On Racah operators

As a generalization of the well-known Racah coefficients (defined for finite-dimensional representations of semisimple Lie groups), we introduce the notion of Racah operators for locally compact groups with “nice” dual space. In the case of the group PSL(2,?), these operators are explicitly indicated.     

Spanning L 2 of a nilpotent Lie group by eigenvectors of invariant differential operators

Let be a non-abelian, connected, simply connected, nilpotent Lie group. We show that the eigenvectors of a finite number of families of left invariant differential operators and their conjugates span a dense subspace of L ² (G). The restriction of the left regular representation to each one of these (left invariant) eigenspaces disintegrates into irreducible unitary representations with multiplicities 0 and 1 only. J. Ludwig and C. Molitor-Braun are supported by the research grant R1F104C09 of the University of Luxembourg.     

Wilson Function Transforms Related to Racah Coefficients

The irreducible -representations of the Lie algebra consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch–Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for , which turn out to be Askey–Wilson functions and Askey–Wilson polynomials.This research was done during my stay at the Department of Mathematics at Chalmers University of Technology and Göteborg University in Sweden, supported by a NWO-TALENT stipendium of the Netherlands Organization for Scientific Research (NWO).     

Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered

We establish the following result.

Theorem. Let be a integrable bounded group representation whose Arveson spectrum is scattered. Then the subspace generated by all eigenvectors of the dual representation is dense in Moreover, the closed subalgebra generated by the operators () is semisimple.

If, in addition, does not contain any copy of then the subspace spanned by all eigenvectors of is dense in Hence, the representation is almost periodic whenever it is strongly continuous.

     

Sharp estimates for Hardy operators on Heisenberg group

In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp (p, p) estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on L^p(Hⁿ) is still p/(p−1). This goes some way to imply that the L^p norms of the Hardy operator are the same despite the domains are intervals on ℝ, balls in ℝⁿ, or ‘ellipsoids’ on the Heisenberg group Hⁿ. By constructing a special function, we find the best constant in the weak type (1,1) inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H¹ to L¹. Moreover, we describe the difference between M^p weights and A^p weights and obtain the characterizations of such weights using the weighted Hardy inequalities.     

Hausdorff operators on the Heisenberg group

This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group H n. The sharp bounds for the strong type(p, p)(1 ≤ p ≤∞) estimates of n-dimensional Hausdorff operators on H n are obtained. The sharp bounds for strong(p, p) estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on H n. The weak type(p, p)(1 ≤ p ≤∞) estimates are also obtained.     

The classification of Leonard triples of Racah type

Let K$\mathbb{K}$ denote an algebraically closed field of characteristic zero. Let V   denote a vector space over K$\mathbb{K}$ with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A  , A^?$A^{?}$, A^ε$A^{\epsilon }$ in End(V)$\mathrm{End}(V)$ such that for each B∈{A,A^?,A^ε}$B\in \{A,A^{?},A^{\epsilon }\}$ there exists a basis for V with respect to which the matrix representing B   is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. In this paper we define a family of Leonard triples said to have Racah type and classify them up to isomorphism. Moreover, we show that each of them satisfies the _Z3${\mathbb{Z}}_3$-symmetric Askey–Wilson relations. As an application, we construct all Leonard triples that have Racah type from the universal enveloping algebra U(_sl2)$U({\mathit{sl}}_2)$.     

On harmonic analysis of causal operators

We follow the group representation theory approach to define causal operators on Banach modules and present some of their spectral properties.     

On the asymptotic distribution of the spectrum of an operator over irreducible representations of its symmetry group

We show that a representation of a finite group in the eigenfunction space of an elliptic operator defined on a Riemannian manifold and commuting with the effective action of the group is asymptotically a multiple of the regular representation of the group.     

Tensor products of subnormal operators

We shall use a -algebra approach to study operators of the form where is subnormal and is normal. We shall determine the spectral properties for these operators, and find the minimal normal extension and the dual operator. We also give a necessary condition for to contain a compact operator and a sufficient condition for the algebraic equivalence of and .

We also consider the existence of a homomorphism satisfying . We shall characterize the operators such that exists for every operator .

The problem of when is unitarily equivalent to is considered. Complete results are given when and are positive operators with finite multiplicity functions and has compact self-commutator. Some examples are also given.

     

A faithfulness criterion for the Gassner representation of the pure braid group

This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, . Long and Paton proved that if a Burau matrix has ones on the diagonal and zeros below the diagonal then is the identity matrix. In this paper, a generalization of Long and Paton's result will be proved. Our main theorem is that if the trace of the image of an element of under the reduced Gassner representation is , then this element lies in the kernel of this representation. Then, as a corollary, we prove that an analogue of the main theorem holds true for the Burau representation of the braid group.

     

Hyponormal Toeplitz operators on the polydisk

By bounded vector-valued functions and block matrix representations of Hankel operators, we completely characterize the hyponormality of Toeplitz operators on the Hardy space of the polydisk.     

Representation of feedback operators for parabolic control problems

In this paper we present results on existence and regularity of integral representations of feedback operators arising from parabolic control problems. The existence of such representations is important for the design of low order compensators and in the placement of sensors. This paper extends earlier results of J. A. Burns and B. B. King to problems with spatial dimensions.     

Norm estimates for functions of two operators on tensor products of Hilbert spaces

Analytic operator valued functions of two operators on tensor products of Hilbert spaces are considered. A precise norm estimate is established. Applications to operator differential equations are also discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplicative representation of bilinear operators

We establish that each lattice bimorphism from the Cartesian product of two vector lattices into a universally complete vector lattice is representable as the product of two lattice homomorphisms defined on the factors. This fact makes it possible to reduce the problem to the linear case and obtain some results on representation of an order bounded disjointness preserving bilinear operator as a strongly disjoint sum of weighted shift or multiplicative operators.     

Algorithms for Computing Generalized U(<Emphasis Type="Italic">N</Emphasis>) Racah Coefficients

Algorithms are developed for computing generalized Racah coefficients for the U(N) groups. The irreducible representations (irreps) of the U(N) groups, as well as their tensor products, are realized as polynomials in complex variables. When tensor product irrep labels as well as a given irrep label are specified, maps are constructed from the irrep space to the tensor product space. The number of linearly independent maps gives the multiplicity. The main theorem of this paper shows that the eigenvalues of generalized Casimir operators are always sufficient to break the multiplicity. Using this theorem algorithms are given for computing the overlap between different sets of eigenvalues of commuting generalized Casimir operators, which are the generalized Racah coefficients. It is also shown that these coefficients are basis independent. Mathematics Subject Classifications (2000) 22E70, 81R05, 81R40.     

良序套张量积的代数中的强不可约算子

设Ｈ为复的可分无限维Ｈｉｌｂｅｒｔ空间，称有界线性算子Ｔ为强不可约的，如果与Ｔ可交换的幂等算子只有０和Ｉ．王宗尧、蒋春澜、纪有清等人证明了在任何一个套的套代数中都存在大量的强不可约算子，并且找到了它们的酉轨道闭包．本文考虑有限个套的张量积的代数中强不可约算子的存在性问题。证明了：对复平面上任何一个连通完备集σ、总存在一个对角算子Ｎ和它的一个范数可以任意小的紧摄动Ｔ＝Ｘ＋Ｋ，使得Ｔ是一个强不可约算子、Ｔ在有限个良序套的张量积的代数中，并且σ（Ｔ）＝σｌｒｅ（Ｔ）＝σ（Ｎ）＝σｌｒｅ（Ｎ）＝σ进一步，文章还对具有单点谱的算子和良序套与正交补为良序套的张量积的代数进行了讨论，得到了一些结果．