# Operator-valued functions that are measurable in the weak operator topology?

$$\newcommand{\H}{\mathcal{H}}$$Let $$\H_1,\H_2$$ be separable Hilbert spaces and $$\mathcal{L}(\H_1,\H_2)$$ the Banach space of bounded linear operators $$\H_1\to\H_2$$. Suppose $$f:\mathbb{R}\to\mathcal{L}(\H_1,\H_2)$$ in the weak operator topology, i.e., $$\forall\,u\in\H_1$$, $$v\in\H_2$$, we have that $$x\mapsto\langle f(x)u,v\rangle_{\H_2}$$ measurable in the usual sense. What can we say about $$f$$? Is $$f$$ strongly measurable in the sense that for any open set $$U\subset\mathcal{L}(\H_1,\H_2)$$ (in the norm topology), $$f^{-1}(U)$$ is measurable in $$\mathbb{R}^n$$?

• Diestel and Uhl's book has examples to show that this is not true. (You can take $H_2$ to be one dimensional to make things simple). – Kavi Rama Murthy May 11 at 12:13