Fall 2014 Problem 2: Define the bounded linear operator K : L^{2}(0, 1) → L^{2}(0, 1) by

(Kf)(x) = ∫_{0}^{1} xy(1 − xy)f(y)dy.

Find the spectrum or K and classify it.

Spring 2016 Problem 2: Let M be a multiplication on L^{2}(ℝ) defined by

Mf(x) = m(x)f(x),

where m(x) is continuous and bounded. Prove that M is a bounded operator on L^{2}(ℝ) and that the spectrum is given by

{m(x) : x ∈ ℝ}^{cl},

where A^{cl} denotes the closure of A. Can M have eigenvalues?

Fall 2003 Problem 5 and Winter 2005 Problem 4: Find the point spectra of R and L the right and left shift operators on ℓ^{2}(ℕ). (That is R(x_{1}, x_{2}, x_{3}, …) = (0, x_{1}, x_{2}, …) and L(x_{1}, x_{2}, x_{3}, …) = (x_{2}, x_{3}, x_{4}, …)).

Fall 2012 Problem 3: Let T be a bounded linear operator on a Hilbert space H. Show that

(a) If ∥T ∥ ≤ 1, then T and its adjoint operator T^{∗} have the same fixed point. i.e. Show that for x ∈ H,

T x = x ⟺ T^{∗}x = x.

(b) Let λ be an eigenvalue of T. Is it true that its complex conjugate \overline{λ} must be an eigenvalue of T^{∗}? Is it true that \overline{λ} must be in the spectrum of T^{∗}? Justify your answers.

Winter 2007 Problem 6: Show that the spectrum of every unitary operator lies on the unit circle.

Winter 2008 Problem 4 Assume that H is a complex seperable Hilbert space and A ∈ B(X) with spectrum σ, resolvent set ρ and resolvent operators R(μ) = (μI − A)^{−1}.

(1) * Show that if μ ∈ ρ and |ν − μ| < ∥R(μ)∥^{−1} then

R(ν) = [I − (μ − ν)R(μ)]^{−1}R(μ).

(2) Show that if μ ∈ ρ then ∥R(μ)∥ ≥ (d(μ, σ))^{−1}.

Fall 2012 Problem 2: Assume that k ∈ Cℝ[0, 1] and define T ∈ B(L^{2}[0, 1]) by setting (T f)(x) = k(x) ∫_{0}^{1} k(t)f(t)dt for any f ∈ L^{2}[0, 1].

(1) Show that T is self-adjoint.

(2) Show that T^{2} = rT for some number r.

(3) Find the spectral radius of T.

Spring 2013 Problem 2: Show that if {T_{i}} ∈ B(X) is a sequence of bounded operators on a Banach space X which all have the same spectrum σ = σ(T_{i}) and converge in norm to T then σ ⊆ σ(T).

Fall 2013 Problem 2: Define T : L^{2}[0, 1] → L^{2}[0, 1] by (T f)(x) = ∫_{0}^{x} f(t)dt for f ∈ L^{2}[0, 1] and x ∈ [0, 1].

(1) * Show that T is bounded.

(2) * Show that T has no eigenvalues.

(3) Find lim_{n→∞} ∥T^{n}∥.