Chapter I.

Fredholm mappings of index zero and linear boundary value problems

1. Definitions . Le t X, Z b e real normed spaces , and denot e b y |- | the correspondin g

norms. A linear mappin g

L\ do m L C X - * Z ,

with ker L = L"

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{0 } an d Im L = L(do m L) , will be calle d a Fredholm mapping if th e

following tw o condition s hold :

(i) ke r L ha s a finite dimension ;

(ii) I m L i s closed an d has a finite codimension .

Recall that th e codimensio n o f I m L i s the dimensio n o f Z/Im I, i.e . the dimensio n o f th e

cokernel coke r L o f L.

When L i s a Fredholm mapping , its (Fredholm) index i s the intege r

Ind L = di m ker L - codi m I m L.

EXAMPLES.

W e now giv e some classical examples of Fredhol m mappings .

(1) 0: X—• Z is a Fredholm mappin g if an d onl y i f X an d Z have finit e dimension ,

in which case

Ind 0 = di m X - di m Z.

(2) / : X —•* • X i s a Fredholm mappin g of inde x zero, and, more generally , the sam e is

true fo r / + C with C: X — • X linea r an d compac t (i.e . such tha t C takes the unit bai l of

X int o a compact set) .

(3) I f L: do m L C X — • Z is one-to-one an d onto , then L i s a Fredholm mappin g of

index zero .

(4) I f X an d Z ar e finit e dimensional , every linear mappin g L: X — • Z i s a Fredhol m

mapping o f index di m X - di m Z.

It follow s no w fro m th e definitio n abov e o f a Fredholm mappin g an d fro m basi c re-

sults of linea r functiona l analysi s that ther e exis t continuou s projector s

P.X-+X, Q.Z-+Z

such tha t

Im P = ke r L, ke r Q = I m L

so tha t

X = ke r L ® ker P, Z = I m L ® Im Q

as topological direc t sums . Consequently , th e restrictio n L

p

o f L t o do m L n ke r P is one-

to-one an d ont o I m L, s o that it s (algebraic) inverse Kp: I m L — • dom L n ke r P is defined .

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http://dx.doi.org/10.1090/cbms/040/03