  We show that every biorthogonal wavelet determines a representation by
operators on Hilbert space satisfying simple identities, which captures the
established relationship between orthogonal wavelets and Cuntz-algebra
representations in that special case. Each of these representations is shown to
have tractable finite-dimensional co-invariant doubly-cyclic subspaces.
Further, motivated by these representations, we introduce a general Fock-space
Hilbert space construction which yields creation operators containing the
Cuntz--Toeplitz isometries as a special case.
