MAT 578 HOMEWORK 4 Due: 9/20/07
Solve three of the following problems.
15. Let Xbe a Banach space, let M,Nbe closed subspaces, and assume that Nis finite
dimensional.
(a) Prove that M+Nis closed.
(b) Let π:X→X/N be the quotient map. Prove that π(M) is closed.
(c) Let Xand Ybe Banach spaces, and let T∈B(X, Y ) be Fredholm. Let X0be a
closed subspace of X. Prove that T X0is closed.
16. For a bounded operator Ton a Banach space X, define the Weyl spectrum of Tto be
σW(T) = ∩σ(T+E) : E∈K(X).
(a) Prove that σW(T) = ∅if Xis finite dimensional.
(b) Prove that σe(T)⊆σW(T) if Xis infinite dimensional.
17. Let Sbe the unilateral rightward shift on `2. Prove that σe(S) = Tand that σW(T) =
D(where Dis the open unit disc in C).
18. Let Xand Ybe Banach spaces.
(a) Let T∈F(X , Y ). Prove that Tis a compact perturbation of an invertible
operator if and only if ind(T) = 0.
(b) Let S,T∈F(X, Y ). Prove that ind(S) = ind(T) if and only if there is an
invertible operator U∈B(Y) such that S−TU ∈K(X, Y ).
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