  We introduce a framework for proving statements about linear operators by
verification of ideal membership in a free algebra. More specifically,
arbitrary first-order statements about identities of morphisms in preadditive
semicategories can be treated. We present a semi-decision procedure for
validity of such formulas based on computations with noncommutative
polynomials. These algebraic computations automatically incorporate linearity
and benefit from efficient ideal membership procedures. In the framework,
domains and codomains of operators are modelled using many-sorted first-order
logic. To eliminate quantifiers and function symbols from logical formulas, we
apply Herbrand's theorem and Ackermann's reduction. The validity of the
resulting formulas is shown to be equivalent to finitely many ideal memberships
of noncommutative polynomials. We explain all relevant concepts and discuss
computational aspects. Furthermore, we illustrate our framework by proving
concrete operator statements assisted by our computer algebra software.
