| Abstract: |
| We present a simple method for proving generation results for strongly continuous semigroups and cosine functions via resolvent decomposition. Let \(A\) be a closed linear operator on a Banach space and suppose that the domain of \(A\) consists of elements \(f\) satisfying a boundary/transmission condition of the form \(\Phi f = 0\) for a linear functional \(\Phi\). In many concrete problems, \(\Phi\) is a finite sum of simpler functionals, say \(\Phi = \sum \phi\). Under a mild nondegeneracy condition on the kernel of \(\lambda - A\), we show that for every \(\lambda\) in the resolvent sets of all \(A_{|\ker \phi}\) we have
\[
(\lambda - A)^{-1} = \sum_{\phi} \alpha_{\lambda,\phi} (\lambda - A_{|\ker \phi})^{-1}
\]
with \(\alpha_{\lambda,\phi} = \phi h / \Phi h\), where \(h\) is a fixed nonzero element of the kernel of \(\lambda - A\). That is, the resolvent of A is an affine (or even convex) combination of simpler resolvents. Combining this identity with standard criteria (the Hille--Yosida theorem for semigroups and the Sova--Da Prato--Giusti theorem for cosine functions) yields a family of generation theorems: if each \(A_{\phi}\) is a generator and the coefficients \(\alpha_{\lambda,\phi}\) satisfy natural regularity conditions, then \(A\) is a generator as well.
As applications, we obtain generation results beyond the purely Markov/Feller setting and prove the existence of cosine functions associated with skew and snapping out Brownian motions on star graphs. Moreover, we show that skew Brownian motion can be approximated by snapping out Brownian motion, illustrating how resolvent decompositions translate structural boundary information into concrete evolution families.
{\bfseries References}
[1] A.~Gregosiewicz, \emph{Resolvent decomposition with applications to semigroups and cosine functions}, Math. Ann. \textbf{391} (2025), 4011--4035. |
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