Harmonic analysis of signed Ruelle transfer operators

Motivated by wavelet analysis, we prove that there is a one-to-one correspondence between the following data: (i) Solutions to R(h) = h where R is a certain non-positive Ruelle transfer operator; (ii) Operators that intertwine a certain class of representations of the C ∗-algebra AN on two unitary generators U, V subject to the relation UVU −1 = V N. This correspondence enables us to give a criterion for the biorthogonality of a pair of scaling functions and calculate all solutions of the equation R(h) = h in some concrete cases.  2002 Elsevier Science (USA). All rights reserved