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Nearly weakly orthonormal sequences, singular value estimates, and Calderon-Zygmund operators. (English) Zbl 0699.47012

The direct way to estimate the singular values of a compact operator is to decompose it as a sum of orthogonal rank one pieces. However, such decompositions can generally not be found in practice. We give a variation of the decomposition using nearly weakly orthonormal (NWO) sequences. The NWO decomposition is easier to do in examples but is still strong enough to give good singular value estimates. We illustrate this by giving sharp trace ideal estimates for the double layer potential and for the first and higher commutators of multiplication operators and Calderon-Zygmund operators. In particular, NWO sequences seem well suited for dealing with operators which depend nonlinearly on their symbol. We also show that the class of operators which admit NWO expansions is well behaved under composition, under conjugation by weights, and under conjugation by changes of variable.
Reviewer: T.Ando

MSC:

47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
47A10 Spectrum, resolvent
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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