Different domains induce different heat semigroups on \(C_0(\Omega)\) (Q2708549)

Let \(\Omega_1, \Omega_2 \subset {\mathbb R}^N\) be two open connected sets which are regular in capacity. For \(j=1,2\) let \((T^j_t)_{t\geq 0}\) be the contraction semigroup generated by the Laplacian operator on \(L^2(\Omega_j)\). If there exists a non-zero lattice homomorphism \(U:L^2(\Omega_1)\rightarrow L^2(\Omega_2)\) such that \(UT^1_t = T^2_t U\) (\(t\geq 0\)), the author proves that \(\Omega_1\) and \(\Omega_2\) are congruent. He establishes the same result if the regularity in capacity is replaced by the regularity in the sense of Wiener and \(C_0\) is taken instead of \(L^2\).NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].