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Let  $  A _ {0} $
and  $  A _ {1} $
be two Banach spaces (cf. [[Banach space|Banach space]]) embedded in a Hausdorff [[Topological vector space|topological vector space]]. Such a pair of spaces is termed a Banach couple or Banach pair. The theory of [[Interpolation of operators|interpolation of operators]] provides a variety of interpolation methods or interpolation functors for generating interpolation spaces with respect to any such couple  $  ( A _ {0} ,A _ {1} ) $, 
namely normed spaces  $  A $(
cf. [[Normed space|Normed space]]) having the property that every [[Linear operator|linear operator]]  $  T : {A _ {0} + A _ {1} } \rightarrow {A _ {0} + A _ {1} } $
such that  $  T : {A _ {j} } \rightarrow {A _ {j} } $
boundedly for  $  j = 0,1 $
also maps  $  A $
to  $  A $
boundedly.

A fundamental problem in interpolation theory is the description of all interpolation spaces with respect to a given Banach pair  $  ( A _ {0} ,A _ {1} ) $. 
In the 1960s, A.P. Calderón [[#References|[a4]]] and B.S. Mityagin [[#References|[a10]]] independently gave characterizations of all interpolation spaces  $  A $
with respect to the particular couple  $  ( A _ {0} ,A _ {1} ) = ( L _ {1} , L _  \infty  ) $. 
Calderón showed that  $  A $
is an interpolation space if and only if it has the following monotonicity property: For every element  $  a \in A $
and every element  $  b \in A _ {0} + A _ {1} $, 
whenever  $  K ( t,b ) \leq  K ( t,a ) $
for all  $  t > 0 $, 
it follows that  $  b \in A $
and  $  \| b \| _ {A} \leq  C \| a \| _ {A} $
for some absolute constant  $  C $.

Here,  $  K ( t,f ) = K ( t,f;A _ {0} ,A _ {1} ) $
denotes the Peetre  $  K $-
functional of  $  f $
with respect to the couple  $  ( A _ {0} ,A _ {1} ) $. 
In this particular case, where the couple is  $  ( L _ {1} ,L _  \infty  ) $, 
there is a concrete formula for  $  K ( t,x ) $(
cf. [[Interpolation of operators|Interpolation of operators]] for further details).

Mityagin's result, though of course ultimately equivalent to Calderón's, is formulated differently, in terms of the effect of measure-preserving transformations and multiplication by unimodular functions on elements of  $  A $.

The work of Calderón and Mityagin triggered a long series of papers by many mathematicians (many of these are listed in [[#References|[a2]]] and in [[#References|[a5]]]) in which it was shown that all the interpolation spaces of many other Banach pairs  $  ( A _ {0} ,A _ {1} ) $
can also be characterized via the Peetre  $  K $-
functionals for those pairs, by a monotonicity condition exactly analogous to the one in Calderón's result above. The Banach pairs  $  ( A _ {0} ,A _ {1} ) $
for which such a characterization holds are often referred to as Calderón couples or Calderón pairs. (They are also sometimes referred to using other terminology, such as Calderón–Mityagin couples,  $  K $-
monotone couples or  $  {\mathcal C} $-
pairs.)

It is also convenient to use the terminology  $  K $-
space for any normed space  $  A $
satisfying  $  A _ {0} \cap A _ {1} \subset  A \subset  A _ {0} + A _ {1} $
as well as the above-mentioned monotonicity property with respect to the  $  K $-
functional for  $  ( A _ {0} ,A _ {1} ) $. 
By the important  $  K $-
divisibility theorem of Yu.A Brudnyi and N.Ya. Kruglyak [[#References|[a2]]], it follows that each such  $  K $-
space necessarily coincides, to within equivalence of norms, with a space of the special form  $  ( A _ {0} ,A _ {1} ) _ {G}  ^ {K} $(
as defined in [[Interpolation of operators|Interpolation of operators]]). Thus, for Calderón pairs, all the interpolation spaces are of this relatively simple form.

So, one can remark that, roughly speaking, for a Banach pair  $  ( A _ {0} ,A _ {1} ) $
to be Calderón, the class of its interpolation spaces has to be relatively small, and correspondingly, the family of linear operators which are bounded on both  $  A _ {0} $
and  $  A _ {1} $
has to be relatively large.

Those Banach pairs which are known to be Calderón include pairs  $  ( L _ {p _ {0}  } ( w _ {0} ) ,L _ {p _ {1}  } ( w _ {1} ) ) $
of weighted  $  L _ {p} $
spaces for all choices of weight functions and for all exponents  $  p _ {0} , p _ {1} \in [ 1, \infty ] $(
the Sparr theorem, [[#References|[a12]]]). Other examples include all Banach pairs of Hilbert spaces, various pairs of Hardy spaces, or of Lorentz or Marcinkiewicz spaces and all  "iterated"  pairs of the form

$$ \tag{a1 }
( A _ {0} ,A _ {1} ) = \left ( ( B _ {0} ,B _ {1} ) _ {\theta _ {0}  ,q _ {0} }  ^ {K} , ( B _ {0} ,B _ {1} ) _ {\theta _ {1}  , q _ {1} }  ^ {K} \right ) .
$$

In this last example  $  ( B _ {0} ,B _ {1} ) $
can be taken to be an arbitrary Banach pair and  $  \theta _ {j} \in ( 0,1 ) $
and  $  q _ {j} \in [ 1, \infty ] $
can be arbitrary numbers. Here,  $  ( B _ {0} ,B _ {1} ) _ {\theta,q }   ^ {K} $
denotes the Lions–Peetre real-method interpolation space, consisting of all elements  $  b \in B _ {0} + B _ {1} $
for which the norm

$$ 
\left \| b \right \| = \left \{ \int\limits _ { 0 } ^  \infty   {( t ^ {- \theta } K ( t,b;B _ {0} ,B _ {1} ) )  ^ {q} }  { {
\frac{dt }{t}
 } } \right \} ^ { {1 / q } }
$$

is finite.

By choosing particular pairs  $  ( B _ {0} ,B _ {1} ) $
one obtains, as special cases of this last result, that various pairs of Besov spaces (cf. [[Imbedding theorems|Imbedding theorems]]) or Lorentz  $  L _ {p,q }  $
spaces or Schatten operator ideals are all Calderón.

In parallel with all these positive results it has also been shown that many Banach pairs fail to be Calderón. These include  $  ( L _ {p} ( \mathbf R  ^ {n} ) , W _ {p}  ^ {1} ( \mathbf R  ^ {n} ) ) $
where  $  p \neq 2 $(
here  $  W _ {p}  ^ {1} $
is a [[Sobolev space|Sobolev space]]) and  $  ( C ( [ 0,1 ] ) , { \mathop{\rm Lip} } ( [ 0,1 ] ) ) $
and also such simple pairs as  $  ( {\mathcal l} _ {1} \oplus {\mathcal l} _ {2} , {\mathcal l} _  \infty  \oplus {\mathcal l} _  \infty  ) $
and  $  ( L _ {1} + L _  \infty  ,L _ {1} \cap L _  \infty  ) $.

In [[#References|[a3]]], Brudnyi and A. Shteinberg consider whether pairs of the form  $  ( F _ {0} ( B _ {0} ,B _ {1} ) ,F _ {1} ( B _ {0} ,B _ {1} ) ) $
are Calderón, where  $  F _ {0} $
and  $  F _ {1} $
are interpolation functors (cf. [[Interpolation of operators|Interpolation of operators]]). Their results for the pair  $  ( B _ {0} ,B _ {1} ) = ( C ( [ 0,1 ] ) , { \mathop{\rm Lip} } ( [ 0,1 ] ) ) $
lead them to conjecture that the above-mentioned result about iterated pairs of the form (a1) cannot be extended, i.e., that  $  ( F _ {0} ( B _ {0} ,B _ {1} ) ,F _ {1} ( B _ {0} ,B _ {1} ) ) $
is Calderón for every Banach pair  $  ( B _ {0} ,B _ {1} ) $
if and only if both functors  $  F _ {j} $
are of the form  $  F _ {j} ( B _ {0} ,B _ {1} ) = ( B _ {0} ,B _ {1} ) _ {\theta _ {j}  ,q _ {j} }  ^ {K} $. 
As they also remark, all Calderón pairs which have so far (1996) been identified are either couples of Banach lattices of measurable functions on a given measure space, or are obtained from such lattice couples as partial retracts or  $  K $-
subcouples. One can ask whether this might in fact be true for all Calderón pairs.

N.J. Kalton [[#References|[a7]]] has given very extensive results about pairs of rearrangement-invariant spaces which are or are not Calderón, including a characterization of all rearrangement-invariant spaces  $  X $
for which  $  ( X,L _  \infty  ) $
is Calderón. Kalton's results, and also the following general negative result from [[#References|[a5]]], suggest that in some sense the Calderón property is very much linked to the spaces of the pair having some sort of  $  L _ {p} $
structure or  "near-Lp"  structure. This result also shows that Sparr's theorem for weighted  $  L _ {p} $
spaces cannot be sharpened: Let  $  {( X _ {0} ,X _ {1} ) } $
be a pair of saturated  $  \sigma $-
order continuous Banach lattices with the Fatou property on the non-atomic measure space  $  ( \Omega, \Sigma, \mu ) $. 
Suppose that at least one of the spaces  $  X _ {0} $
and  $  X _ {1} $
does not coincide, to within equivalence of norms, with a weighted  $  L  ^ {p} $
space on  $  \Omega $. 
Then there exist weight functions  $  {w _ {j} } : \Omega \rightarrow {( 0, \infty ) } $
for  $  j = 0,1 $
such that the weighted Banach pair  $  ( X _ {0} ( w _ {0} ) ,X _ {1} ( w _ {1} ) ) $
is not Calderón.

In most known examples of Banach pairs  $  ( A _ {0} ,A _ {1} ) $
which are not Calderón, this happens because the complex interpolation spaces  $  [ A _ {0} ,A _ {1} ] _  \alpha  $(
see [[Interpolation of operators|Interpolation of operators]]) are not  $  K $-
spaces. But M. Mastyło and V.I. Ovchinnikov have found examples (see [[#References|[a9]]]) of non-Calderón couples for which all the spaces  $  [ A _ {0} ,A _ {1} ] _  \alpha  $
are  $  K $-
spaces.

The notion of Calderón couples can also be considered in the wider context of operators  $  T $
mapping from the spaces of one Banach pair  $  ( A _ {0} ,A _ {1} ) $
to a possibly different Banach pair  $  ( B _ {0} ,B _ {1} ) $. 
In such a context one says that  $  A $
and  $  B $
are relative interpolation spaces if every linear mapping  $  T : {A _ {0} + A _ {1} } \rightarrow {B _ {0} + B _ {1} } $
which maps  $  A _ {j} $
boundedly into  $  B _ {j} $
for  $  j = 0,1 $
also maps  $  A $
boundedly into  $  B $. 
(In the notation of [[Interpolation of operators|Interpolation of operators]],  $  \{ A _ {0} ,A _ {1} ,A \} $
is an interpolation triple relative to  $  \{ B _ {0} ,B _ {1} ,B \} $.) 
One says that  $  A $
and  $  B $
are relative  $  K $-
spaces if, for all  $  a \in A $
and  $  b \in B _ {0} + B _ {1} $, 
the  $  K $-
functional inequality

$$ 
K ( t,b;B _ {0} ,B _ {1} ) \leq  K ( t,a;A _ {0} ,A _ {1} )  \textrm{ for  all  }  t > 0
$$

implies that  $  b \in B $
with  $  \| b \| _ {B} \leq  C \| a \| _ {A} $.

$  ( A _ {0} ,A _ {1} ) $
and  $  ( B _ {0} ,B _ {1} ) $
are said to be relative Calderón couples if  $  A $
and  $  B $
are relative interpolation spaces if and only if they are relative  $  K $-
spaces. J. Peetre has shown (see [[#References|[a6]]]) that if  $  ( B _ {0} ,B _ {1} ) $
is any pair of weighted  $  L _  \infty  $
spaces, then  $  ( A _ {0} ,A _ {1} ) $
and  $  ( B _ {0} ,B _ {1} ) $
are relative Calderón couples for all Banach pairs  $  ( A _ {0} ,A _ {1} ) $. 
Dually, if  $  ( A _ {0} ,A _ {1} ) $
is an arbitrary pair of weighted  $  L _ {1} $
spaces, then  $  ( A _ {0} ,A _ {1} ) $
and  $  ( B _ {0} ,B _ {1} ) $
are relative Calderón couples for all Banach pairs  $  ( B _ {0} ,B _ {1} ) $
satisfying a mild  "closure"  condition. This latter result is another consequence of the Brudnyi–Kruglyak  $  K $-
divisibility theorem.

Finally, given that there are so many cases of couples whose interpolation spaces cannot be all characterized by a Calderón-style condition, one must also seek alternative ways to characterize interpolation spaces. See [[#References|[a11]]] and [[#References|[a8]]] for some special cases. (Cf. also [[#References|[a1]]].)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Arazy,   M. Cwikel,   "A new characterization of the interpolation spaces between $L^p$ and $L^q"  ''Math. Scand.'' , '''55'''  (1984)  pp. 253–270</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  Y.A. Brudnyi,   N.Ja. Krugljak,   "Real interpolation functors" , North-Holland  (1991)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  Y. Brudnyi,   A. Shteinberg,   "Calderón couples of Lipschitz spaces"  ''J. Funct. Anal.'' , '''131'''  (1995)  pp. 459–498</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  A.P. Calderón,   "Spaces between $L^1$ and $L^\infty$ and the theorem of Marcinkiewicz"  ''Studia Math.'' , '''26'''  (1966)  pp. 273–299</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Cwikel,   P. Nilsson,   "Interpolation of weighted Banach lattices" , ''Memoirs'' , Amer. Math. Soc.  (to appear)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Cwikel,   J. Peetre,   "Abstract $K$ and $J$ spaces"  ''J. Math. Pures Appl.'' , '''60'''  (1981)  pp. 1–50</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  N.J. Kalton,   "Calderón couples of re-arrangement invariant spaces"  ''Studia Math.'' , '''106'''  (1993)  pp. 233–277</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Maligranda,   V.I. Ovchinnikov,   "On interpolation between $L^1 + L^\infty$ and $L^1 \cap L^\infty$"  ''J. Funct. Anal.'' , '''107'''  (1992)  pp. 343–351</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Mastyło,   V.I. Ovchinnikov,   "On the relation between complex and real methods of interpolation"  ''Studia Math.''  (to appear)  (Preprint Report 056/1996, Dept. Math. Comput. Sci. Adam Mickiewicz Univ., Poznan, 1996)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  B.S. Mityagin,   "An interpolation theorem for modular spaces" , ''Proc. Conf. Interpolation Spaces and Allied Topics in Analysis, Lund, 1983'' , ''Lecture Notes in Mathematics'' , '''1070''' , Springer  (1984)  pp. 10–23  (In Russian)  ''Mat. Sbornik'' , '''66'''  (1965)  pp. 472–482</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  V.I. Ovchinnikov,   "On the description of interpolation orbits in couples of $L_p$ spaces when they are not described by the $K$-method. Interpolation spaces and related topics" , ''Israel Math. Conf. Proc. Bar Ilan University'' , '''5''' , Amer. Math. Soc.  (1992)  pp. 187–206</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  G. Sparr,   "Interpolation of weighted $L^p$ spaces"  ''Studia Math.'' , '''62'''  (1978)  pp. 229–271</TD></TR></table>