This paper is an investigation of factorable operators, Riesz bases and Weyl-Heisenberg frames on G with respect to a function-valued inner product, the so called phi-bracket product, on L2(G) where G is a locally compact abelian group and phi is a topological isomorphism on G. We investigate phi-factorable operators on L2(G) and extend Riesz Representation Theorems for these operators. Also we introduce phi-Riesz bases and obtain a characterization of them in L2(G). Finally we show that several well known theorems for Weyl-Heisenberg frames in L2(R) remain valid in L2(G), and they are unified under the aspects of group theory, in connection with the phi-bracket product.

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