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HILLE - YOSIDA Cantate Theorem, Proof, and Applications

HILLE - YOSIDA Cantate Theorem, Proof, and Applications. CHORAL [Theorem]. (1) Let A be closed and linear with dense domain in X. It generates a semigroup and we can all relax. It does so if it fulfills - that is known to everybody. the condition of Hille and Yosida. RECITATIV [Proof].

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HILLE - YOSIDA Cantate Theorem, Proof, and Applications

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  1. HILLE - YOSIDA Cantate Theorem, Proof, and Applications

  2. CHORAL [Theorem]

  3. (1) Let A be closed and linear with dense domain in X.

  4. It generates a semigroup and we can all relax.

  5. It does so if it fulfills - that is known to everybody

  6. the condition of Hille and Yosida.

  7. RECITATIV [Proof]

  8. For the proof of the just heard Theorem of Hille and of Yosida

  9. we first remark the following:

  10. The existence of the semigroupis shown by bounded approximation of A.

  11. Thus, a sequence of uniformly continuous semigroups is obtained, which converges to the semigroup T.

  12. It remains to show, that A is the generator of this semigroup.

  13. This, however, is left to the reader as an exercise.

  14. CHORAL [Quod erat demonstrandum]

  15. ARIE [Applications]

  16. Oh, what a lovely theorem! It can be used every day.

  17. By this theorem, generators are identified by properties of theirresolvents.

  18. Many bad PDE´s are solved by that theorem.

  19. In physics and also in space travel there is no successwithout Hille- Yosida.

  20. Therefore, the Theorem is beautiful and true.

  21. FINALE [Applications II]

  22. Let us now generate without hesitation; approximation,and rescaling is now trivial.

  23. 2. Now, everybody can solve the problems, good or evil,approximation, and rescaling is now trivial.

  24. 3. Long live Hille and Yosida, since they showed us this theorem; approximation, and rescaling is now trivial.

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