 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-7/3/g/solution

Use the stated Fredholm definition of the essential spectrum, which applies to general bounded operators:

Σ_{e} (L) = {z \in C : L - z I is not Fredholm} .

For each

z

, part (f) applied to

L - z I

says that

L - z I

is Fredholm exactly when

L + K - z I

is Fredholm. Taking the complement of these equivalent Fredholm conditions gives

Σ_{e} (L + K) = Σ_{e} (L) .

This is the general Weyl theorem for bounded compact perturbations. Individual isolated eigenvalues can change; the Fredholm obstruction, and hence this essential spectrum, cannot.

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