Calderón-Toeplitz operator

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