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Regularized cosine functions and polynomials of group generators. (English) Zbl 0973.47032

Summary: Let \(iA_j\) \((1\leq j\leq n)\) be commuting generators of bounded strongly continuous groups and \(P(A)=\sum_{|\alpha |\leq m} a_\alpha A^\alpha\) \((A^\alpha =A_1^{\alpha_1} \cdots A_n^{\alpha_n})\). It is proven that \(\overline {P(A)}\) generates an exponentially bounded regularized cosine function and an \(l\)-times integrated cosine function under suitable conditions on the polynomial \(P(\xi)\). Then, these results are applied to the partial differential operators \(P(D)\) \((iA_j= \partial/ \partial x_j\), \(1\leq j\leq n)\) on spaces like \(L^p(\mathbb{R}^n)\) \((1\leq p< \infty)\), \(C_0(\mathbb{R}^n)\) and \(BUC (\mathbb{R}^n)\).

MSC:

47D09 Operator sine and cosine functions and higher-order Cauchy problems
47D62 Integrated semigroups
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