Problems for elementary spectral theory 1. Let A : ( → ® be given by A(x1, 12, ..., Tn, ...) = T1, Xn 2'3 4... 4 Is this operator invertible? If the answer is "yes", find the inverse. 2. Consider the...

3

Extracted text: Problems for elementary spectral theory 1. Let A : ( → ® be given by A(x1, 12, ..., Tn, ...) = T1, Xn 2'3 4... 4 Is this operator invertible? If the answer is "yes", find the inverse. 2. Consider the following operator A: L2(0, 1) → L²(0, 1), (2) fx = (x)f Does A have eigenvalues? Find the spectrum of A. 3. Find the spectrum of the following operator A : (² → (²: A(x1,., xn, ...) = (x1 + x2, x2 + x3, ..., Xn + Xn+1, …..). Hint. Use the spectral mapping theorem. 4. Let A : e → ² be given by A(x1,..., In, ...) = 3 I +u Prove that o(A) = {0}. Hint. Use Gelfand's theorem.