Exercise 9, § 1

We denote by E the complex Hilbert space $L^2([0,1])$ of square integrable functions on $[0,1]$ with respect to the Lebesgue measure $\mu$.

Let $u_0\in \mathscr{L}(E)$ be the Volterra endomorphism on E, such that

for all $f\in L^2([0,1])$ and almost every $x\in [0,1]$.

a) The endomorphism $u_0$ is compact and Sp($u_0$) $=\{0\}$.

b) The endomorphism $u^*_0u_0$ is of finite trace; calculate its trace and deduce the value of the series

$+\infty$

$\sum\frac{1}{n^2}$.

$n=1$

c) Let $v_0$ be the endomorphism of rank 1 of E defined by

where 1 denotes the class of the constant function 1. Let $u=u_0-v_0$. Now prove that $u$ is a Hilbert–Schmidt operator, and calculate the kernel N such that

for $f\in E$ and almost every $x\in [0,1]$.

d) Determine the eigenvalues of $u$ and their spectral multiplicity. (Calculate $u(e_n)$, where $e_n(t) =e^{2i\pi nt}$ for all $n\in \mathbf{Z}.$)

e) Calculate the trace of $u^*u$ and deduce once more the value of the series of general term $1/n^2$ for $n\geqslant 1$.

f) Generalize these arguments to prove that, if $k\geqslant 1$ is an integer strictly positive, then

$+\infty$

$\sum\frac{1}{n^{2k}}\in \pi^{2k}\mathbf{Q}^*$.

$n=1$