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# lecture16_17 - Lectures 16 and 17 16.3 Fredholm Operators A...

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Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact op- erators is the closure of the set of finite rank operator in operator norm. In this sense compact operator are similar to the finite dimensional case. One property of finite rank operators that does not generalize to this setting is theorem from linear algebra that if T : X Y is a linear transformation of finite dimensional vector spaces then dim ( ker ( T )) dim ( Coker ( T )) = dim ( X ) dim ( Y ). 35 Of course if X or Y is infinite dimensional then the right hand side of equal- ity does not make sense however the stability property that the equality implies could be generalized. This brings us to the study of Fredholm operators. It turns out that many of the operators arising naturally in geometry, the Laplacian, the Dirac operator etc give rise to Fredholm operators. The following is mainly from H ¨ormander Definition 16.13. Let X and Y be Banach spaces and let T : X Y be a bounded linear operator. T is said to be Fredholm if the following hold. 1. ker ( T ) is finite dimensional. 2. Ran ( T ) is closed. 3. Coker ( T ) is finite dimensional. If T is Fredholm define the index of T denoted Ind ( T ) to be the number dim ( ker ( T )) dim ( Coker ( T )) First let us show that the closed range condition is redundant. Lemma 16.14. Let T : X Y be a operator so that the range admits a closed complementary subspace. Then the range of T is closed. Proof: C be a closed complement for the range. We can assume that T is injective since ker ( T ) is a closed subspace and hence X / ker ( T ) is a Banach space so we can replace T by the induced map from this quotient. Now consider the map S : X C Y defined by S ( x , c ) = T ( x ) + c . S is bounded linear isomorphism and hence by the open mapping theorem S is a topological isomorphism. Thus Ran ( T ) = S ( X ⊕ { 0 } ) is closed. . An important result that will be used over and over again is the openness of invertibility in the operator norm.

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