Exercise 3, § 4

Let $(a_{ij})_{(i,j) \in I \times I}$ be a countably infinite double family of complex numbers. Assume that there exist two numbers $\beta > 0$, $\gamma > 0$, and a family $(p_i)_{i \in I}$ of numbers $> 0$ satisfying the relations ()

for all $i, j$ in $I$. a) Show that there exists a continuous endomorphism $u$ from $\ell^2_c(I)$, with norm $\leq (\beta \gamma)^{1/2}$, such that for all $x = (x_i)_{i \in I}$ in $\ell^2_c(I)$, we have $u(x) = y$, where $y = (y_i)$ is given by $y_i = \sum_j a_{ij} x_j$ (for $x = (x_i)$ and $y = (y_i)$ in $\mathbf{C}^I$), put $v_{ij} = |x_i| (p_j |a_{ij}| / p_i)^{1/2}$, $w_{ij} = |y_j| (p_i |a_{ij}| / p_j)^{1/2}$, and find a bound for $\sum_{i,j} v_{ij} w_{ij}$. b) For $I$, take the set of all integers $\geq 1$, and let $a_{ij} = (i + j)^{-1}$. Show that the conditions () are satisfied with $p_i = i^{-1/2}$ and $\beta = \gamma = \pi$ (these being the best possible constants) (compare the series in () with an integral). In an analogous manner, treat the case where $I = \mathbf{N}$ and $a_{ij} = (i + j + 1)^{-1}$ ("Hilbert's matrix"). \ c) Let $\mathscr{H} = \mathbf{H}^2(D)$ be the Hardy space, consisting of all functions $f(z) = \sum_{n=0}^\infty a_n z^n$, analytic in the open disc $D : |z| < 1$ and such that $\|f\|^2 = \sum_n |a_n|^2 < +\infty$; then $\|f\|$ is a norm on $\mathscr{H}$, for which $\mathscr{H}$ is isomorphic to $\ell^2_c(\mathbf{N})$. Given two functions $f, g$ in $\mathscr{H}$, show that the function $t \mapsto f(t) g(t)$ of the real variable $t$ is integrable on $[0, 1]$ with respect to the Lebesgue measure and that the formula $B(f, g) = \int_0^1 f(t) g(t) \, dt$ defines a continuous bilinear form on $\mathscr{H} \times \mathscr{H}$ (consider $B(f, f) = \int_0^1 f(t)^2 \, dt$ for a function $f \in \mathscr{H}$ of the form $f(z) = \sum_{n=0}^N a_n z^n$ with $a_n \geq 0$ for $0 \leq n \leq N$; use Cauchy's theorem to establish the relation from which we get $B(f, f) \leq \frac{1}{2} \int_{-\pi}^\pi |f(e^{i\theta})|^2 d\theta = \pi \|f\|^2$. In this way, get the result of b), according to which the Hilbert's matrix defines an endomorphism of norm $\leq \pi$ of the hilbertian space $\ell^2_c(\mathbf{N})$.)