Exercise 14, § 4

Let E be a complex Hilbert space, T a Hilbert-Schmidt endomorphism of E. (a) Let R and L be the positive Hilbert-Schmidt endomorphisms such that $R^2 = T^*T$ and $L^2 = TT^*$ (V, p. 56, cor. 3); put $R = \operatorname{abs}(T)$, and say that it is the “absolute value” of T (cf. V, p. 74, exerc. 4); one has $L = \operatorname{abs}(T^*)$. Now prove that one has $\operatorname{Ker}(T) = \operatorname{Ker}(R)$ and $L(E) = \overline{T(E)}$. There exists one and only one isometry V of $R(E)$ onto $T(E)$ such that $T = VR$; if V is extended by continuity to $\overline{R(E)}$, then to an operator $U \in \mathscr{L}(E)$ by taking $U.x = 0$ in the orthogonal supplement of $R(E)$, one also has $T = UR$ (polar decomposition of T). One has $R = U^*T = U^*UR = RU^*U$ and $L = URU^*, \ T = LU^*$. If T belongs to $\mathscr{L}^1(E)$, the same is true of $R = \operatorname{abs}(T)$, and T is the product of two Hilbert-Schmidt endomorphisms. (b) If T belongs to $\mathscr{L}^1(E)$, now prove that one has $\operatorname{Tr}(\operatorname{abs}(T)) = \sup_i (\sum | \langle a_i|T.b_i \rangle |)$, where in the right-hand member $(a_i)$ and $(b_i)$ run through the set of orthonormal bases of E (use the polar decomposition of T). Prove that if one puts $\|T\|_1 = \operatorname{Tr}(\operatorname{abs}(T))$, $\|T\|_1$ is a norm on the space $\mathscr{L}^1(E)$, such that $\|T\|_2 \leq \|T\|_1$. (c) Conversely, if $T \in \mathscr{L}(E)$ is such that, for every pair $((a_i), (b_i))$ of orthonormal bases of E, the sum $\sum_i |\langle a_i|T.b_i \rangle|$ is finite, T belongs to $\mathscr{L}^1(E)$ (first remark that T is a Hilbert-Schmidt endomorphism, and use the polar decomposition of T). (d) Let $(T_v)$ be a sequence of Hilbert-Schmidt endomorphisms (resp. of elements of $\mathscr{L}^1(E)$) such that, for every pair of points x, y of E, the sequence $(\langle x|T_v.y \rangle)$ is convergent towards $\langle x|T.y \rangle$, where T is a linear mapping of E into itself; suppose moreover that the sequence of norms $\|T_v\|_2$ (resp. $\|T_v\|_1$) is bounded. Prove that T is a Hilbert-Schmidt endomorphism (resp. an element of $\mathscr{L}^1(E)$) (use b) and c)). (e) Deduce from d) that the space $\mathscr{L}^1(E)$ is a Banach space for the norm $\|T\|_1$. (f) For an endomorphism $T \in \mathscr{L}(E)$ to belong to $\mathscr{L}^1(E)$, it is necessary and sufficient that, for at least one orthonormal basis $(e_i)$ of E, the sum $\sum_i \|T.e_i\|$ is finite (with the notations of a), remark that $|\langle e_i|R.e_i \rangle| \leq \|T.e_i\|$. (g) In the space $\ell^2_{\mathbf{C}}$, let $(e_n)$ be the canonical orthonormal basis, and let $a = \sum_{n=0}^{\infty} \frac{1}{n+1} e_n$; if F is the subspace $\mathbf{C}.a$ of dimension 1, the orthogonal projector $p_F$ is of finite trace, but the series $\sum \|p_F.e_n\|$ is not convergent.