  We introduce a new approach to the classification of operator identities,
based on basic concepts from the theory of algebraic operads together with
computational commutative algebra applied to determinantal ideals of matrices
over polynomial rings. We consider operator identities of degree 2 (the number
of variables in each term) and multiplicity 1 or 2 (the number of operators in
each term), but our methods apply more generally. Given an operator identity
with indeterminate coefficients, we use partial compositions to construct a
matrix of consequences, and then use computer algebra to determine the values
of the indeterminates for which this matrix has submaximal rank. For
multiplicity 1 we obtain six identities, including the derivation identity. For
multiplicity 2 we obtain eighteen identities and two parametrized families,
including the left and right averaging identities, the Rota-Baxter identity,
the Nijenhuis identity, and some new identities which deserve further study.
