Prove an equality between two different expressions involving the Lag operator

Question

I'm trying to show that

$$G(L) = \sigma^2(\sum_{j=0}^\infty \psi_{j}^2 + \sum_{h=1}^\infty\sum_{j=0}^\infty \psi_j \psi_{j+h}(L^h-L^{-h}))$$

is equal to:

$$\sigma^2(\sum_{j=0}^\infty\psi_jL^j)(\sum_{h=0}^\infty\psi_hL^{-h})$$

with no avail, any help?

(L is an operator, specifically the lag operator)