Semigroups of operators and polynomials of generators of bounded strongly continuous groups. (English) Zbl 0815.47049

Summary: Let \(iA_ j\) \((1\leq j\leq n)\) be commuting generators of bounded strongly continuous groups, \(P(A)= \sum_{| \alpha| \leq m} a_ \alpha A^ \alpha\) \((A^ \alpha= A_ 1^{\alpha_ 1} \dots A_ n^{\alpha_ n})\). By a constructive method, we show that \(P(A)\) generates an analytic semigroup, integrated semigroup or \(C\)-semigroup under different conditions. For \(P(A)= \sum_{| \alpha |\leq m} a_ \alpha (t) A^ \alpha\), we also construct the evolution operator or \(C\)-evolution operator and then show that the initial value problem is well posed. These results can be applied to many differential operators on \(L^ p (\mathbb{R}^ n)\) \((1\leq p< \infty)\), \(C_ 0 (\mathbb{R}^ n)\) and \(UC_ b (\mathbb{R}^ n)\).


47D03 Groups and semigroups of linear operators
34G10 Linear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
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