Exercise 3, § 4

Let $(a_{ij})_{(i,j) \in I \times I}$ be a countably infinite double family of complex numbers. Suppose 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) Now prove that there exists a continuous endomorphism $u$ of $\ell^2_c(I)$, of norm $\leq (\beta \gamma)^{1/2}$, such that for every $x = (x_i)_{i \in I}$ in $\ell^2_c(I)$, one has $u(x) = y$, where $y = (y_i)$ is given by $y_i = \sum_j a_{ij} x_j$ (for $x = (x_i)$ and $y = (y_i)$ in $\mathbf{C}^{(I)}$, set $v_{ij} = |x_i| (p_j |a_{ij}| / p_i)^{1/2}$, $w_{ij} = |y_j| (p_i |a_{ij}| / p_j)^{1/2}$ and majorize $\sum_{i,j} v_{ij} w_{ij}$).

b) Take for $I$ the set of integers $\geq 1$ and put $a_{ij} = (i + j)^{-1}$. Now prove that the conditions () are satisfied with $p_i = i^{-1/2}$ and $\beta = \gamma = \pi$ (these constants being the best possible) (compare the series in () with an integral). Treat in an analogous manner the case where $I = \mathbf{N}$ and $a_{ij} = (i + j + 1)^{-1}$ (“Hilbert matrix”).

\* c) Let $\mathscr{H} = \mathbf{H}^2(D)$ be the Hardy space, formed by the functions $f(z) = \sum_{n=0}^\infty a_n z^n$ holomorphic in the open disk $D : |z| < 1$ and such that $\|f\|^2 = \sum |a_n|^2 < +\infty ; \|f\|$ is therefore 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 real variable function $t \mapsto f(t) g(t)$ is integrable on $[0, 1]$ for 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

whence $B(f, f) \leq \frac{1}{2} \int_{-\pi}^\pi |f(e^{i\theta})|^2 d\theta = \pi \|f\|^2$. Recover in this way the result of b) according to which the Hilbert matrix defines an endomorphism of norm $\leq \pi$ of the Hilbert space $\ell^2_c(\mathbf{N})$. \*