Definition 4

We say that a mapping $\Phi$ of $E \times F$ into G is right sesquilinear for J if it satisfies conditions (1), (2), (3) (def. 1, No. 1) as well as (7) $\Phi(x, by) = \Phi(x, y) \cdot b^J$ for all $x \in E,\ y \in F$ and $b \in B$.

If J is the identity (which requires that B be commutative), we recover the notion of a bilinear mapping.

Let $(e_i)_{i \in I}$ and $(f_k)_{k \in K}$ be two families of elements of E and F, and let $(a_i)_{i \in I}$ and $(b_k)_{k \in K}$ be elements of A and B which are zero except for a finite number of them. Then

As in the case of a bilinear mapping, the elements $\Phi(e_i, f_k)$ determine $\Phi$ uniquely when $(e_i)$ and $(f_k)$ are generating systems, and may be chosen arbitrarily when $(e_i)$ and $(f_k)$ are bases of E and F; when $(e_i)$ and $(f_k)$ are finite bases, $(\Phi(e_i, f_k))$ is called the matrix of $\Phi$ with respect to these bases.

As for bilinear mappings, one defines on the set of right sesquilinear mappings (for $J$) from $E \times F$ into $G$ a bimodule structure over the centres of $A$ and $B$. One defines the notion of inverse image of a sesquilinear mapping by the same formula as for a bilinear mapping. We shall moreover see that the study of sesquilinear mappings can be reduced to that of bilinear mappings.