Generalized Resolvent Equations and Unsymmetric Dirichlet Spaces

Extract

Let (X, , μ) and (X, , μ′) be measure spaces with the measures μ and μ′ totally finite. Suppose {U_λ: λ > 0} is a family of positive (i.e., ϕ ≧ 0 ⇒ U_λϕ ≧ 0) continuous linear operators from L²(X, dμ′) to L²(X,dμ) with the following additional properties: if ϕ ≧ 0 then U_λϕ is non-decreasing as λ increases, while λ⁻¹U_λϕ is nonincreasing.

A family {M_λ:λ > 0} of continuous linear operators from L²(X, dμ) to L²(X, dμ′) satisfies the “generalized resolvent equation” relative to {U_λ} if

(0.1)

for positive λ and v. If U_λ = λI, then (0.1) is just the well-known resolvent equation. The family {M_λ} is called submarkov if M_λ is a positive operator and

(0.2)

it is conservative if

(0.3)