Random Operator-Valued Frames in Hilbert Spaces

TL;DR

This paper introduces new random operator-valued frames in Hilbert spaces through iterative methods, demonstrating convergence properties and providing conditions for constructing Parseval frames with strong probabilistic guarantees.

Contribution

It presents novel iterative algorithms for constructing random operator-valued frames, including a Kaczmarz-like method and a residual-weighted iteration, with proven convergence and frame properties.

Findings

Convergence of the Kaczmarz-like iteration in mean square and almost surely.

Residual-weighted iteration satisfies an exact identity and converges geometrically.

Constructs almost-sure Parseval frames for sequences of positive contractions.

Abstract

We study strongly measurable random bounded operators on separable Hilbert spaces and analyze two simple iterations driven by independent random positive contractions. The first, a Kaczmarz-like iteration, converges in mean square and almost surely and produces a random operator-valued frame. In the projection case it yields a Parseval identity. The second, a residual-weighted iteration, enjoys an exact step-by-step identity: the accumulated analysis terms plus a residual equal the identity operator. Under a mild mean-coercivity condition, the residual shrinks at a geometric rate in expectation, vanishes almost surely, and admits nonasymptotic tail bounds. As a result, the construction delivers an almost-sure Parseval frame for any independent sequence of positive contractions, not only projections.