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arXiv:2610.33821

Banach-Cesaro positive operator-valued measure

Sviatoslav V. Dzhenzher

math.FAquant-phSubmitted Oct 24, 2026• Announced Oct 27, 2026
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Abstract

We construct a family of bounded, finitely additive, covariant positive operator-valued measures (POVMs) on the real line by applying generalised Banach-Cesaro asymptotic averaging to locally finite covariant POVMs. In contrast to the strictly shift-invariant case, where countably additive operator measures are restricted to scalar multiples of a single fixed operator, the introduced covariant setting allows for non-commuting averaged operator values while successfully normalising the total mass. We develop an integration theory for complex-valued functions by taking the completion of simple integrable functions with respect to the continuous operator norm induced by the positive integral of their absolute values. The integration spaces for the original and the averaged measures are shown to possess distinct null classes. Finally, we introduce the operator-valued Fourier transform on these integration spaces and analyze the core algebraic constraints imposed by the covariance property on the exponential multipliers.

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8 pages

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In the nonrelativistic setting with finitely many canonical degrees of freedom, a shift-covariant phase space observable is uniquely characterized by a positive operator of trace one and, in turn, by the Fourier-Weyl transform of this operator. We study three properties of such observables, and characterize them in terms of the zero set of this transform. The first is informational completeness, for which it is necessary and sufficient that the zero set has dense complement. The second is a version of informational completeness for the Hilbert-Schmidt class, equivalent to the zero set being of measure zero, and the third, known as regularity, is equivalent to the zero set being empty. We give examples demonstrating that all three conditions are distinct. The three conditions are the special cases for p = 1, 2, ∞ of a more general notion of p-regularity defined as the norm density of the span of translates of the operator in the Schatten-p class. We show that the relation between zero sets and p-regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis.