Math 642 Final Exam Review – Spring 2018

Operators on Hilbert space

• Unbounded operators
  • Be able to define the terms and be able to show the results below.
    • Operators. Densely defined, closed & closable, extensions, adjoint, self adjoint, and resolvent operator $R_\lambda(L):=(L - \lambda I)^{-1}$.
      • Be able to prove that if $L$ is densely defined, then $L^\ast$ is closed. Know that if $L$ is both closed and densely defined, then so is $L^\ast$.
    • Resolvent set and spectrum for a closed, densely defined operator $L$
      • Resolvent set $\rho(L)=\{\lambda\in \mathbb C: R_\lambda \in \mathcal B(\mathcal H)\}$ or $R(L-\lambda I)=\mathcal H$.
      • Spectrum $\sigma(L)=\rho^\complement(L)$. The discrete spectrum $\sigma_d$ consists of all eigenvalues of $L$. The residual spectrum $\sigma_r$ consists of all $\lambda$ such that $L-\lambda I$ is one-to-one and $\overline{R(L-\lambda I)} \ne \mathcal H$. The continuous spectrum $\sigma_c$ is composed of all $\lambda$ such that $L-\lambda I$ is one-to-one and $\overline{R(L-\lambda I)} = \mathcal H$. Finally, $\sigma=\sigma_d\cup \sigma_r\cup \sigma_c$, where $\sigma_d$, $\sigma_r$ and $\sigma_c$ are disjoint.
  • Resolvent operators
    • Be able to prove that $R_\lambda(L)$ is analytic, bounded on the resolvent set.
    • First Resolvent Identity: Be able to show this: If $L$ is a closed, densely defined operator on a Hilbert space $\mathcal H$, then, for $\lambda, \lambda'\in \rho(L)$,
      $ R_{\lambda}(L)-R_{\lambda'}(L) = (\lambda-\lambda')R_\lambda(L)R_{\lambda'}(L). $
• Self adjoint operators.
  • Be able to show that if $L=L^\ast$, then $\sigma_r= \emptyset$, and both $\sigma_d$ and $\sigma_c$ are subsets of $\mathbb R$.
  • Spectral theorem for $L=L^\ast$
    • Be able to define the term spectral family, E_λ. Be able to state the spectral theorem.
    • Stone's formula, Green's functions, and spectral transform. Be able to find the spectral transform in simple cases — Fourier transform, Fourier sine and cosine transforms.

Fourier transforms

• Definition of transform and inverse transform. (Use whichever sign convention you want, just be consistent.)
• Be able to establish simple properties (Theorem 7.2).
• Be able to compute transforms and inverses of transforms, using contour integration if necessary. Know the convolution theorem and be able to establish simple L¹ properties of convolutions — e.g., f,g ∈ L¹ implies that f∗g ∈ L¹.
• Be able to prove or sketch a proof for each of these:
  • The convolution theorem
  • The Plancheral/Parseval theorem
  • ``Useful'' form of Plancheral/Parseval's Theorem.   $\int_{\mathbb R} f(u)\hat g(u)du = \int_{\mathbb R} \hat f(u)g(u)du$.
  • The Shannon Sampling Theorem

Schwartz space and tempered distributions

• Schwartz space $\mathcal S$. (See Feldman's online notes).
  • Definition and notation
  • Semi-norm and (equivalent) metric space topologies
  • Know that the Fourier transform is a bijective map from $\mathcal S$ into itself.
• Tempered distributions, $\mathcal S'$. (See Feldman's online notes.)
  1. Definition and notation, derivatives of distributions, multiplication of a distribution by C^∞ functions increasing polynomially
  2. Know the continuity test for a linear functional to be a tempered distribution (Feldman, Theorem 6, pg. 7)
  3. The Fourier transform of a tempered distribution is defined via Parseval's identity,
     $\int_{\mathbb R} T(u)\hat \phi(u)du = \int_{\mathbb R} \hat T(u)\phi(u)du$
     
  4. Examples of Fourier transforms of tempered distributions

Updated 4/27/2018 (fjn).