Exercise 2, § 4

Let $A = (\alpha_{mn})_{(m,n) \in \mathbf{N} \times \mathbf{N}}$ be a double sequence of complex numbers, which we also call 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; let $A.x$ be the point $(y_m)$ of the product space $\mathbf{C}^\mathbf{N}$, then $x \mapsto A.x$ 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. Let $E_n$ denote the subspace of $\mathbf{C}^{(\mathbf{N})}$ generated by the first $n$ vectors of the canonical basis, $P_n$ the canonical projection from $\mathbf{C}^\mathbf{N}$ onto $E_n$; when $E_n$ is assigned the norm induced by that of the space $\ell_2^\mathbf{C}(\mathbf{N})$, $\|u\|$ denotes the norm of the linear mapping $u$ from the finite dimensional hilbertian space $E_n$ into itself. a) In order that the image of $\mathbf{C}^{(\mathbf{N})}$ under 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 from $\ell_2^\mathbf{C}(\mathbf{N})$ into itself, it is necessary and sufficient that the norms $\|P_n A P_n\|$ are bounded. This implies that the rows and the columns of $A$ belong to $\ell_2^\mathbf{C}(\mathbf{N})$ (exerc. 1). b) Let $A^*$ denote 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 $\ell_2^\mathbf{C}(\mathbf{N})$), then the series $\beta_{mn} = \sum_p \overline{\alpha}_{pm} \alpha_{pn}$ are absolutely convergent, and we put $A^*A = (\beta_{mn})$. Show that for $x \mapsto A.x$ to extend to a continuous linear mapping $u$ from $\ell_2^\mathbf{C}(\mathbf{N})$ into itself, it is necessary and sufficient that the norms $\|P_n (A^*A) P_n\|$ are bounded (we have $\langle P_n (A^*A) P_n.x|x \rangle = \|AP_n.x\|^2$ for all $x \in E_n$). Then $x \mapsto A^*A.x$ extends to a positive hermitian mapping $u^*u$ from $\ell_2^\mathbf{C}(\mathbf{N})$ 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 then we put $XY = (\zeta_{mn})$. We say that a power $X^k$ ($k$ integer > 1) is defined if $X^{k-1}$ and $X^{k-1}X$ are defined and then we put $X^k = X^{k-1}X$; in this case, $X^p X^q = X^k$ 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, show that for all $x \in E_n$, we have $\langle (P_n A^*A P_n)^2.x|x \rangle \leq \langle P_n (A^*A)^2 P_n.x|x \rangle$, and deduce that $\|P_n A^*A P_n\|^2 \leq \|P_n (A^*A)^2 P_n\|$. d) In order that an infinite matrix A be such that the image of $\mathbf{C}^{(\mathbf{N})}$ under $x \mapsto A.x$ is contained in $\ell_2^\mathbf{C}(\mathbf{N})$ and that $x \mapsto A.x$ extends to a continuous linear mapping from $\ell_2^\mathbf{C}(\mathbf{N})$ into itself, it is necessary and sufficient that the following three conditions are 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) we have

where $(A^*A)^n_{mm}$ denotes the term with 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 $E_n$, then $\|C\| \leq n \sup_{1 \leq i \leq n} |C_{ii}|$, by considering the trace of $C$ and by diagonalizing $C$. Using the inequality proved in c), show that for every integer $k > 1$, if conditions (i), (ii) and (iii) are satisfied.)