We review the notion of a vector space, basis and dimension, linear transformations between vector spaces, dual vector spaces and transformations, spectral decomposition for normal operators (which includes symmetric, Hermitian, orthogonal, and unitary operators), and determinants. Along the way we review direct-sum decompositions, bilinear forms and inner product spaces, adjoints, characterist...

We define exact functors from categories of Harish-Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they...

A theory of non-unitary unbounded similarity transformation operators is developed. To this end the class of J-unitary operators U is introduced. These operators are similar to unitary operators in their algebraic aspects but differ in their topological properties. It is shown how J-unitary operators are related to so-called J-biorthonormal systems and J-selfadjoint projections. Families {Uα} o...

We prove several de Finetti theorems for the unitary dual group, also called Brown algebra. Firstly, we provide a finite theorem characterizing $R$-diagonal elements with an identical distribution. This is surprising, since it applies to sequences in contrast classical and quantum groups; also, does not involve any known independence notion. Secondly, considering infinite $W^*$-probability spac...

Abstract. Dual Gramian analysis is one of the fundamental tools developed in a series of papers [37, 40, 38, 39, 42] for studying frames. Using dual Gramian analysis, the frame operator can be represented as a family of matrices composed of the Fourier transforms of the generators of (generalized) shiftinvariant systems, which allows us to characterize most properties of frames and tight frames...

The quantum deformation of the group of motions of the plane and its Pontryagin dual are described in details. It is shown that the Pontryagin dual is a quantum deformation of the group of transformations of the plane generated by translations and dilations. An explicite expression for the unitary bicharacter describing the Pontryagin duality is found. The Heisenberg commutation relations are w...

Deterministic classical dynamical systems have an ergodic hierarchy, from through mixing, to Bernoulli that are "as random as a coin-toss". Dual-unitary circuits been recently introduced solvable models of many-body nonintegrable quantum chaotic having hierarchy properties. We extend this include the apex putative which is Bernoulli, in sense correlations single and two-particle observables van...

The shadow V ′ of a self-dual vertex operator superalgebra V is defined as the direct sum of the irreducible modules of its even vertex operator subalgebra V(0) not contained in V = V(0)⊕V(1). We describe the self-dual “very nice” unitary rational vertex operator superalgebras V of rank c whose shadows have the largest possible minimal weights c 8 or c 8 −1. The results are analogous to and imp...