A Hörmander type multiplier theorem for multilinear operators

In this paper, we prove a Hörmander type multiplier theorem for multilinear operators. As a corollary, we can weaken the regularity assumption for multilinear Fourier multipliers to assure the boundedness.     

多线性Fourier乘子算子的加权估计——献给陆善镇教授75华诞

本文考虑多线性Fourier乘子算子在加权Lebesgue空间的乘积空间上的性质,利用多线性Fourier乘子算子的核估计以及多线性奇异积分算子的加权理论,建立多线性Fourier乘子算子的（关于多重Ap/r（R^mn）权函数以及关于一般权函数的）两个加权估计.     

Classification and geometric aspects of vector valued Fourier transforms

It is shown that for any locally compact abelian group ?? and 1 ≤ p ≤ 2, the Fourier type p norm with respect to ?? of a bounded linear operator T between Banach spaces, denoted by ‖T |???^??_p‖, satisfies ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the direct product of ?₂, ?₃, ?₄, … It is also shown that if ?? is not of bounded order then Cⁿ_p ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the circle group, n is a onnegative integer and C^p = . From these inequalities, for any locally compact abelian group ?? ‖T |???^??₂‖ ≤ ‖T |???^??₂‖, and moreover if ?? is not of bounded order then ‖T |???^??₂‖ = ‖T |???^??₂‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A higher rank version of Kitai's theorem

We prove that each linear action of on an infinite-dimensional Banach space generated by compact operators cannot be hypercyclic. This result generalizes a theorem of Kitai for the case of Z₊ actions. Contrary to the case of infinite dimension, a hypercyclic action of on C is given.     

Fourier multipliers and spectral measures in Banach function spaces

It is classical that amongst all spaces L^p (G), 1 ≤ p ≤ ∞, for , or say, only L² (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L² (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L² (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L² (G); this fails for every p ≠ 2. We show that this special status of L² (G) amongst the spaces L^p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).      

Weighted Compact Commutator of Bilinear Fourier Multiplier Operator

Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W~s(R~(2n)) ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R~n) functions is a compact operator from L~(p1)(R~n, w_1) × L~(p2)(R~n, w_2) to L~p(R~n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R~(2n)).     

On hyperinvariant subspaces of contraction operators on a Banach space whose spectrum contains the unit circle

In this paper, we prove that every operator in a class of contraction operators on a Banach space whose spectrum contains the unit circle has a nontrivial hyperinvariant subspace. This research is supported by the Natural Science Foundation of P. R. China (No. 10771039)     

Operator weak amenability of the Fourier algebra

We show that for any locally compact group , the Fourier algebra is operator weakly amenable.

     

A Coburn type theorem for Toeplitz operators on the Dirichlet space

A celebrated theorem of Coburn asserts that, on the setting of the Hardy space, if a Toeplitz operator is nonzero, then either it is one-to-one or its adjoint operator is one-to-one. In this paper, we show that an analogous result holds for Toeplitz operators acting on the Dirichlet space.     

Weyl's theorem and hypercyclic/supercyclic operators

Necessary and sufficient conditions for hypercyclic/supercyclic Banach space operators T to satisfy are proved.     

Group invariance and <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounded operators

The Hilbert and Riesz transforms can be characterized up to scalar as the translation invariant operators that satisfy additionally certain relative invariance of conformal transformation groups. In this article, we initiate a systematic study of translation invariant operators from group theoretic viewpoints, and formalize a geometric condition that characterizes specific multiplier operators uniquely up to scalar by means of relative invariance of affine subgroups. After providing some interesting examples of multiplier operators having “large symmetry”, we classify which of these examples can be extended to continuous operators on L ^p (R ⁿ ) (1 < p < ∞). T. Kobayashi was partially supported by Grant-in-Aid for Scientific Research 18340037, Japan Society for the Promotion of Science. A. Nilsson was partially supported by Japan Society for the Promotion of Science.     

Harmonic operators: the dual perspective

The study of harmonic functions on a locally compact group G has recently been transferred to a “non-commutative” setting in two different directions: Chu and Lau replaced the algebra L ^∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L ^∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand, Jaworski and the first author replaced L ^∞(G) by to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action of σ on VN(G) can be extended to . We study the corresponding space of “σ-harmonic operators”, i.e., fixed points in under the action of σ. We show, under mild conditions on either σ or G, that is in fact a von Neumann subalgebra of . Our investigation of relies, in particular, on a notion of support for an arbitrary operator in that extends Eymard’s definition for elements of VN(G). Finally, we present an approach to via ideals in , where denotes the trace class operators on L ²(G), but equipped with a product different from composition, as it was pioneered for harmonic functions by Willis. M. Neufang was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach. V. Runde was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach.     

Marcinkiewicz multiplier theorem and the Sunouchi operator for Ciesielski–Fourier series

Some classical results due to Marcinkiewicz, Littlewood and Paley are proved for the Ciesielski–Fourier series. The Marcinkiewicz multiplier theorem is obtained for L_p spaces and extended to Hardy spaces. The boundedness of the Sunouchi operator on L_p and Hardy spaces is also investigated.     

A full-invariant theorem and some applications

Let (X,τ₁) and (Y,τ₂) be two Hausdorff locally convex spaces with continuous duals X′ and Y′, respectively, L(X,Y) be the space of all continuous linear operators from X into Y, K(X,Y) be the space of all compact operators of L(X,Y). Let WOT and UOT be the weak operator topology and uniform operator topology on K(X,Y), respectively. In this paper, we characterize a full-invariant property of K(X,Y); that is, if the sequence space λ has the signed-weak gliding hump property, then each λ-multiplier WOT-convergent series ∑_iT_i in K(X,Y) must be λ-multiplier convergent with respect to all topologies between WOT and UOT if and only if each continuous linear operator T :(X,τ₁)→(λ^β,σ(λ^β,λ)) is compact. It follows from this result that the converse of Kalton's Orlicz–Pettis theorem is also true.     

Sharp Fourier type and cotype with respect to compact semisimple Lie groups

Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.

     

Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T∈B(X) is hereditarily polaroid, T∈HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T∈B(X) has SVEP and R∈B(X) is a Riesz operator which commutes with T, then T+R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T+Q and T^∗+Q^∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T+Q satisfies Weyl's theorem. If A∈B(X) is an algebraic operator which commutes with the polynomially HP operator T, then T+N is polaroid and has SVEP, f(T+N) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ(T+N), and f^∗(T+N) satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T+N).     

The amenability constant of the Fourier algebra

For a locally compact group , let denote its Fourier algebra and its dual object, i.e., the collection of equivalence classes of unitary representations of . We show that the amenability constant of is less than or equal to and that it is equal to one if and only if is abelian.

     

A Bessel function multiplier

We obtain nearly sharp estimates for the norms of certain convolution operators.