IN 1932, Stone [1] gave the first result concerning C<sub>0</sub> semigroups generated by an unbounded operator,which says that a linear operator is the infinitesimal generator of a C<sub>0</sub&g...IN 1932, Stone [1] gave the first result concerning C<sub>0</sub> semigroups generated by an unbounded operator,which says that a linear operator is the infinitesimal generator of a C<sub>0</sub> group of unitary operators on a Hilbert space if and only if it is skew-adjoint. This result has been applied extensively to linear partial differential equations （PDEs） with a law of conservation. Hille [2] then discovered the generation theorem of the Hille-Yosida type for a C<sub>0</sub> group on a Banach space. There are also some conditions under which a C<sub>0</sub> semigroup on a Banach space can be embedded in a C<sub>0</sub> group. We refer the readers to ref. [3] for details of the results mentioned above.展开更多
A method of approaching to the infinite dimensional linear operators by the finite dimensional operators is discussed. It is shown that,for every infinite dimensional operator A and every natural number n, there exist...A method of approaching to the infinite dimensional linear operators by the finite dimensional operators is discussed. It is shown that,for every infinite dimensional operator A and every natural number n, there exists an n dimensional optimal approximation to A. The norm error is found and the necessary and sufficient condition for such n dimensional optimal approximations to be unique is obtained.展开更多
文摘IN 1932, Stone [1] gave the first result concerning C<sub>0</sub> semigroups generated by an unbounded operator,which says that a linear operator is the infinitesimal generator of a C<sub>0</sub> group of unitary operators on a Hilbert space if and only if it is skew-adjoint. This result has been applied extensively to linear partial differential equations （PDEs） with a law of conservation. Hille [2] then discovered the generation theorem of the Hille-Yosida type for a C<sub>0</sub> group on a Banach space. There are also some conditions under which a C<sub>0</sub> semigroup on a Banach space can be embedded in a C<sub>0</sub> group. We refer the readers to ref. [3] for details of the results mentioned above.
文摘A method of approaching to the infinite dimensional linear operators by the finite dimensional operators is discussed. It is shown that,for every infinite dimensional operator A and every natural number n, there exists an n dimensional optimal approximation to A. The norm error is found and the necessary and sufficient condition for such n dimensional optimal approximations to be unique is obtained.
