Exercise 2, § 4

Let $A = (\alpha_{mn})_{(m,n) \in \mathbf{N} \times \mathbf{N}}$ be a double sequence of complex numbers, also called an infinite matrix. For every point $x = (x_n)$ of the direct sum space $\mathbf{C}^{(\mathbf{N})}$, the sums $y_m = \sum_n \alpha_{mn} x_n$ are defined, and we denote by $A.x$ the point $(y_m)$ of the product space $\mathbf{C}^\mathbf{N}$, so that $x \mapsto A.x$ is a linear mapping from $\mathbf{C}^{(\mathbf{N})}$ into $\mathbf{C}^\mathbf{N}$, and every linear mapping from $\mathbf{C}^{(\mathbf{N})}$ into $\mathbf{C}^\mathbf{N}$ is of this form. We denote by $E_n$ the subspace of $\mathbf{C}^{(\mathbf{N})}$ generated by the first $n$ vectors of the canonical basis, by $P_n$ the canonical projection of $\mathbf{C}^\mathbf{N}$ onto $E_n$; when $E_n$ is provided with the norm induced by that of the space $\ell_2^\mathbf{C}(\mathbf{N})$, we denote by $\|u\|$ the norm of a linear mapping $u$ of the finite-dimensional hilbertian space $E_n$ into itself. a) In order that the image of $\mathbf{C}^{(\mathbf{N})}$ by the mapping $x \mapsto A.x$ be contained in $\ell_2^\mathbf{C}(\mathbf{N})$ and that this mapping extend to a continuous linear mapping of $\ell_2^\mathbf{C}(\mathbf{N})$ into itself, it is necessary and sufficient that the norms $\|P_n A P_n\|$ be bounded. This implies that the rows and columns of $A$ belong to $\ell_2^\mathbf{C}(\mathbf{N})$ (exerc. 1). b) We denote by $A^*$ the infinite matrix $(\alpha'_{mn})$, where $\alpha'_{mn} = \overline{\alpha}_{nm}$. If the columns of $A$ belong to $\ell_2^\mathbf{C}(\mathbf{N})$ (in other words, if $x \mapsto A.x$ maps $\mathbf{C}^{(\mathbf{N})}$ into $\ell_2^\mathbf{C}(\mathbf{N})$), the series $\beta_{mn} = \sum_p \overline{\alpha}_{pm} \alpha_{pn}$ are absolutely convergent, and we set $A^*A = (\beta_{mn})$. Now prove that in order that $x \mapsto A.x$ extend to a continuous linear mapping $u$ of $\ell_2^\mathbf{C}(\mathbf{N})$ into itself, it is necessary and sufficient that the norms $\|P_n(A^*A)P_n\|$ be bounded (one has $\langle P_n(A^*A)P_n.x|x \rangle = \|AP_n.x\|^2$ for every $x \in E_n$). Then $x \mapsto A^*A.x$ extends to the positive hermitian mapping $u^*u$ of $\ell_2^\mathbf{C}(\mathbf{N})$ into itself. c) For two infinite matrices $X = (\xi_{mn})$, $Y = (\eta_{mn})$, we say that the product $XY$ is defined if the series $\zeta_{mn} = \sum_p \xi_{mp} \eta_{pn}$ are absolutely convergent, and we then set $XY = (\zeta_{mn})$. We say that a power $X^k$ ($k$ integer > 1) is defined if $X^{k-1}$ and $X^{k-1}X$ are defined and we then set $X^k = X^{k-1}X$; in this case one has $X^p X^q = X^k$ for every pair of integers $p, q$ such that $p + q = k$. If $A$ is an infinite matrix whose columns are in $\ell_2^\mathbf{C}(\mathbf{N})$ and if the product $(A^*A)^2$ is defined, now prove that for every $x \in E_n$, one has $\langle (P_n A^*A P_n)^2.x|x \rangle \leq \langle P_n (A^*A)^2 P_n.x|x \rangle$, and deduce that one has $\|P_n A^*A P_n\|^2 \leq \|P_n (A^*A)^2 P_n\|$. d) In order that an infinite matrix $A$ be such that the image of $\mathbf{C}^{(\mathbf{N})}$ by $x \mapsto A.x$ be contained in $\ell_2^\mathbf{C}(\mathbf{N})$ and that $x \mapsto A.x$ extend to a continuous linear mapping of $\ell_2^\mathbf{C}(\mathbf{N})$ into itself, it is necessary and sufficient that the following three conditions be satisfied: (i) the rows and columns of $A$ are in $\ell_2^\mathbf{C}(\mathbf{N})$; (ii) the powers $(A^*A)^k$ are defined for every integer $k > 1$; (iii) one has

where $(A^*A)_{mn}^n$ denotes the term of indices $(m, m)$ of the matrix $(A^*A)^n$. (Observe that if $C$ is the matrix with respect to the canonical basis of $E_n$ of a positive hermitian endomorphism of $E_n$, one has $\|C\| \leq n \cdot \sup_{1 \leq i \leq n} |C_{ii}|$ by considering the trace of $C$ and diagonalizing $C$. Using the inequality proved in c), now prove that one has for every integer $k > 1$