Semigroups for Generalized Birth-and-Death Equations in lp Spaces.

We prove the existence of C₀ semigroups related to some birth-and-death type infinite systems of ODEs with possibly unbounded coefficients, in the scale of spaces l^p, $1\leq p<\infty.$ For some particular cases we also provide a characterization of the spectra of their generators. For the proof of the generation theorem in the case p > 1 we extend the Chernoff perturbation result ([9]) on relatively bounded perturbations of generators. The results presented here have been used in [5] and they play important role for analysing chaoticity of dynamical systems considered there. As a by-product of our approach we obtain a result related to the classical Shubin theorem [20]. We show that this theorem, saying that for a class of bounded infinite matrices the spectrum of the corresponding maximal operator in l^p is independent on $p\in [1,\infty),$ cannot be extended to unbounded matrices. [ABSTRACT FROM AUTHOR]