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Register a new userThe SHAI property for the operators on L^p
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A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for L^p (0, 1) for 1 < p < infty, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1 < p < infty.
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In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $phi$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_phi$ are orthogonal. When the order $n$ of $phi$ is $2$ or $3$, we show that $M_phi$ is reducible on $D$ if and only if $phi$ is equivalent to $z^n$. When the order of $phi$ is $4$, we determine the reducing subspaces for $M_phi$, and we see that in this case $M_phi$ can be reducible on $D$ when $phi$ is not equivalent to $z^4$. The same phenomenon happens when the order $n$ of $phi$ is not a prime number. Furthermore, we show that $M_phi$ is unitarily equivalent to $M_{z^n} (n > 1)$ on $D$ if and only if $phi = az^n$ for some unimodular constant $a$.
Let $T^n$ denote the n-dimensional torus. The class of the bounded operators on $L^2(T^n)$ with analytic orbit under the action of $T^n$ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on $T^n$ whose discrete symbol $(a_j)_{jin Z^n}$ is uniformly analytic, in the sense that there exists $C>1$ such that the derivatives of $a_j$ satisfy $|partial^alpha a_j(x)|leq C^{1+|alpha|}alpha!$ for all $xin T^n$, all $jin Z^n$ and all $alphain N^n$. This implies that this class of pseudodifferential operators is a spectrally invariant *-subalgebra of the algebra of all bounded operators on $L^2(T^n)$.
In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on $L^p$-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the $s$-nuclearity, $0<s leq 1,$ of periodic and discrete pseudo-differential operators. To accomplish this, we classify those $s$-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on $sigma$-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato-Ponce inequality and to examine the $s$-nuclearity of multilinear Bessel potentials as well as the $s$-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.
In this paper, we show that under a mild condition, a principal submodule of the Bergman module on a bounded strongly pseudoconvex domain with smooth boundary in $mathbb{C}^n$ is $p$-essentially normal for all $p>n$. This is a significant improvement of the results of the first author and K. Wang, where the same result is shown to hold for polynomial-generated principal submodules of the Bergman module on the unit ball $mathbb{B}_n$ of $mathbb{C}^n$. As a consequence of our main result, we prove that the submodule of $L_a^2(mathbb{B}_n)$ consisting of functions vanishing on a pure analytic subsets of codimension $1$ is $p$-essentially normal for all $p>n$.
Let $D^alpha, alpha>0$, be the Vladimirov-Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity $D^alpha D^{-alpha}f=f$ was known only for the case where $f$ has a compact support. Following a result by Samko about the fractional Laplacian of real analysis, we extend the above identity in terms of $L^p$-convergence of truncated integrals. Differences between real and non-Archimedean cases are discussed.
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