# Math 641-600 — Fall 2018

## Current Assignment

Assignment 10 - Due Wednesday, November 14, 2018.

• Read sections 3.3-3.5, and my notes on Compact Operators, and on the Closed Range Theorem.
• Do the following problems.

1. Section 3.3: 1 (Assume the appropriate operators are closed and that λ is real.)

2. Section 3.4: 2(b)

3. Consider the Hilbert space $\mathcal H=\ell^2$ and let $S=\{x=(x_{1}\ x_{2}\ x_3\ \ldots)\in \ell^2: \sum_{n=1}^\infty (n^2+1)|x_n|^2 <1\}$. Show that $S$ is a precompact subset of $\ell^2$.

4. Let $S$ be a bounded subset (not a subspace!) of a Hilbert space $\mathcal H$. Show that $S$ is precompact if and only if every sequence in $S$ has a convergent subsequence. (Note: If $S$ is just precompact, the limit point of the sequence may not be in $S$, because $S$ may not be closed.)

5. Show that every compact operator on a Hilbert space is bounded.

6. Consider the finite rank (degenerate) kernel
k(x,y) = φ1(x)ψ1(y) + φ2(x)ψ2(y), where φ1 = 6x-3, φ2 = 3x2, ψ1 = 1, ψ2 = 8x − 6.
Let Ku= ∫01 k(x,y)u(y)dy. Assume that L = I-λ K has closed range,
1. For what values of λ does the integral equation
u(x) - λ∫01 k(x,y)u(y)dy =f(x)
have a solution for all f ∈ L2[0,1]?
2. For these values, find the solution u = (I − λK)−1f — i.e., find the resolvent.
3. For the values of λ for which the equation does not have a solution for all f, find a condition on f that guarantees a solution exists. Will the solution be unique?

7. In the following, H is a Hilbert space and B(H) is the set of bounded linear operators on H. Let L be in B(H) and let N:= sup {|< Lu, u>| : u ∈ H, ||u|| = 1}.

Updated 11/7/2018.