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| Médium: | Recurso digital |
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Zenodo
2026
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| On-line přístup: | https://doi.org/10.5281/zenodo.19879889 |
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- <p>This paper does not introduce a new scalar Fourier-multiplier semigroup theorem. Starting<br>from standard branchwise L2 Fourier-multiplier semigroup ingredients, we prove a compatibil-<br>ity and normal-form realization theorem for one fixed real branch algebra A = R[u]/(u4 − 1)<br>on L2(Rd; A). The contribution is to show that the intrinsic branch reduction of A, the<br>three reduced multiplier semigroups, the reconstructed A-valued flow, the relative symbol-<br>sum generator, the graph-core comparison with the componentwise strong prototype, the<br>coordinate-exact branch identities in the fixed branch-adapted Hilbert norm, and the bounded-<br>bulk observation pushforward all refer to the same operator-theoretic object. Thus the result<br>is a branch-algebraic compatibility theorem inside a standard L2 multiplier setting, not a<br>replacement for scalar multiplier semigroup theory or for general vector-valued multiplier<br>theory.<br>The central operator-domain theorem is the maximal finite-a.e. symbol-sum graph-<br>core identity. The non-tautological point is not scalar multiplier generation itself, and<br>not a theorem about arbitrary unbounded sums, but the comparison between the raw<br>componentwise prototype and the closed relative symbol-sum multiplier realization inside<br>the prescribed L2 multiplier model. The former may have a strictly smaller domain because<br>it asks for a(D) and b(D) separately, while the latter is governed by the reduced sums<br>a + b, a − b, a + ib.<br>The paper proves, before the reconstruction theorem is used, that the raw prototype is<br>nevertheless a graph core for the closed maximal finite-a.e. symbol-sum realization. The<br>intrinsic plus, minus, and realified complex branches reduce the formal multiplier<br>Gform = IA ⊗ a(D) + Lu ⊗ b(D)<br>to the branch symbols λ+ = a + b, λ− = a − b, and the complex-coordinate representative<br>λc = a + ib. Throughout the paper the multiplier symbols are ordinary measurable finite-<br>a.e. complex-valued functions; extended-valued symbols or symbols singular on sets of<br>positive measure are not part of the theorem package. The last symbol is the transported<br>scalar coordinate of the real two-dimensional Ac-branch through Ic; the precise normal-form<br>convention is fixed once in Proposition 3.5.<br>Under the reduced growth hypotheses, the three branch operators generate C0-semigroups<br>by standard multiplier-semigroup arguments. These hypotheses are used as a sufficient<br>Hilbert-space multiplier package for this branch-adapted L2 theory, not as a minimal or<br>classification-level condition for all possible multiplier realizations. For the prescribed L2<br>multiplier formula itself, Lemma 2.11 records the matching near-necessity: boundedness of<br>etλ(D) at one positive time is equivalent to an upper essential bound on ℜλ. Their prescribed<br>direct-sum assembly gives the branch-coordinate A-valued flow. The same flow is also<br>identified explicitly as the L2 Fourier multiplier with finite-dimensional matrix symbol<br>exp(t(a(ξ)IAC + b(ξ)Lu)),<br>whose branch-coordinate blocks are etλ+ , etλ− , and, after the global real c-branch coordinate<br>map, etλc . The reconstructed generator agrees with the relative symbol-sum Fourier-multiplier<br>1<br>realization. The reconstruction theorem itself is stated without a global uniqueness claim; a<br>separate corollary records only the internal uniqueness inside the imposed branch-compatible<br>class. All exact operator-norm and graph-norm identities are stated with respect to the<br>fixed branch-adapted Hilbert norm; under an arbitrary equivalent coefficient norm, the<br>corresponding statements become two-sided estimates with equivalence constants rather than<br>literal isometries.<br>For the observation layer, we restrict throughout to bounded bulk maps<br>O ∈ L(L2(Rd; A), Z).<br>This layer is deliberately modest: it records bounded-bulk pushforward, masking, separated-<br>channel bookkeeping, and inverse-on-range consequences of sensor nondegeneracy estimates<br>that are assumed or verified separately. It does not prove an exact observability theorem,<br>and it does not derive sensor lower bounds from the multiplier dynamics alone. To keep this<br>conditional hypothesis from being merely a canonical projector in disguise, Section 6 includes<br>a Fourier-frame branch-sensor example: a finite family of multiplier windows satisfying a<br>pointwise frame lower bound gives a concrete bounded-below separated branch sensor by<br>Plancherel. All observation estimates involving ω+, ω−, and ωc are upper-envelope estimates<br>transported through the chosen bounded sensor; they are not datum-specific actual observed<br>growth rates and they are not asserted to be sharp after observation. Boundary observations,<br>point sensors, trace maps, unbounded admissible observations, time-dependent sensors,<br>control-theoretic observability estimates, variable coefficients, nonlinear perturbations, and<br>inverse identification are outside this paper</p>