Resolvent decomposition with applications to semigroups and cosine functions

Abstract For an operator whose domain is the kernel of a functional that is a finite sum of simpler functionals, we show that the resolvent of the operator can be decomposed as an affine combination of resolvents associated with these simpler functionals. Specifically, given an operator A and a finite number of functionals $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> defined on the domain of A , we prove that the resolvent $$(\lambda - A_{|\ker \sum \phi })^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>ker</mml:mo> <mml:mo>∑</mml:mo> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> is an affine combination of resolvents $$(\lambda - A_{|\ker \phi })^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>ker</mml:mo> <mml:mi>ϕ</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> , provided that the sum of the functionals is nontrivial on the kernel of $$\lambda - A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> . We use this result to derive generation theorems for semigroups and cosine functions.As applications, we prove that there are cosine functions associated with skew and snapping out Brownian motions on star graphs, and that skew Brownian motion can be approximated by snapping out Brownian motion.