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Evolution systems and perturbations of generators of strongly continuous groups

Authors:

Email author" target="_blank">Ralph?deLaubenfels Email author

Institution:

(1) Scientia Research Institute, 22000 Jordan Run Road, Guysville, Ohio 45735, USA

Abstract:

Suppose A generates a strongly continuous linear group $\{ e^{tA} \} _{t \in {\mathbf{R}}}$ on a Banach space X and B is a linear operator on X. It is shown that an extension of $$(A + B)$$

generates a strongly continuous semigroup if and only if the family of operators $\{ e^{ - rA} Be^{rA} \} _{r \geq 0}$ has an appropriate evolution system. This produces simple sufficient conditions for an extension of $$(A + B)$$

to generate a strongly continuous semigroup, including

(1)

$\pm A,B$ being m-dissipative and $r \mapsto e^{ - rA} Be^{rA} x \in C^1 (0,\infty ),X),$ for all x in the domain of B; or

(2)

$\pm A,B$ being m-dissipative and $\{ e^{ - rA} Be^{rA} \}$ being a commuting family of operators with

$\left\{ {x \in \bigcap\limits_{r \geq 0} {\mathcal{D}(e^{ - rA} Be^{rA} )|r} \mapsto e^{ - rA} Be^{rA} x \in C(0,\infty ),X)} \right\}$

dense. This is applied to many differential operators; for at least one class of applications, the semigroup is generated by the closure of $$(A + B)$$

and the equivalence between semigroups and evolution systems enables us to construct it explicitly. In all the applications, including the sufficient conditions (1) and (2) above, the semigroup generated by an extension of $$(A + B)$$

is given by the Trotter product formula

$T(t)x = \mathop {\lim }\limits_{n \to \infty } (e^{\frac{t} {n}B} e^{\frac{t} {n}A} )^n x\quad (t \geq 0,x \in X).$

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 47A55 47D03 47F05 Secondary 47D06 34G10

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