Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$.

By the spectral theorem, there are scalars $\lambda_{1,a}\geqslant \lambda_{2,a}\geqslant\cdots\geqslant0$ and an orthonormal basis $\Phi_a = \{\varphi_{1,a},\varphi_{2,a},\dots\}$ of $H$ such that $a(\varphi_{k,a}) = \lambda_{k,a}\varphi_{k,a}$ for all $k$.

Questions:

1. Are the mappings $a\mapsto \lambda_{k,a}$ measurable?
2. Is it possible to choose $\Phi_a$ such that the mappings $a\mapsto\varphi_{k,a}$ are measurable?

Remark: on $\mathbb{R}$ and $H$ I'm considering the Borel $\sigma$-fields. On $A$ I can accept any “reasonable” $\sigma$-field.