Research proof index

𝓜 μ (f + g) extends 𝓜 μ f + 𝓜 μ g. In general the domains do not match: ψ ∈ (𝓜 μ f + 𝓜 μ g).domain amounts to MemHS (f • ψ) μ and MemHS (g • ψ) μ whereas ψ ∈ (𝓜 μ (f + g)).domain is equivalent to the weaker condition MemHS ((f + g) • ψ) μ. See mulOperator_add_eq for a sufficient condition to ensure equality.

The adjoint of a multiplication operator is again a multiplication operator.

(𝓜 μ g).domain = ⊤ is a sufficient condition to ensure equality in mulOperator_compRestricted_ge.

𝓜 μ (f • g) extends 𝓜 μ f * 𝓜 μ g. In general the domains do not match: ψ ∈ (𝓜 μ f * 𝓜 μ g).domain amounts to MemHS (g • ψ) μ and MemHS (f • g • ψ) μ whereas ψ ∈ (𝓜 μ (f • g)).domain only requires MemHS (f • g • ψ) μ. See mulOperator_compRestricted_eq for a sufficient condition to ensure equality.

The multiplication operator corresponding to a function of temperate growth contains all Schwartz maps in its domain.

The multiplication operator corresponding to a μ-a.e. strongly measurable function is densely defined.