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 $$UC_ b (\mathbb{R}^ n)$$.

MSC:

 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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