Representations of Super $W(2,2)$ algebra $\mathfrak{L}$

In paper, we study the representation theory of super $W(2,2)$ algebra $\mathfrak{L}$. We prove that $\mathfrak{L}$ has no mixed irreducible modules and give the classification of irreducible modules of intermediate series. We determinate the conjugate-linear anti-involution of $\mathfrak{L}$ and give the unitary modules of intermediate series.