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An [[Integral operator|integral operator]] depending on two function parameters, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c1100201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c1100202.png" />, and defined by the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c1100203.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c1100204.png" /> is the inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c1100205.png" /> (the space of square-integrable functions), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c1100206.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c1100207.png" />.
+
An [[Integral operator|integral operator]] depending on two function parameters, $  b $
 +
and  $  \psi $,  
 +
and defined by the formula
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c1100208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c1100209.png" /> satisfying the admissibility condition (i.e., for almost-every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002010.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002012.png" /> being the [[Fourier transform|Fourier transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002013.png" />), the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002014.png" /> becomes the identity. The formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002015.png" /> is known as the Calderón reproducing formula.
+
$$
 +
T _ {b, \psi }  ( f ) = \int\limits _ { 0 } ^  \infty  \int\limits _ {\mathbf R  ^ {n} } {b ( u,s ) \left \langle  {f, \psi _ {u,s }  } \right \rangle \psi _ {u,s }  }  {d \mu ( u,s ) } ,
 +
$$
  
The name "Calderón–Toeplitz operator" comes from the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002016.png" /> (for admissible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002017.png" />) is unitarily equivalent to the Toeplitz-type operator
+
where $ \langle  {\cdot, \cdot } \rangle $
 +
is the inner product in  $  L  ^ {2} ( \mathbf R  ^ {d} ) $(
 +
the space of square-integrable functions),  $  d \mu ( u,s ) = s ^ {- ( d + 1 ) }  du  ds $,
 +
and  $  \psi _ {u,s }  ( x ) = s ^ {- d/2 } \psi ( { {( x - u ) } / s } ) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002018.png" /></td> </tr></table>
+
For  $  b \equiv 1 $
 +
and  $  \psi \in L  ^ {2} ( \mathbf R  ^ {d} ) $
 +
satisfying the admissibility condition (i.e., for almost-every  $  \xi \in \mathbf R  ^ {d} $
 +
one has  $  \int _ {0}  ^  \infty  {| { {\widehat \psi  } ( s \xi ) } |  ^ {2} }  { {{ds } / s } } = 1 $,
 +
$  {\widehat \psi  } $
 +
being the [[Fourier transform|Fourier transform]] of  $  \psi $),
 +
the operator  $  T _ {b, \psi }  $
 +
becomes the identity. The formula  $  T _ {1, \psi }  ( f ) = f $
 +
is known as the Calderón reproducing formula.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002019.png" /> denotes the operator of multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002021.png" /> is the orthogonal projection from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002022.png" /> onto its closed subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002023.png" />, called the space of Calderón transforms.
+
The name  "Calderón–Toeplitz operator" comes from the fact that  $  T _ {b, \psi }  $(
 +
for admissible  $  \psi $)
 +
is unitarily equivalent to the Toeplitz-type operator
 +
 
 +
$$
 +
{P _  \psi  M _ {b} } : {W _  \psi  } \rightarrow {W _  \psi  } ,
 +
$$
 +
 
 +
where  $  M _ {b} $
 +
denotes the operator of multiplication by $  b $
 +
and $  P _  \psi  $
 +
is the orthogonal projection from $  L  ^ {2} ( \mathbf R  ^ {n} \times ( 0, \infty ) , d \mu ) $
 +
onto its closed subspace $  W _  \psi  = \{ {\langle  {f, \psi _ {u,s }  } \rangle } : {f \in L  ^ {2} ( \mathbf R  ^ {d} ) } \} $,  
 +
called the space of Calderón transforms.
  
 
Calderón–Toeplitz operators were introduced by R. Rochberg in [[#References|[a4]]] as a wavelet counterpart of Toeplitz operators defined on Hilbert spaces of holomorphic functions. They are the model operators that fit nicely in the context of wavelet decomposition of function spaces and almost diagonalization of operators (cf. also [[Wavelet analysis|Wavelet analysis]]). They also are an effective time-frequency localization tool [[#References|[a1]]].
 
Calderón–Toeplitz operators were introduced by R. Rochberg in [[#References|[a4]]] as a wavelet counterpart of Toeplitz operators defined on Hilbert spaces of holomorphic functions. They are the model operators that fit nicely in the context of wavelet decomposition of function spaces and almost diagonalization of operators (cf. also [[Wavelet analysis|Wavelet analysis]]). They also are an effective time-frequency localization tool [[#References|[a1]]].
  
Properties of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002024.png" /> for fixed, sufficiently smooth, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002025.png" /> are:
+
Properties of the mapping $  b \mapsto T _ {b, \psi }  $
 +
for fixed, sufficiently smooth, $  \psi $
 +
are:
  
1) (correspondence principle, [[#References|[a5]]]). Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002026.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002027.png" /> is bounded, self-adjoint and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002028.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002029.png" /> be the spectral projection associated with the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002030.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002031.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002032.png" /> so that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002034.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002035.png" /> disjoint hyperbolic balls of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002036.png" />, then the dimension of the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002037.png" /> is at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002038.png" />.
+
1) (correspondence principle, [[#References|[a5]]]). Suppose that 0 \leq  b \leq  1 $.  
 +
Then $  T _ {b, \psi }  $
 +
is bounded, self-adjoint and 0 \leq  T _ {b, \psi }  \leq  1 $.  
 +
Let $  P _ {\lambda, \epsilon }  $
 +
be the spectral projection associated with the interval $  ( \lambda - \epsilon, \lambda + \epsilon ) $.  
 +
For any $  \epsilon > 0 $
 +
there is an $  R = R ( \epsilon ) $
 +
so that if $  \lambda \in [ 0,1 ] $
 +
and $  b \equiv \lambda $
 +
on $  N $
 +
disjoint hyperbolic balls of radius $  R $,  
 +
then the dimension of the range of $  P _ {\lambda, \epsilon }  $
 +
is at least $  N $.
  
2) ([[#References|[a2]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002040.png" />.
+
2) ([[#References|[a2]]]). Let $  b \geq  0 $
 +
and $  {\widetilde{b}  } ( u,s ) = \langle  {T _ {b, \psi }  \psi _ {u,s }  , \psi _ {u,s }  } \rangle $.
  
i) (boundedness). The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002041.png" /> is bounded if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002042.png" /> is bounded.
+
i) (boundedness). The operator $  T _ {b, \psi }  $
 +
is bounded if and only if $  {\widetilde{b}  } $
 +
is bounded.
  
ii) (compactness). The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002043.png" /> is compact if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002044.png" /> at infinity.
+
ii) (compactness). The operator $  T _ {b, \psi }  $
 +
is compact if and only if $  {\widetilde{b}  } \rightarrow 0 $
 +
at infinity.
  
iii) (Schatten ideal behaviour). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002045.png" /> is compact, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002046.png" />,
+
iii) (Schatten ideal behaviour). If $  T _ {b, \psi }  $
 +
is compact, then for $  p > 0 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002047.png" /></td> </tr></table>
+
$$
 +
\sum _ { n } \left | {s _ {n} ( T _ {b, \psi }  ) } \right |  ^ {p} \cong \int\limits _ { 0 } ^  \infty  \int\limits _ {\mathbf R  ^ {d} } {\left | { {\widetilde{b}  } ( u,s ) } \right |  ^ {p} }  {d \mu ( u,s ) } ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002048.png" /></td> </tr></table>
+
$$
 +
s _ {n} ( T _ {b, \psi }  ) = \inf  \left \{ {\left \| {T _ {b, \psi }  - A _ {n} } \right \| } : {A _ {n}  n \textrm{  \AAh  dimensional  } } \right \}
 +
$$
  
and the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002049.png" /> means that the quotient is bounded above and below with constants independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002050.png" />.
+
and the symbol $  \cong $
 +
means that the quotient is bounded above and below with constants independent of $  b $.
  
The eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002051.png" /> can be estimated as follows ([[#References|[a6]]], [[#References|[a3]]]).
+
The eigenvalues of $  T _ {b, \psi }  $
 +
can be estimated as follows ([[#References|[a6]]], [[#References|[a3]]]).
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002053.png" /> have compact support, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002054.png" /> being smooth with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002056.png" /> both non-negative, and suppose that the support of the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002057.png" /> is contained in a cube of side length one. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002058.png" /> be the non-increasing rearrangement (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002059.png" />) of the sequence
+
Suppose that $  b $,  
 +
$  \psi $
 +
have compact support, $  b ( u,s ) = b _ {1} ( u ) b _ {2} ( s ) $
 +
being smooth with $  b _ {1} $,  
 +
$  b _ {2} $
 +
both non-negative, and suppose that the support of the kernel of $  T _ {b, \psi }  $
 +
is contained in a cube of side length one. Let $  a _ {n}  ^ {M} $
 +
be the non-increasing rearrangement (with respect to $  m $)  
 +
of the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002060.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty  {b _ {2} ( s ) \left | { {\widehat \psi  } ( sMm ) } \right |  ^ {2} }  { {
 +
\frac{ds }{s}
 +
} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002061.png" /> is a natural number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002062.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002063.png" /> are integers. There are positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002065.png" /> and a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002066.png" /> such that
+
where $  M $
 +
is a natural number, $  m = ( m _ {1} \dots m _ {d} ) $,  
 +
and $  m _ {1} \dots m _ {d} $
 +
are integers. There are positive constants c $,  
 +
$  C $
 +
and a natural number $  M $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002067.png" /></td> </tr></table>
+
$$
 +
c a _ {n}  ^ {M} \leq  s _ {n} ( T _ {b, \psi }  ) \leq  C a _ {n}  ^ {1} .
 +
$$
  
In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002068.png" /> the eigenvalues satisfy two-sided estimates: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110020/c11002069.png" />.
+
In particular, for $  \psi = \chi _ {( 0,1 ) }  - \chi _ {( - 1,0 ) }  $
 +
the eigenvalues satisfy two-sided estimates: $  s _ {n} \cong n ^ {- 2 } $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Daubechies,  "Ten lectures on wavelets" , ''CBMS-NSF Regional Conf. Ser.'' , '''6''' , SIAM  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Nowak,  "On Calderón–Toeplitz operators"  ''Monatsh. Math.'' , '''116'''  (1993)  pp. 49–72</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Nowak,  "Some eigenvalue estimates for wavelet related Toeplitz operators"  ''Colloq. Math.'' , '''LXV'''  (1993)  pp. 149–156</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Rochberg,  "Toeplitz and Hankel operators, wavelets, NWO sequences and almost diagonalization of operators"  W.B. Arveson (ed.)  R.G. Douglas (ed.) , ''Proc. Symp. Pure Math.'' , '''51''' , Amer. Math. Soc.  (1990)  pp. 425–444</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Rochberg,  "A correspondence principle for Toeplitz and Calderón–Toeplitz operators"  M. Cwikel (ed.)  etAAsal. (ed.) , ''Israel Math. Conf. Proc.'' , '''5'''  (1992)  pp. 229–243</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Rochberg,  "Eigenvalue estimates for Calderón–Toeplitz operators"  K. Jarosz (ed.) , ''Lecture Notes in Pure and Appl. Math.'' , '''136''' , M. Dekker  (1992)  pp. 345–357</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Daubechies,  "Ten lectures on wavelets" , ''CBMS-NSF Regional Conf. Ser.'' , '''6''' , SIAM  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Nowak,  "On Calderón–Toeplitz operators"  ''Monatsh. Math.'' , '''116'''  (1993)  pp. 49–72</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Nowak,  "Some eigenvalue estimates for wavelet related Toeplitz operators"  ''Colloq. Math.'' , '''LXV'''  (1993)  pp. 149–156</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Rochberg,  "Toeplitz and Hankel operators, wavelets, NWO sequences and almost diagonalization of operators"  W.B. Arveson (ed.)  R.G. Douglas (ed.) , ''Proc. Symp. Pure Math.'' , '''51''' , Amer. Math. Soc.  (1990)  pp. 425–444</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Rochberg,  "A correspondence principle for Toeplitz and Calderón–Toeplitz operators"  M. Cwikel (ed.)  etAAsal. (ed.) , ''Israel Math. Conf. Proc.'' , '''5'''  (1992)  pp. 229–243</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Rochberg,  "Eigenvalue estimates for Calderón–Toeplitz operators"  K. Jarosz (ed.) , ''Lecture Notes in Pure and Appl. Math.'' , '''136''' , M. Dekker  (1992)  pp. 345–357</TD></TR></table>

Latest revision as of 06:29, 30 May 2020


An integral operator depending on two function parameters, $ b $ and $ \psi $, and defined by the formula

$$ T _ {b, \psi } ( f ) = \int\limits _ { 0 } ^ \infty \int\limits _ {\mathbf R ^ {n} } {b ( u,s ) \left \langle {f, \psi _ {u,s } } \right \rangle \psi _ {u,s } } {d \mu ( u,s ) } , $$

where $ \langle {\cdot, \cdot } \rangle $ is the inner product in $ L ^ {2} ( \mathbf R ^ {d} ) $( the space of square-integrable functions), $ d \mu ( u,s ) = s ^ {- ( d + 1 ) } du ds $, and $ \psi _ {u,s } ( x ) = s ^ {- d/2 } \psi ( { {( x - u ) } / s } ) $.

For $ b \equiv 1 $ and $ \psi \in L ^ {2} ( \mathbf R ^ {d} ) $ satisfying the admissibility condition (i.e., for almost-every $ \xi \in \mathbf R ^ {d} $ one has $ \int _ {0} ^ \infty {| { {\widehat \psi } ( s \xi ) } | ^ {2} } { {{ds } / s } } = 1 $, $ {\widehat \psi } $ being the Fourier transform of $ \psi $), the operator $ T _ {b, \psi } $ becomes the identity. The formula $ T _ {1, \psi } ( f ) = f $ is known as the Calderón reproducing formula.

The name "Calderón–Toeplitz operator" comes from the fact that $ T _ {b, \psi } $( for admissible $ \psi $) is unitarily equivalent to the Toeplitz-type operator

$$ {P _ \psi M _ {b} } : {W _ \psi } \rightarrow {W _ \psi } , $$

where $ M _ {b} $ denotes the operator of multiplication by $ b $ and $ P _ \psi $ is the orthogonal projection from $ L ^ {2} ( \mathbf R ^ {n} \times ( 0, \infty ) , d \mu ) $ onto its closed subspace $ W _ \psi = \{ {\langle {f, \psi _ {u,s } } \rangle } : {f \in L ^ {2} ( \mathbf R ^ {d} ) } \} $, called the space of Calderón transforms.

Calderón–Toeplitz operators were introduced by R. Rochberg in [a4] as a wavelet counterpart of Toeplitz operators defined on Hilbert spaces of holomorphic functions. They are the model operators that fit nicely in the context of wavelet decomposition of function spaces and almost diagonalization of operators (cf. also Wavelet analysis). They also are an effective time-frequency localization tool [a1].

Properties of the mapping $ b \mapsto T _ {b, \psi } $ for fixed, sufficiently smooth, $ \psi $ are:

1) (correspondence principle, [a5]). Suppose that $ 0 \leq b \leq 1 $. Then $ T _ {b, \psi } $ is bounded, self-adjoint and $ 0 \leq T _ {b, \psi } \leq 1 $. Let $ P _ {\lambda, \epsilon } $ be the spectral projection associated with the interval $ ( \lambda - \epsilon, \lambda + \epsilon ) $. For any $ \epsilon > 0 $ there is an $ R = R ( \epsilon ) $ so that if $ \lambda \in [ 0,1 ] $ and $ b \equiv \lambda $ on $ N $ disjoint hyperbolic balls of radius $ R $, then the dimension of the range of $ P _ {\lambda, \epsilon } $ is at least $ N $.

2) ([a2]). Let $ b \geq 0 $ and $ {\widetilde{b} } ( u,s ) = \langle {T _ {b, \psi } \psi _ {u,s } , \psi _ {u,s } } \rangle $.

i) (boundedness). The operator $ T _ {b, \psi } $ is bounded if and only if $ {\widetilde{b} } $ is bounded.

ii) (compactness). The operator $ T _ {b, \psi } $ is compact if and only if $ {\widetilde{b} } \rightarrow 0 $ at infinity.

iii) (Schatten ideal behaviour). If $ T _ {b, \psi } $ is compact, then for $ p > 0 $,

$$ \sum _ { n } \left | {s _ {n} ( T _ {b, \psi } ) } \right | ^ {p} \cong \int\limits _ { 0 } ^ \infty \int\limits _ {\mathbf R ^ {d} } {\left | { {\widetilde{b} } ( u,s ) } \right | ^ {p} } {d \mu ( u,s ) } , $$

where

$$ s _ {n} ( T _ {b, \psi } ) = \inf \left \{ {\left \| {T _ {b, \psi } - A _ {n} } \right \| } : {A _ {n} n \textrm{ \AAh dimensional } } \right \} $$

and the symbol $ \cong $ means that the quotient is bounded above and below with constants independent of $ b $.

The eigenvalues of $ T _ {b, \psi } $ can be estimated as follows ([a6], [a3]).

Suppose that $ b $, $ \psi $ have compact support, $ b ( u,s ) = b _ {1} ( u ) b _ {2} ( s ) $ being smooth with $ b _ {1} $, $ b _ {2} $ both non-negative, and suppose that the support of the kernel of $ T _ {b, \psi } $ is contained in a cube of side length one. Let $ a _ {n} ^ {M} $ be the non-increasing rearrangement (with respect to $ m $) of the sequence

$$ \int\limits _ { 0 } ^ \infty {b _ {2} ( s ) \left | { {\widehat \psi } ( sMm ) } \right | ^ {2} } { { \frac{ds }{s} } } , $$

where $ M $ is a natural number, $ m = ( m _ {1} \dots m _ {d} ) $, and $ m _ {1} \dots m _ {d} $ are integers. There are positive constants $ c $, $ C $ and a natural number $ M $ such that

$$ c a _ {n} ^ {M} \leq s _ {n} ( T _ {b, \psi } ) \leq C a _ {n} ^ {1} . $$

In particular, for $ \psi = \chi _ {( 0,1 ) } - \chi _ {( - 1,0 ) } $ the eigenvalues satisfy two-sided estimates: $ s _ {n} \cong n ^ {- 2 } $.

References

[a1] I. Daubechies, "Ten lectures on wavelets" , CBMS-NSF Regional Conf. Ser. , 6 , SIAM (1992)
[a2] K. Nowak, "On Calderón–Toeplitz operators" Monatsh. Math. , 116 (1993) pp. 49–72
[a3] K. Nowak, "Some eigenvalue estimates for wavelet related Toeplitz operators" Colloq. Math. , LXV (1993) pp. 149–156
[a4] R. Rochberg, "Toeplitz and Hankel operators, wavelets, NWO sequences and almost diagonalization of operators" W.B. Arveson (ed.) R.G. Douglas (ed.) , Proc. Symp. Pure Math. , 51 , Amer. Math. Soc. (1990) pp. 425–444
[a5] R. Rochberg, "A correspondence principle for Toeplitz and Calderón–Toeplitz operators" M. Cwikel (ed.) etAAsal. (ed.) , Israel Math. Conf. Proc. , 5 (1992) pp. 229–243
[a6] R. Rochberg, "Eigenvalue estimates for Calderón–Toeplitz operators" K. Jarosz (ed.) , Lecture Notes in Pure and Appl. Math. , 136 , M. Dekker (1992) pp. 345–357
How to Cite This Entry:
Calderón-Toeplitz operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Calder%C3%B3n-Toeplitz_operator&oldid=17003
This article was adapted from an original article by K. Nowak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article