Mordell proved that the Ramanujan tau function, expressing the coefficients of the Ramanujan form, is a multiplicative function: The idea goes back to earlier work of Adolf Hurwitz, who treated algebraic correspondences between modular curves which realise some individual Hecke operators.
Modular forms are particular kinds of functions of a lattice, subject to conditions making them analytic functions and homogeneous with respect to homotheties, as well as moderate growth at infinity; these conditions are preserved by the summation, and so Hecke operators preserve the space of modular forms of a given weight.
This can be rewritten in the form which leads to the formula for the Fourier coefficients of Tm(f(z)) = Σ bnqn in terms of the Fourier coefficients of f(z) = Σ anqn: One can see from this explicit formula that Hecke operators with different indices commute and that if a0 = 0 then b0 = 0, so the subspace Sk of cusp forms of weight k is preserved by the Hecke operators.
In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional.
The presence of this commutative operator algebra plays a significant role in the harmonic analysis of modular forms and generalisations.