Let $T:(Z,\|\cdot\|_{Z})\to(H,\|\cdot\|_{H})$ be a continuous linear operator between a separable Banach space, $Z$, and a separable Hilbert space, $H$. Assume that both $Z$ and $H$ are infinite-dimensional.
Let $f\in Z^{*}$ be a continuous linear functional. Does it follow that the image set
$$T(\{z\in Z\;|\;f(z)\le 0\})$$
is Borel measurable in $H$?
Of note: I am particularly interested in the case in which $Z$ is not a pre-dual (thus, I cannot rely on constructions that exploit weak* compactness).