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Search: All articles in the CJM digital archive with keyword unitary

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1. CJM Online first

Clouâtre, Raphaël
Unitary Equivalence and Similarity to Jordan Models for Weak Contractions of Class $C_0$
We obtain results on the unitary equivalence of weak contractions of class $C_0$ to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product.

Keywords:weak contractions, operators of class $C_0$, Jordan model, unitary equivalence
Categories:47A45, 47L55

2. CJM 2012 (vol 66 pp. 700)

He, Jianxun; Xiao, Jinsen
Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two
Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$.

Keywords:Radon transform, wavelet transform, free nilpotent Lie group, unitary representation, inversion formula, sub-Laplacian
Categories:43A85, 44A12, 52A38

3. CJM 2012 (vol 65 pp. 600)

Kroó, A.; Lubinsky, D. S.
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials
We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain - in particular this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex, under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.

Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions
Categories:42C05, 42C99, 42B05, 60B20

4. CJM 2009 (vol 62 pp. 439)

Sundhäll, Marcus; Tchoundja, Edgar
On Hankel Forms of Higher Weights: The Case of Hardy Spaces
In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundhäll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.

Keywords:Hankel forms, Schatten—von Neumann classes, Bergman spaces, Hardy spaces, Besov spaces, transvectant, unitary representations, Möbius group
Categories:32A25, 32A35, 32A37, 47B35

5. CJM 2008 (vol 60 pp. 1067)

Kariyama, Kazutoshi
On Types for Unramified $p$-Adic Unitary Groups
Let $F$ be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field $F_0$, and let $G$ be a symplectic group over $F$ or an unramified unitary group over $F_0$. Following the methods of Bushnell--Kutzko for $\GL(N,F)$, we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in $G$. In particular, we obtain an irreducible supercuspidal representation of $G$ like $\GL(N,F)$.

Keywords:$p$-adic unitary group, type, supercuspidal representation, Hecke algebra
Categories:22E50, 22D99

6. CJM 2006 (vol 58 pp. 262)

Biswas, Indranil
Connections on a Parabolic Principal Bundle Over a Curve
The aim here is to define connections on a parabolic principal bundle. Some applications are given.

Keywords:parabolic bundle, holomorphic connection, unitary connection
Categories:53C07, 32L05, 14F05

7. CJM 2005 (vol 57 pp. 598)

Kornelson, Keri A.
Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group
Differential operators $D_x$, $D_y$, and $D_z$ are formed using the action of the $3$-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space $L^2(S)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.

Keywords:discrete Heisenberg group,, unitary representation,, local solvability,, difference operator
Categories:43A85, 22D10, 39A70

8. CJM 2003 (vol 55 pp. 91)

Choi, Man-Duen; Li, Chi-Kwong; Poon, Yiu-Tung
Some Convexity Features Associated with Unitary Orbits
Let $\mathcal{H}_n$ be the real linear space of $n\times n$ complex Hermitian matrices. The unitary (similarity) orbit $\mathcal{U} (C)$ of $C \in \mathcal{H}_n$ is the collection of all matrices unitarily similar to $C$. We characterize those $C \in \mathcal{H}_n$ such that every matrix in the convex hull of $\mathcal{U}(C)$ can be written as the average of two matrices in $\mathcal{U}(C)$. The result is used to study spectral properties of submatrices of matrices in $\mathcal{U}(C)$, the convexity of images of $\mathcal{U} (C)$ under linear transformations, and some related questions concerning the joint $C$-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

Keywords:Hermitian matrix, unitary orbit, eigenvalue, joint numerical range
Categories:15A60, 15A42

9. CJM 2002 (vol 54 pp. 571)

Li, Chi-Kwong; Poon, Yiu-Tung
Diagonals and Partial Diagonals of Sum of Matrices
Given a matrix $A$, let $\mathcal{O}(A)$ denote the orbit of $A$ under a certain group action such as \begin{enumerate}[(4)] \item[(1)] $U(m) \otimes U(n)$ acting on $m \times n$ complex matrices $A$ by $(U,V)*A = UAV^t$, \item[(2)] $O(m) \otimes O(n)$ or $\SO(m) \otimes \SO(n)$ acting on $m \times n$ real matrices $A$ by $(U,V)*A = UAV^t$, \item[(3)] $U(n)$ acting on $n \times n$ complex symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$, \item[(4)] $O(n)$ or $\SO(n)$ acting on $n \times n$ real symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$. \end{enumerate} Denote by $$ \mathcal{O}(A_1,\dots,A_k) = \{X_1 + \cdots + X_k : X_i \in \mathcal{O}(A_i), i = 1,\dots,k\} $$ the joint orbit of the matrices $A_1,\dots,A_k$. We study the set of diagonals or partial diagonals of matrices in $\mathcal{O}(A_1,\dots,A_k)$, {\it i.e.}, the set of vectors $(d_1,\dots,d_r)$ whose entries lie in the $(1,j_1),\dots,(r,j_r)$ positions of a matrix in $\mathcal{O}(A_1, \dots,A_k)$ for some distinct column indices $j_1,\dots,j_r$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of $A_1,\dots,A_k$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}(A_1,\dots,A_k)$ of complex Hermitian matrices under the action of unitary similarities are also discussed.

Keywords:orbit, group actions, unitary, orthogonal, Hermitian, (skew-)symmetric matrices, diagonal, singular values
Categories:15A42, 15A18

10. CJM 2002 (vol 54 pp. 634)

Weber, Eric
Frames and Single Wavelets for Unitary Groups
We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson's theorem to apply to frames generated by the action of the group. Within this setup we use Stone's theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multi\-resolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on $L^2({\mathbb R})$.

Keywords:wavelet, multiresolution analysis, unitary group representation, frame
Categories:42C40, 43A25, 42C15, 46N99

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