**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 $L^\ast$.

- Be able to
- 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.

- Operators. Densely defined, closed & closable, extensions,
adjoint, self adjoint, and resolvent operator $R_\lambda(L):=(L -
\lambda I)^{-1}$.
- 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). $

- Be able to

- Be able to define the terms and be able to show the results
below.
- 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.

- Be able to define the term spectral family,
E

- Be able to

**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 L ^{1}properties of convolutions — e.g., f,g ∈ L^{1}implies that f∗g ∈ L^{1}. - 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.)
- Definition and notation, derivatives of distributions,
multiplication of a distribution by C
^{∞}functions increasing polynomially - Know the continuity test for a linear functional to be a tempered distribution (Feldman, Theorem 6, pg. 7)
- The Fourier transform of a tempered distribution is defined via
Parseval's identity,

$\int_{\mathbb R} T(u)\hat \phi(u)du = \int_{\mathbb R} \hat T(u)\phi(u)du$

- Examples of Fourier transforms of tempered distributions

- Definition and notation, derivatives of distributions,
multiplication of a distribution by C

Updated 4/27/2018 (fjn).