On the inverse best approximation property of systems of subspaces of a Hilbert space

Abstract

Let $H$ be a Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$. Denote by $P_k$ the orthogonal projection onto $H_k$, $k=1,2,...,n$. Following Patrick L. Combettes and Noli N. Reyes, we will say that the system of subspaces $H_1,...,H_n$ possesses the inverse best approximation property (IBAP) if for arbitrary elements $x_1\in H_1,...,x_n\in H_n$ there exists an element $x\in H$ such that $P_{k}x=x_k$ for all $k=1,2,...,n$. We provide various new necessary and sufficient conditions for a system of $n$ subspaces to possess the IBAP. Using the main characterization theorem, we study properties of the systems of subspaces which possess the IBAP, obtain a sufficient condition for a system of subspaces to possess the IBAP, and provide examples of systems of subspaces which possess the IBAP. These results are applied to a problem of probability theory. Let $(\Omega ,\mathcal{F},\mu )$ be a probability space and ${\mathcal{F}}_1,...,{\mathcal{F}}_n$ be sub-$\sigma$-algebras of $\mathcal{F}$. We will say that the collection ${\mathcal{F}}_1,...,{\mathcal{F}}_n$ possesses the inverse marginal property (IMP) if for arbitrary random variables ${\xi }_1,...,{\xi }_n$ such that (1) ${\xi }_k$ is ${\mathcal{F}}_k$-measurable, $k=1,2,...,n$; (2) $E|{\xi }_k{|}^2<\infty$, $k=1,2,...,n$; (3) $E{\xi }_1=E{\xi }_2=...=E{\xi }_n$, there exists a random variable $\xi$ such that $E|\xi {|}^2<\infty$ and $E(\xi |{\mathcal{F}}_k)={\xi }_k$ for all $k=1,2,...,n$. We will show that a collection of sub-$\sigma$-algebras possesses the IMP if and only if the system of corresponding marginal subspaces possesses the IBAP. We consider two examples; in the first example $\Omega =\mathbb{N}$, in the second example $\Omega =[a,b)$. For these examples we establish relations between the IMP, the IBAP, closedness of the sum of marginal subspaces and "fast decreasing" of tails of the measure $\mu$.