Covariant Representations for Matrix-valued Transfer Operators

Abstract

Motivated by the multivariate wavelet theory, and by the spectral theory of transfer operators, we construct an abstract affine structure and a multiresolution associated to a matrix-valued weight. We describe the one-toone correspondence between the commutant of this structure and the fixed points of the transfer operator. We show how the covariant representation can be realized on R n if the weight satisfies some low-pass condition.

Groupoid methods in wavelet analysis

“…But we note that some of the recent work of Dutkay and Roysland [15,16] can be formulated in the setting we are describing by taking more complicated bundles and representations.…”

Representations of Polynomial Covariance Type Commutation Relations by Linear Integral Operators on $$L_p$$ Over Measure Spaces

“…A description of the structure of representations for the relation (1) and more general families of self-adjoint operators satisfying such relations by bounded and unbounded self-adjoint linear operators on a Hilbert space using reordering formulas for functions of the algebra elements and operators satisfying covariance commutation relation, functional calculus and spectral representation of operators and interplay with dynamical systems generated by iteration of maps involved in the commutation relations have been considered in [4,[8][9][10]12,13,[18][19][20][21][22][39][40][41][42][43][44][46][47][48][49][50][52][53][54][55][56][57][58][59][60][62][63][64][65][66].…”