Compactness properties of perturbed sub-stochastic \(C_0\)-semigroups on \(L^1(\mu)\) with applications to discreteness and spectral gaps (Q2829379)

Let \(\{U(t): t \geq 0\}\) be a positive \(C_0\)-semigroups of contractions in \(L^1(\Omega,\mu)\) with generator \(T\), where \((\Omega,\mu)\) is an abstract measure space. Denote by \(\{U_V(t): t \geq 0\}\) the (appropriately defined) perturbed semigroup generated by \(T_V :=T - V\), where \(V : (\Omega,\mu)\to {\mathbb R}\) is a nonnegative measurable potential. The main object of the paper under review is to investigate resolvent compactness of \(T_V\), compactness of the perturbed semigroup \(\{U_V(t): t \geq 0\}\), and existence of spectral gaps for perturbed generators \(T_V\). There is an extensive literature devoted to these problems in the Hilbert space \(L^2(\Omega,\mu)\) and the spaces \(L^p(\Omega,\mu)\), \(1<p<\infty\), but the case \(p=1\) was not investigated in previous studies.NEWLINENEWLINEThe main results are formulated in terms of properties of the sublevel sets \(\Omega_M := \{x: V (x) \leq M\}\). In particular, it is proved that weak compactness of \((T-\lambda)^{-1}:L^1(\Omega;\mu)\to L^1(\Omega_M;\mu)\) implies resolvent compactness of the perturbed resolvent \((T_V-\lambda)^{-1}\). The results are concretised further for metric measure spaces \((\Omega,d,\mu)\). The existence of a metric \(d\) allows to define ``thinness at infinity'' of sublevel sets in the sense that, for some point \(\overline{y}\in\Omega\) and all \(r > 0\), NEWLINE\[NEWLINE \mu\{\Omega_M \cap B(y;r)\}\to 0 \quad\text{as}\quad d(y,\overline{y})\to+\infty,NEWLINE\]NEWLINE where \(B(y; r)\) is the ball centered at \(y\) with radius \(r\). If sublevel sets \(\Omega_M\) are ``thin at infinity'', the author proves resolvent compactness of \(T_V\) if \((T-\lambda)^{-1}\) is an integral operator with kernel \(G(x,y)\leq f(d(x,y))\), where \(f : {\mathbb R}_+ \to {\mathbb R}_+\) is nonincreasing and such that (for large \(r\)) \(r \to f(r)\sup\limits_{x\in\Omega} \mu(B(x,r+1))\) is nonincreasing and integrable at infinity.NEWLINENEWLINEIf \((T_V-\lambda)^{-1}\) is not compact, the author investigates the existence (and the size) of spectral gaps on \(L^1(\Omega,d,\mu)\). It is shown how spectral gaps occur when sublevel sets are not ``thin at infinity with respect to \(\left(U(t)\right)_{t\geq 0}\)''. For example, it is proved that \(T_V\) has a spectral gap if the kernel \(G(x,y)\) satisfies the estimate NEWLINE\[NEWLINE\sup\limits_{M>0} \lim\limits_{C\to +\infty} \sup\limits_{y\in\Omega}\int\limits_{\{x\in\Omega_M; d(x,x_0)\geq C\}} G(x,y)\, \mu(dx) < \frac{1}{1-s(T_V)}NEWLINE\]NEWLINE for some \(x_0\in\Omega\), where \(s(T_V):=\sup\{\operatorname{Re} \lambda: \lambda\in\sigma(T_V)\}\).NEWLINENEWLINEIndefinite potentials are also dealt with. The author illustrates the relevance of some aspects of the abstract results by giving compactness and spectral results on convolution semigroups, magnetic Schrödinger semigroups, weighted Laplacians (in particular, the Poincaré inequality for probability measures \(e^{\Phi(x)}dx\) on \({\mathbb R}^N\)), and Witten Laplacians on 1-forms.