By H B. Phillips
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Extra info for Differential Equations, 1937
We recall t h a t a b o u n d e d o p e r a t o r A is said to be : (i) semi-Fredholm iff Ker A is f i n i t e d i m e n s i o n a l a n d Rg A is closed ; (it) Fredholm iff A is s e m i = F r e d h o l m a n d t h e c o d i m e n s i o n of Rg A is f i n i t e ; A F r e d h o l m o p e r a t o r m a y also be called a n index operator, or a noetherian operator (in e n g l i s h t r a n s l a t i o n s of russian texts). We w a n t to d e t e r m i n e c o n d i t i o n s to h a v e (FI), resp.
2m one has: d l m P x = d t m Q x if: (f2) actually contains all s w h i c h are not h a l f - i n t e g e r s . }. e. ,2m-1 and ~m(O) contains all integers ~ A d e v o t e d to t h e Laplace o p e r a t o r . 15) by a double a p p l i c a t i o n : • t o L, • to e a c h '~-'x" To u n d e r s t a n d t h a t , it is e n o u g h to n o t i c e t h a t : • t h e s i n g u l a r c h a i n s (x t ..... xj) of fix, w h e n c o m p l e t e d b y x : (x,x I..... xj), c o n s t i t u t e t h e singular c h a i n s of f , • t h e split o p e r a t o r s associated to ~£'x on fix are split o p e r a t o r s associated to L on f . §7 S t r o n g l y elliptic s y s t e m s We extend to strongly elliptic systems, the results w e have stated in ~4, 5, 6 .
A d e v o t e d to t h e Laplace o p e r a t o r . 15) by a double a p p l i c a t i o n : • t o L, • to e a c h '~-'x" To u n d e r s t a n d t h a t , it is e n o u g h to n o t i c e t h a t : • t h e s i n g u l a r c h a i n s (x t ..... xj) of fix, w h e n c o m p l e t e d b y x : (x,x I..... xj), c o n s t i t u t e t h e singular c h a i n s of f , • t h e split o p e r a t o r s associated to ~£'x on fix are split o p e r a t o r s associated to L on f . §7 S t r o n g l y elliptic s y s t e m s We extend to strongly elliptic systems, the results w e have stated in ~4, 5, 6 .