math.FA

Ordered realizations of discrete POVMs are studied through a residual transform generated by sequential tests. One application of the transform replaces each coordinate by the effect obtained after all earlier tests have failed, and appends the remaining mass as a terminal outcome. Under natural hypotheses, iterating the transform produces a collapsed POVM whose non-escape coordinates are the parts of the original effects that survive all earlier tests. The resulting collapse map gives an equivalence relation on ordered POVM realizations. Its range and fibers are characterized. The range consists of collapsed POVMs, whose non-escape coordinates are mutually orthogonal and whose support projections strongly sum to the identity. The fiber over a collapsed POVM consists of all ordered realizations with the same residually visible compressions. In particular, different ordered realizations, including ones with different off-diagonal coupling data, can have the same collapsed image. After collapse, the non-escape coordinates are fixed under further residual iteration. The remaining dynamics takes place in the escape effect, which is fragmented by a universal scalar functional calculus.

We establish the following result, confirming a conjecture of Jean Esterle. For each closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ with the following property: if $T$ is a contraction on a Hilbert space such that $\sigma(T)\subset E$ and $\|T^{-n}\|=O(u_n)$ as $n\to\infty$, then $T$ is a unitary operator. A key tool used in the proof is a result generalizing the well-known fact that closed subsets $E$ of the real axis of Lebesgue measure zero are removable for bounded holomorphic functions. We show that such sets remain removable even for certain unbounded holomorphic functions of moderate growth near $E$, where the notion of `moderate' depends on $E$.