Summary: V. P. Zahariuta [in Stud. Math. 46, 201-221 (1973; Zbl 0261.46003)] used the theory of Fredholm operators to develop a method to classify Cartesian products of locally convex spaces. In this work we modify his method to study the isomorphic classification of Cartesian products of the kind \(E_0^p(a)\times E_\infty^q(b)\) where \(1\leq p,q<\infty\), \(p\neq q\), \(a= (a_n)_{n=1}^\infty\) and \(b= (b_n)_{n=1}^\infty\) are sequences of positive numbers and \(E_p^0(a)\), \(E_\infty^q(b)\) are respectively \(\ell^p\)-finite and \(\ell^q\)-infinite type power series spaces.