Toeplitz operators on Fock-Sobolev spaces with positive measure symbols

We discuss Toeplitz operators on Fock-Sobolev space with positive measure symbols.By FockCarleson measure,we obtain the characterizations for boundedness and compactness of Toeplitz operators.We also give some equivalent conditions of Schatten p-class properties of Toeplitz operators by Berezin transform.     

〈2, 1〉-Compact operators

We consider the class of the continuous L _2,1 linear operators in L ₂ that are sums of the operators of multiplication by bounded measurable functions and the operators sending the unit ball of L ₂ into a compact subset of L ₁. We prove that a functional equation with an operator from L _2,1 is equivalent to an integral equation with kernel satisfying the Carleman condition. We also prove that if T ∈ L _2,1 and VTV ^?1 ∈ L _2,1 for all unitary operators V in L ₂ then T = α1 + C, where α is a scalar, 1 is the identity operator in L ₂, and C is a compact operator in L ₂.     

单位球的Bergman空间上Schatten类加权复合算子

本文研究了单位球的Bergman空间上Schatten类加权复合算子,得到了这种加权复合算子属于Schatten-Von Neumann.理想S_p的几个充要条件.作为推论给出了Wφ,φ是一个Hilbert- Schmidt算子的充要条件是∫_(Bn)(|ψ(ω)|~2)/((1-|φ(ω)|~2)~(n 1)dV(ω)<∞..     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.     

Weighted composition operators on weighted Bergman spaces of bounded symmetric domains

In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator W _φψ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators.     

Schatten p-norm inequalities related to an extended operator parallelogram law

Let C_p be the Schatten p-class for p>0. Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: if A={A₁,A₂,…,A_n} and B={B₁,B₂,…,B_n} are two sets of operators in, then C₂

     

Paraproducts and Hankel operators of Schatten class via p-John-Nirenberg Theorem

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten-von Neumann class S_p, if and only if its symbol is in the dyadic Besov space B_p^d. Our main tools are a product formula for paraproducts and a “p-John-Nirenberg-Theorem” due to Rochberg and Semmes.We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols.Using an averaging technique by Petermichl, we retrieve Peller's characterizations of scalar and vector Hankel operators of Schatten-von Neumann class S_p for 1<p<∞. We then employ vector techniques to characterise little Hankel operators of Schatten-von Neumann class, answering a question by Bonami and Peloso.Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts.     

Inverse spectral problem for integro-differential operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ? which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L ₂[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ? and by two pairs of boundary conditions coinciding at one of the finite points.     

Orthogonality in Classes

 Let denote the von Neumann–Schatten class, its norm and let be an elementary operator defined by . We shall characterize those operators which are orthogonal to the range of in the sense that for all . The main results of this paper are: If and (i) if A, C, respectively B, D are commuting normal operators with , or (ii) if A, B are contractions and , then is orthogonal to the range of if and only of S is in the kernel of . Furthermore, in both cases, the algebraic direct sum satisfies .     

Boundedness of pseudodifferential operators on modulation spaces

We study classes of pseudodifferential operators which are bounded on large collections of modulation spaces. The conditions on the operators are stated in terms of the L^p,q estimates for the continuous Gabor transforms of their symbols. In particular, we show how these classes are related to the class of operators of Gröchenig and Heil, which is bounded on all modulation spaces.     

Weighted Hardy and singular operators in Morrey spaces

We study the weighted boundedness of the Cauchy singular integral operator SΓ in Morrey spaces Lp,λ(Γ) on curves satisfying the arc-chord condition, for a class of “radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces Lp,λ(0,?), ?>0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.     

Characterization of pseudodifferential operators in Hölder-Zygmund spaces

We consider pseudodifferential operators with symbols of the Hörmander class S _{1, δ} ^m , 0 ≤ δ < 1, in Hölder-Zygmund spaces on ? ⁿ and obtain a Beals-type characterization of such operators. By way of application, we show that the inverse of a pseudodifferential operator invertible in a Hölder-Zygmund space is itself a pseudodifferential operator, and hence, the spectra of a pseudodifferential operator in the space L ₂ and in the Hölder-Zygmund spaces coincide as sets.     

Time-Frequency analysis of localization operators

We study a class of pseudodifferential operators known as time-frequency localization operators, Anti-Wick operators, Gabor-Toeplitz operators or wave packets. Given a symbol a and two windows ?₁,?₂, we investigate the multilinear mapping from to the localization operator A_a^?₁,?₂ and we give sufficient and necessary conditions for A_a^?₁,?₂ to be bounded or to belong to a Schatten class. Our results are formulated in terms of time-frequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.     

Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds

In this Note, we present criteria on both symbols and integral kernels ensuring that the corresponding operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given. A notion of invariant operator and its full symbol associated with an elliptic operator are introduced. Some applications to the study of r  -nuclearity on L^p$L^p$ spaces are also obtained.     

Orthogonality in Classes

 Let denote the von Neumann–Schatten class, its norm and let be an elementary operator defined by . We shall characterize those operators which are orthogonal to the range of in the sense that for all . The main results of this paper are: If and (i) if A, C, respectively B, D are commuting normal operators with , or (ii) if A, B are contractions and , then is orthogonal to the range of if and only of S is in the kernel of . Furthermore, in both cases, the algebraic direct sum satisfies . (Received 9 February 2000; in revised form 21 February, 2001)     

On composition operators onH p -spaces in several variables

In this paper, we prove that a composition operator onH ^p (B) is Fredholm if and only if it is invertible if and only if its symbol is an automorphism onB, and give the representation of the spectra of a class of composition operators. In addition, using composition operator, we discuss intertwining Toeplitz operators. Supported by NNSF and PDSF     

Stability of localized operators

Let ?p, 1?p?∞, be the space of all p-summable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the ?p-stability of the convolution operator Ca for different 1?p?∞ are equivalent to each other, i.e., if Ca has ?p-stability for some 1?p?∞ then Ca has ?q-stability for all 1?q?∞. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the ?p-stability (or Lp-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.     

On Hille-Tamarkin operators and Schatten classes

We determine the smallest Schatten class containing all integral operators with kernels inL _p(L_{p', q})^symm, where 2 <p∞ and 1≦q≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1].     

Products of operator ideals and extensions of Schatten classes

We study relations between Schatten classes and product operator ideals, where one of the factors is the Banach ideal Π_E,2 of (E, 2)‐summing operators, and where E is a Banach sequence space with ?₂ ? E. We show that for a large class of 2‐convex symmetric Banach sequence spaces the product ideal Π_E,2 ○ ??^a_q,s is an extension of the Schatten class ??_F with a suitable Lorentz space F. As an application, we obtain that if 2 ≤ p, q < ∞, 1/r = 1/p + 1/q and E is a 2‐convex symmetric space with fundamental function λ_E(n) ≈? n^1/p, then Π_E,2 ○ Π_q is an extension of the Schatten class ??r,q (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Berezin and Garding Inequalities

Let ? be a convex function on ?, let ?(σ) be a pseudodifferential operator with symbol σ, let Λ_σ be the set of its eigenvalues, and let m(λ) be the multiplicity of an eigenvalue λ ∈ Λ_σ. Under certain natural assumptions about properties of pseudodifferential operators, we prove that \(\sum {_{\lambda \in \Lambda _\sigma } m(\lambda )} \varphi (\lambda ) \leqslant \operatorname{Tr} L(\varphi (\sigma )) + R\), where R is an error term of the same order as the remainder term in the Garding inequality.