Random Operator-Valued Frames in Hilbert Spaces
James Tian
TL;DR
The paper develops an operator-theoretic, randomized framework for constructing operator-valued frames via i.i.d. random positive contractions. It proves convergence to the identity through a Kaczmarz-like iteration and establishes a pathwise energy decomposition that yields random Parseval frames in projection cases, with nonasymptotic, anytime bounds. A residual-weighted scheme is introduced, delivering an exact telescoping identity and geometric residual decay under a mean-coercivity condition, enabling almost-sure Parseval frames for general contractions and providing explicit rates and lower frame bounds. The theory is illustrated with rank-one projections and extended to fractal settings such as Cantor-type measures, highlighting robust, adaptive normalization in a broad Hilbert-space context. These results connect randomization, frame theory, and dilation theory to produce constructive, scalable random frames with strong probabilistic guarantees.
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.