Abstract
Some recent results on n-times integrated C-semigroups and C-cosine functions of hermitian and positive operators on Banach spaces are discussed. The following are some interesting properties: 1) A hermitian n-times integrated C-semigroup T(ּ) (resp. C-cosine function C(ּ)) is (n — 1)-th continuously differentiable in operator norm on [0, ∞) and T(n-1) (•) (resp. C (n-1)(ּ)) is a norm continuous hermitian once integrated C-semigroup (resp. C-cosine function) if n ≥ 2; 2) A hermitian C-semigroup is infinitely differentiable in operator norm on (0, ∞); 3) A hermitian C-cosine function is norm continuous either at all points or at none point of [0, ∞); 4) A positive C-semigroup (resp. C-cosine function) which dominates C is infinitely differentiable in operator norm on [0, ∞); moreover, if it is nondegenerate, then its generator A must be bounded and \(T(t) = \sum\nolimits_{n = 0}^\infty {\frac{{{t^n}}}{{n!}}} {A^n}C (resp. C(t) = \sum\nolimits_{n = 0}^\infty {\frac{{{t^{2n}}}}{{(2n)!}}{A^n}C);} 5) \) Hermitian n-times integrated semigroups and hermitian n-times integrated cosine functions are exponentially bounded.
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Li, YC., Shaw, SY. (2000). Integrated C-Semigroups and C-Cosine Functions of Hermitian and Positive Operators. In: Balakrishnan, A.V. (eds) Semigroups of Operators: Theory and Applications. Progress in Nonlinear Differential Equations and Their Applications, vol 42. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8417-4_17
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DOI: https://doi.org/10.1007/978-3-0348-8417-4_17
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