Calderón-Toeplitz operator

An integral operator depending on two function parameters, and , and defined by the formula

where is the inner product in (the space of square-integrable functions), , and .

For and satisfying the admissibility condition (i.e., for almost-every one has , being the Fourier transform of ), the operator becomes the identity. The formula is known as the Calderón reproducing formula.

The name "Calderón–Toeplitz operator" comes from the fact that (for admissible ) is unitarily equivalent to the Toeplitz-type operator

where denotes the operator of multiplication by and is the orthogonal projection from onto its closed subspace , 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 for fixed, sufficiently smooth, are:

1) (correspondence principle, [a5]). Suppose that . Then is bounded, self-adjoint and . Let be the spectral projection associated with the interval . For any there is an so that if and on disjoint hyperbolic balls of radius , then the dimension of the range of is at least .

2) ([a2]). Let and .

i) (boundedness). The operator is bounded if and only if is bounded.

ii) (compactness). The operator is compact if and only if at infinity.

iii) (Schatten ideal behaviour). If is compact, then for ,

where

and the symbol means that the quotient is bounded above and below with constants independent of .

The eigenvalues of can be estimated as follows ([a6], [a3]).

Suppose that , have compact support, being smooth with , both non-negative, and suppose that the support of the kernel of is contained in a cube of side length one. Let be the non-increasing rearrangement (with respect to ) of the sequence

where is a natural number, , and are integers. There are positive constants , and a natural number such that

In particular, for the eigenvalues satisfy two-sided estimates: .

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