Let $(H,(\cdot,\cdot)_H)$ and $(Q,(\cdot,\cdot)_Q)$ two Hilbert separable spaces s.t $H\subset Q$ and let $B:H\to Q$ a bounded and linear operator. Let $\sigma,\tau\in H$ two fixed elements.
My question is find $\chi$ in terms of $\sigma, \tau$ and $B$ such that
$(B(\sigma),B(\tau))_Q=(\chi,\tau)_Q$
EDIT: The relation between those inner products is
$(\sigma,\tau)_H=(\sigma,\tau)_Q+(B\sigma,B\tau)_Q$
I believe that we must use the Riesz representation theorem or the Riesz operator.
In the situation Where $H\subset Q$, and the inner product on $H$ is given by
$$(\sigma,\tau)_H = (\sigma,\tau)_Q + (B(\sigma),B(\tau))_Q,$$
you can in general not express the linear form
$$\lambda \colon \tau \mapsto (B(\sigma), B(\tau))_Q$$
(change the roles of $\sigma$ and $\tau$ if your inner products are linear in the first and antilinear in the second argument) as a $Q$-inner product, since $\tau \mapsto (\chi,\tau)_Q$ is (for all $\chi$) a continuous linear form on $Q$, but $\tau \mapsto (B(\sigma),B(\tau))_Q$ is in general not continuous in the subspace topology.