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30 Aug 2019: Part III, Paper 15. 3. 3. (i) Define Fredholm map, Fredholm operator, Fredholm index. ... iii) Prove that s As is Fredholm.[You may quote the Closed Range Lemma without proof].

17 Jun 2019: Define a Fredholm operator T , on a Hilbert space H, and define the index of T. ... not using anytheorems about Fredholm operators) that, for each k Z, there is a Fredholm operator Son H with ind S = k.

30 Aug 2019: 2. 1. (a) Define what is meant by a Fredholm operator of index n on Hilbert space. ... b) Show that an operator is Fredholm if and only if it is invertible modulo thecompact operators.

17 Jun 2019: Define what it meansto say that T is (i) compact, and (ii) Fredholm. ... Prove that S and T are Fredholm ifand only if both ST and T S are Fredholm.

27 Apr 2019: Choosing the Green’s function that satisfies ahomogeneous Dirichlet boundary condition on the x-axis, the boundary integral equations giverise to the following Fredholm integral equation of the first kind.

29 Jan 2019: σess(T) = {z C : T zI is not Fredholm}, ress(T) = sup{|z| : z σess(T)}.

30 Aug 2019: range(L) is closed. A complex number ξ is said to in the essential spectrum of L iff (L ξ) is not Fredholm. ... Prove that L isFredholm iff L K is Fredholm. (g) Consider a bounded operator L and a compact operator K on H.

17 Jun 2019: Show that theSturm–Liouville equation can be written as a so-called Fredholm integral equation of theform. ... iv) Derive the Fredholm integral equation for the Sturm–Liouville equation on theinterval [0,1].

30 Aug 2019: 3 Using Mahler’s method, show that the Fredholm series. n=0 zln , where l is an. ... Prove that, for l > 3 and z = 12 , the transcendence of the Fredholm series can beobtained directly from the Thue-Siegel-Roth theorem.

17 Jun 2019: 17C Mathematical Methods. (i) State the Fredholm alternative for Fredholm integral equations of the second kind.