A lifting theorem for symmetric commutants

Let $T_{1},\dots , T_{n}\in B(\mathcal{H})$ be bounded operators on a Hilbert space $\mathcal{H}$ such that $T_{1} T_{1}^{*}+\cdots + T_{n} T_{n}^{*}\leq I_{\mathcal{H}}$. Given a symmetry $j$ on $\mathcal{H}$, i.e., $j^{2}=j^{*} j=I_{\mathcal{H}}$, we define the $j$-symmetric commutant of $\{T_{1},\dots , T_{n}\}$ to be the operator space

In this paper we obtain lifting theorems for symmetric commutants. The result extends the Sz.-Nagy-Foias commutant lifting theorem ($n=1, j=I_{\mathcal{H}}$), the anticommutant lifting theorem of Sebestyén ( $n=1, j=-I_{\mathcal{H}}$), and the noncommutative commutant lifting theorem ( $j=I_{\mathcal{H}}$). Sarason's interpolation theorem for $H^{\infty }$ is extended to symmetric commutants on Fock spaces.