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L₁-uniqueness of degenerate elliptic operators

Derek W. Robinson;  Adam Sikora — 2011

Studia Mathematica

Let Ω be an open subset of ℝ d with 0 ∈ Ω. Furthermore, let H Ω = - ∑ i , j = 1 d ∂ i c i j ∂ j be a second-order partial differential operator with domain C c ∞ ( Ω ) where the coefficients c i j ∈ W l o c 1 , ∞ ( Ω ̅ ) are real, c i j = c j i and the coefficient matrix C = ( c i j ) satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If ∫ 0 ∞ d s s d / 2 e - λ μ ( s ) ² < ∞ for some λ > 0 where μ ( s ) = ∫ 0 s d t c ( t ) - 1 / 2 then we establish that H Ω is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup. Moreover...

A C * -algebraic Schoenberg theorem

Ola Bratteli;  Palle E. T. Jorgensen;  Akitaka Kishimoto;  Donald W. Robinson — 1984

Annales de l'institut Fourier

Let 𝔄 be a C * -algebra, G a compact abelian group, τ an action of G by * -automorphisms of 𝔄 , 𝔄 τ the fixed point algebra of τ and 𝔄 F the dense sub-algebra of G -finite elements in 𝔄 . Further let H be a linear operator from 𝔄 F into 𝔄 which commutes with τ and vanishes on 𝔄 τ . We prove that H is a complete dissipation if and only if H is closable and its closure generates a C 0 -semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative definite...

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