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Register a new userSchatten Class and nuclear pseudo-differential operators on homogeneous spaces of compact groups
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Given a compact (Hausdorff) group $G$ and a closed subgroup $H$ of $G,$ in this paper we present symbolic criteria for pseudo-differential operators on compact homogeneous space $G/H$ characterizing the Schatten-von Neumann classes $S_r(L^2(G/H))$ for all $0<r leq infty.$ We go on to provide a symbolic characterization for $r$-nuclear, $0< r leq 1,$ pseudo-differential operators on $L^{p}(G/H)$-space with applications to adjoint, product and trace formulae. The criteria here are given in terms of the concept of matrix-valued symbols defined on noncommutative analogue of phase space $G/H times widehat{G/H}.$ Finally, we present applications of aforementioned results in the context of heat kernels.
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We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<infty$ of the Hankel operators $H_ f$ and $H_{bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.
A full description of the membership in the Schatten ideal $S_ p(A^2_{omega})$ for $0<p<infty$ of the Toeplitz operator acting on large weighted Bergman spaces is obtained.
This paper is devoted to wavelet analysis on adele ring $bA$ and the theory of pseudo-differential operators. We develop the technique which gives the possibility to generalize finite-dimensional results of wavelet analysis to the case of adeles $bA$ by using infinite tensor products of Hilbert spaces. The adele ring is roughly speaking a subring of the direct product of all possible ($p$-adic and Archimedean) completions $bQ_p$ of the field of rational numbers $bQ$ with some conditions at infinity. Using our technique, we prove that $L^2(bA)=otimes_{e,pin{infty,2,3,5,...}}L^2({bQ}_{p})$ is the infinite tensor product of the spaces $L^2({bQ}_{p})$ with a stabilization $e=(e_p)_p$, where $e_p(x)=Omega(|x|_p)in L^2({bQ}_{p})$, and $Omega$ is a characteristic function of the unit interval $[0,,1]$, $bQ_p$ is the field of $p$-adic numbers, $p=2,3,5,...$; $bQ_infty=bR$. This description allows us to construct an infinite family of Haar wavelet bases on $L^2(bA)$ which can be obtained by shifts and multi-delations. The adelic multiresolution analysis (MRA) in $L^2(bA)$ is also constructed. In the framework of this MRA another infinite family of Haar wavelet bases is constructed. We introduce the adelic Lizorkin spaces of test functions and distributions and give the characterization of these spaces in terms of wavelet functions. One class of pseudo-differential operators (including the fractional operator) is studied on the Lizorkin spaces. A criterion for an adelic wavelet function to be an eigenfunction for a pseudo-differential operator is derived. We prove that any wavelet function is an eigenfunction of the fractional operator. These results allow one to create the necessary prerequisites for intensive using of adelic wavelet bases and pseudo-differential operators in applications.
A pair of functions defined on a set X with values in a vector space E is said to be disjoint if at least one of the functions takes the value 0 at every point in X. An operator acting between vector-valued function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. We characterize compact and weakly compact disjointness preserving operators between spaces of Banach space-valued differentiable functions.
A multiplicative Hankel operator is an operator with matrix representation $M(alpha) = {alpha(nm)}_{n,m=1}^infty$, where $alpha$ is the generating sequence of $M(alpha)$. Let $mathcal{M}$ and $mathcal{M}_0$ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator $M(alpha) in mathcal{M}$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(beta) in mathcal{M}_0$, $$|M(alpha) - N(beta)|_{mathcal{B}(ell^2(mathbb{N}))} = inf left {|M(alpha) - K |_{mathcal{B}(ell^2(mathbb{N}))} , : , K colon ell^2(mathbb{N}) to ell^2(mathbb{N}) textrm{ compact} right}.$$ Intimately connected with this result, it is then proven that the bidual of $mathcal{M}_0$ is isometrically isomorphic to $mathcal{M}$, $mathcal{M}_0^{ast ast} simeq mathcal{M}$. It follows that $mathcal{M}_0$ is an M-ideal in $mathcal{M}$. The dual space $mathcal{M}_0^ast$ is isometrically isomorphic to a projective tensor product with respect to Dirichlet convolution. The stated results are also valid for small Hankel operators on the Hardy space $H^2(mathbb{D}^d)$ of a finite polydisk.
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