On wandering vector multipliers for unitary groups

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system acting on a separable Hilbert space into itself. It is proved that the wandering vector multipliers for a unitary group form a group, which gives a positive answer for a problem of Han and Larson. Furthermore, non-abelian unitary groups of order 6 are considered. We prove that the wandering vector multipliers of such a unitary group can not generate . This negatively answers another of their problems.

     

Wandering vectors for irrational rotation unitary systems

An abstract characterization for those irrational rotation unitary systems with complete wandering subspaces is given. We prove that an irrational rotation unitary system has a complete wandering vector if and only if the von Neumann algebra generated by the unitary system is finite and shares a cyclic vector with its commutant. We solve a factorization problem of Dai and Larson negatively for wandering vector multipliers, and strengthen this by showing that for an irrational rotation unitary system , every unitary operator in is a wandering vector multiplier. Moreover, we show that there is a class of wandering vector multipliers, induced in a natural way by pairs of characters of the integer group , which fail to factor even as the product of a unitary in and a unitary in . Incomplete maximal wandering subspaces are also considered, and some questions are raised.

     

Wandering r-tuples for unitary systems

The properties of the set Wr(U) of all complete wandering r-tuples for a system U of unitary operators acting on a Hilbert space are investigated by parameterizing Wr(U) in terms of a fixed wandering r-tuple Ψ and the set of all unitary operators which locally commute with U at Ψ. The special case of greatest interest is the system 〈D,T〉 of dilation (by 2) and translation (by 1) unitary operators acting on L²(R), for which the complete wandering r-tuples are precisely the orthogonal multiwavelets with multiplicity r. We also give some examples for its application.     

<Emphasis Type="Italic">K</Emphasis>-fusion Frames and the Corresponding Generators for Unitary Systems

Motivated by K-frames and fusion frames, we study K-fusion frames in Hilbert spaces. By the means of operator K, frame operators and quotient operators, several necessary and sufficient conditions for a sequence of closed subspaces and weights to be a K-fusion frame are obtained, and operators preserving K-fusion frames are discussed. In particular, we are interested in the K-fusion frames with the structure of unitary systems. Given a unitary system which has a complete wandering subspace, we give a necessary and sufficient condition for a closed subspace to be a K-fusion frame generator.     

Wandering Operators for Unitary Systems of Hilbert Spaces

In this paper, we introduce the concept of complete wandering operators for a system \({{\mathcal {U}}}\) of unitary operators acting on a Hilbert space, which can be viewed as an abstract mathematical model for \(g\)-orthonormal bases of Hilbert spaces and operator-valued wavelets for \(L^2(R)\). The idea comes from Dai and Larson’s work (Mem Am Math Soc 134:640 1998), where the wandering vectors for a unitary system are introduced as an abstract model for orthogonal wavelets. The topological and algebraical properties of the set \({{\mathcal {W}}}({{\mathcal {U}}})\) of all complete wandering operators for a unitary system \({{\mathcal {U}}}\) are studied. In particular, properties of the local commutant of \({{\mathcal {U}}}\) are established. A parametrization formula for \({{\mathcal {W}}}({{\mathcal {U}}})\) and some interesting algebraic properties of complete wandering operators for a unitary system are obtained. The special case of greatest interest is the wavelet system \(\{{{\mathcal {U}}}_{D,T}\}\). We pay certain attention on studying this more structured unitary system and some structural theorems are established. Lots of properties of the wandering vectors for a unitary system are extended to the more general case, i.e. the wandering operators for a unitary system. However, operator-valued case is more complicated. We also give some examples to illustrate our results. Our works show that wavelet theory and frame theory are deeply connected with operator theory.     

On the finite dimensional unitary representations of Kazhdan groups

We use A. Weil's criterion to prove that all finite dimensional unitary representations of a discrete Kazhdan group are locally rigid. It follows that any such representation is unitarily equivalent to a unitary representation over some algebraic number field.

     

Characterizations of solution sets of convex vector minimization problems

Complete dual characterizations of the weak and proper optimal solution sets of an infinite dimensional convex vector minimization problem are given. The results are expressed in terms of subgradients, Lagrange multipliers and epigraphs of conjugate functions. A dual condition characterizing the containment of a closed convex set, defined by a cone-convex inequality, in a reverse-convex set, plays a key role in deriving the results. Simple Lagrange multiplier characterizations of the solution sets are also derived under a regularity condition. Numerical examples are given to illustrate the significance of the results.     

The Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems

In this paper, the Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems is developed. A minimal and a maximal operators, GKN-sets, and a boundary space for the system are introduced. Algebraic characterizations of the domains of self-adjoint extensions of the minimal operator are given. A close relationship between the domains of self-adjoint extensions and the GKN-sets is established. It is shown that there exist one-to-one correspondences among the set of all the self-adjoint extensions, the set of all the d-dimensional Lagrangian subspaces of the boundary space, and the set of all the complete Lagrangian subspaces of the boundary space.     

Some results on multipliers and numerical multiplier groups of difference sets

In this paper, we use number theoretic methods to study multipliers and numerical multiplier groups of difference sets. We obtain a relation between the decomposition group of a prime divisor of the order of a difference set and the numerical multiplier group, this gives rise to some results concerning the numerical multiplier groups of difference sets. Also we give two characterizations of strong multipliers of a subset in an abelian group which have some applications in difference sets.     

On discrete frames associated with semidirect products

For groups which are the semidirect product of some vector group with a unimodular group we prove that the existence of a discrete frame obtained from an at-most countable set of vectors through the action of a given unitary representation implies that the representation in use has to be square-integrable.     

A domain embedding preconditioner for the Lagrange multiplier system

Finite element approximations for the Dirichlet problem associated to a second-order elliptic differential equation are studied. The purpose of this paper is to discuss domain embedding preconditioners for discrete systems. The essential boundary condition on the interior interface is removed by introducing Lagrange multipliers. The associated discrete system, with a saddle point structure, is preconditioned by a block diagonal preconditioner. The main contribution of this paper is to propose a new operator, constructed from the -inner product, for the block of the preconditioner corresponding to the multipliers.

     

Vector optimization problems with nonconvex preferences

In this paper, some vector optimization problems are considered where pseudo-ordering relations are determined by nonconvex cones in Banach spaces. We give some characterizations of solution sets for vector complementarity problems and vector variational inequalities. When the nonconvex cone is the union of some convex cones, it is shown that the solution set of these problems is either an intersection or an union of the solution sets of all subproblems corresponding to each of these convex cones depending on whether these problems are defined by the nonconvex cone itself or its complement. Moreover, some relations of vector complementarity problems, vector variational inequalities, and minimal element problems are also given. While this paper was being revised in September 2006, Professor Alex Rubinov (the second author of the paper) left us due to the illness. This is a very sad news to us. We dedicate this paper to the memory of Professor Rubinov as a mathematician and truly friend.     

The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism

It is known that the set of all solutions of a commutant lifting and other interpolation problems admits a Redheffer linear-fractional parametrization. The method of unitary coupling identifies solutions of the lifting problem with minimal unitary extensions of a partially defined isometry constructed explicitly from the problem data. A special role is played by a particular unitary extension, called the central or universal unitary extension. The coefficient matrix for the Redheffer linear-fractional map has a simple expression in terms of the universal unitary extension. The universal unitary extension can be seen as a unitary coupling of four unitary operators (two bilateral shift operators together with two unitary operators coming from the problem data) which has special geometric structure. We use this special geometric structure to obtain an inverse theorem (Theorem 8.4) which characterizes the coefficient matrices for a Redheffer linear-fractional map arising in this way from a lifting problem. The main tool is the formalism of unitary scattering systems developed in Boiko et al. (Operator theory, system theory and related topics (Beer-Sheva/Rehovot 1997), pp. 89–138, 2001) and Kheifets (Interpolation theory, systems theory and related topics, pp. 287–317, 2002)     

Special Moufang sets, their root groups and their {micro}-maps

We prove Timmesfeld's conjecture that special abstract rankone groups are quasisimple. We give two characterizations ofthe root groups in special Moufang sets: a normal subgroup ofthe point stabilizer is a root group if it is either regular,or nilpotent and transitive. We prove that if a root group ofa special Moufang set contains an involution, then it is ofexponent 2. We also show that the root groups are abelian ifand only if the so-called µ-maps are involutions.     

Characterizations of the Benson Proper Efficiency for Nonconvex Vector Optimization

Under generalized cone-subconvexlikeness for vector-valued mappings in locally-convex Hausdorff topological vector spaces, a Gordan-form alternative theorem is derived. Some characterizations of the Benson proper efficiency under this generalized convexity are established in terms of scalarization, Lagrangian multipliers, saddle-point criterion, and duality.     

On Beurling-type theorems in weighted and Bergman spaces

We prove that analytic operators satisfying certain series of operator inequalities possess the wandering subspace property. As a corollary, we obtain Beurling-type theorems for invariant subspaces in certain weighted and Bergman spaces.

     

Electrostatics, Hyperbolic Geometry and Wandering Vectors

A family of planar discrete electrostatic systems on the unitcircle with finitely atomic external fields is considered. Thegeometry of particles in the external field yielding a givenminimum energy configuration is studied. As an application,the wandering vectors of the shift operator in the Dirichletspaces associated with finitely atomic measures are also studied.In particular, the zero locus of a wandering vector is discussed.     

Multi-frame vectors for unitary systems

In this paper, the set of all complete multi-normalized tight frame vectors NF ^r (U) with multiplicity r and the set of all complete multi-frame vectors F ^r (U) with multiplicity r for a system U of unitary operators acting on a separable Hilbert space are characterized in terms of co-isometries and surjective operators in (U), the set of all operators which locally commute with U at Ψ ^r , a fixed complete wandering r-tuple for U. Then we study the linear combinations of multi-frame vectors for U and establish some conditions under which these combinations are still the same type of multi-frame vectors for U. Finally, we establish some interesting properties for multi-frame vectors when U is a unitary group. All these results have potential applications in the theory of multi-Gabor systems and multi-wavelet systems.     

The limit sets of Schottky quasiconformal groups are uniformly perfect

In this paper we study Schottky quasiconformal groups. We show that the limit sets of Schottky quasiconformal groups are uniformly perfect, and that the limit set of a given discrete non-elementary quasiconformal group has positive Hausdorff dimension.