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The monograph (71), §A gives an interesting account of Fredholm theory for linear operators on linear spaces with no reference to topology. Related work is due to Kroh (54). P3izori-Oblak (59) studies elements of a Banach algebra whose left regular representations are Fredholm operators. If one is concentrating on an ideal F the choice of the ideal is important. This is exhibited by Yang (98) who studies operators on a Banach space invertible modulo the closed ideal of weakly compact operators.

So there exists t4i(xzy)n = i(x)C # 0 are two linear spaces we write such so dim(X) = dim(Y) xAy / (0) to mean that either the spaces are both infinite dimensional or they have the same finite dimension. 1 (i) LEMMA. Let there exist e, f E Min(A) u, v E A and R such that be a right ideal of A, then f = uev; (ii) dim(eAf) = 1; (iii) (iv) Proof. Since dim(Re) = dim(Rf); dim(Ae/Re) = dim(Af/Rf). (i) Af Choose a non-zero v c eAf. We have observed that eAf 34 (0). is a minimal left ideal, Af = Av, so f = uv for some u E A.

Is a Fredholm operator on n(R) = rank(q) = rank(q) < therefore A which A with Fredholm operators on We use the Barnes idempotents ker(x) so is invertible since the latter ideal is closed in k(h(soc(A))). invertibility modulo Proof. 4 invertibility modulo soc(A) is equivalent to A = (D(A)). 6 x c $(A)<-; (We often implicitly make this assumption, otherwise modulo soc(A). A D(A). 2 shows that the converse of this theorem is false. 7 of DEFINITION. If we define the nuZZity, defect and index X E O(A) x by nul (x) Now if = def (x) n (x) , x E (A) , R q, p d (x) , ind (x) is a Fredholm operator on nullity, defect and index; where = further, = i (x).

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A course of higher mathematics, vol. 1 by Smirnov V.I.

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