Existence conditions for a positive semigroup to possess a positive eigenvalue (Q1916588)

Let \(T(t)\) be a positive semigroup belonging to the class \(C_0\) on \(L_p(D)\), \(1\leq p<\infty\), with \(D\subset\mathbb{R}^k\). We denote by \(r_t\) the spectral radius of the operator \(T(t)\). Theorem 1. Suppose that for some \(t>0\) there exists a positive \(\beta_t\) less than \(r_t\) such that the ring \(S(\beta_t, r_t)= \{\beta_t\leq |\lambda|\leq r_t\}\) contains a finite number of points of the spectrum of the operator \(T(t)\), and that these points of the spectrum are normal points. Then: 1) The operators \(T(s)\) and \(T^*(s)\), \(s>0\), possess nonnegative eigenfunctions \(\psi_0\in L_p\), \(\psi^*_0\in L_q\), \(q^{-1}+ p^{-1}= 1\) at the point \(r_s\). 2) For all \(s>0\), \(r^{1/2}_s=\text{const}\). 3) For all \(s>0\), there is a finite number \(N(s)\) of points of the spectrum of the operator \(T(s)\) in the ring \(S(\beta^{s/t}_t, r_s)\), with \(\sup_sN(s)<\infty\), and these points of the spectrum are normal. A positive semigroup \(T(t)\) is said to be irreducible if for any \(x\in L_p\), \(x\geq 0\), \(|x|\neq 0\); \(y\in L_q\), \(y\geq 0\), \(|y|\neq 0\), there exists a number \(s\) such that \(\langle y,T(s)x\rangle>0\). Theorem 2. Suppose that the semigroup \(T(t)\) is irreducible and that the conditions of Theorem 1 are satisfied. Then: 1) The eigenfunctions \(\psi_0\) and \(\psi^*_0\) are positive a.e. 2) For \(s>0\), the eigenvalue \(r_s\) is of algebraic multiplicity one. 3) There exists a number \(\delta_t>0\), such that for \(s>0\), the point \(r_s\) is the only eigenvalue of the operator \(T(s)\) outside a circle of radius \((r_t- \delta_t)^{s/t}\). Theorems 1 and 2 may be used whenever the operators \(T(s)\) are not ``\(u^0\) bounded''.