometric lifting of  $  T _ {1} $. 
By applying the commutant lifting theorem, there exists an operator  $  B $
from  $  H  ^ {2} $
into  $  L  ^ {2} $
satisfying  $  UB = BT $, 
$  \| B \| = \| A \| $
and  $  PB = A $. 
Therefore, the error  $  d _  \infty  = \| A \| $, 
and there exists an  $  h \in H  ^  \infty  $
such that  $  d _  \infty  = \| {f - h } \| _  \infty  $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Foias,   A.E. Frazho,   "The commutant lifting approach to interpolation problems" , ''Operator Theory: Advances and Applications'' , '''44''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Foias,   H. Özbay,   A. Tannenbaum,   "Robust control of infinite-dimensional systems" , Springer  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Sarason,   "Generalized interpolation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110320/c11032080.png" />"  ''Trans. Amer. Math. Soc.'' , '''127'''  (1967)  pp. 179–203</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Sz.-Nagy,   C. Foias,   "Dilatation des commutants d'opérateurs"  ''C.R. Acad. Sci. Paris'' , '''A266'''  (1968)  pp. 493–495</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Sz.-Nagy,   C. Foias,   "Harmonic analysis of operators on Hilbert space" , North-Holland  (1970)</TD></TR></table>