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Infinitesimal generators and quasi-units in potential theory

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  • Physical Sciences: Mathematics
  • Mathematics


The underlying structure is taken as a strongly superharmonic cone [unk], defined as a partially ordered abelian semigroup with identity 0 which admits a multiplication by nonnegative scalars and satisfies two fundamental axioms of a potentialtheoretic character. In terms of a fixed nonzero element e there is introduced on [unk] a one-parameter family of nonlinear operators Sλ (λ ≥ 0) closely connected with the abstract theory of quasibounded and singular elements. The semigroup {Sλ} admits an infinitesimal generator A, and the elements invariant under A, called quasi-units, generalize the Yosida quasi-units in the theory of Riesz spaces. Quasi-units in [unk] are studied, both from a potentialtheoretic and a function-alanalytic viewpoint, culminating in a spectral representation theorem for quasi-bounded elements which extends the classical Freudenthal spectral theorem of Riesz space theory.

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