It is known that a measure $\mu\in M(\mathbb{T},B(H))$ of finite variation, can be identified with a bounded linear operator $F_\mu: C(\mathbb{T},H\otimes H)\to \mathbb{C}$, where $H$ is a Hilbert space, $\mathbb{T}$ is the Torus and $B(H)$ represents the bounded linear operators from $H$ to $H$. Also, the norm of this operator $F_\mu$ is precisely the variation of $\mu$.
I could find a proof of this, for example, in Diestel Uhl and Dunford Schwartz, actually for more general cases.
My question is: which would be the identification if we consider $ L^2(\mathbb{T},H\otimes H)$ as the range of the operator? Might it be somehow related with the 2-variation of $\mu$?