Abstract. We give criteria for the membership of Hankel operators on the Hardy space on the disc in the Dixmier class, and establish estimates for their Dixmier trace. In contrast to the situation in the Bergman space setting, it turns out that there exist Dixmier-class Hankel operators which are not measurable (i.e. their Dixmier trace depends on the choice of the underlying Banach limit), as well as Dixmier-class Hankel operators which do not belong to the (1, ∞) Schatten-Lorentz ideal. A related question concerning logarithmic interpolation of Besov spaces is also discussed.
Abstract. Let G n,k (K) be the Grassmannian manifold of k-dimensional K-subspaces in K n where K = R, C, H is the field of real, complex or quaternionic numbers. Foras an integration over all ξ ⊂ η. When k + k ′ ≤ n we give an inversion formula in terms of the Gårding-Gindikin fractional integration and the Cayley type differential operator on the symmetric cone of positive k × k matrices over K. This generalizes the recent results of Grinberg-Rubin for real Grassmannians.
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