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Preliminaries

Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space and consider the bounded linear operators from $\mathcal{H}$ to $\mathcal{H}$: $\mathfrak{B}(\mathcal{H})$. Recall the definition of the Hilbert-Schmidt norm of an operator $\mathscr{A}$:

$$ \lVert \mathscr{A} \rVert_{HS}^2 = \textrm{Tr}(A^*A) = \sum_i \lVert A e_i \rVert^2 $$

where $(e_i)_{i=1}^\infty$ is a complete orthonormal system for $\mathcal{H}$. An operator with finite HS-norm is said to be Hilbert-Schmidt and we denote the set of all Hilbert-Schmidt operators from $\mathcal{H}$ to $\mathcal{H}$ by $\mathfrak{B}_2(\mathcal{H})$.

The Bochner mean of a random variable defined on a Banach space $\mathcal{X}$ (if it exists) is given by a limit of simple functions, in the sense that choosing some approximating sequence of integrable simple functions $s_n$ satisfying

$$ \lim_{n \to \infty} \int_\Omega \lVert X-s_n \rVert_\mathcal{X} \, \mathrm{d}P = 0 $$

we define

$$ E(X) = \int_\Omega X \, \mathrm{d}P = \lim_{n \to \infty} \int_\Omega s_n \, \mathrm{d}P $$

The Pettis mean of a random variable defined on a Banach space $\mathcal{X}$ (if it exists) is defined to satisfy:

$$ \varphi(E(X)) = E(\varphi(X)) \quad \forall \varphi \in \mathcal{X}^* $$

Question

Let $(\Omega, \mathbb{F}, P)$ be a probability space and consider some random operator $\mathscr{X}: \Omega \to \mathfrak{B}(\mathcal{H})$. My question is about the interaction between evaluating a random operator and finding the expectation of the random operator.

For which of the expectations defined above (if any) does it hold that:

$$ E(\mathscr{X}(h)) = E(\mathscr{X})(h) \quad \forall h \in \mathcal{H} $$

Does it maybe hold if we assume that $\mathscr{X}$ takes values in $\mathfrak{B}_2(\mathcal{H})$ (since this space is also separable and a Hilbert space so the Bochner and Pettis integrals coincide)?

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