Exercise 14, § 4

Let E be a complex hilbertian space, and T a Hilbert-Schmidt endomorphism of E. a) Let R and L be positive Hilbert-Schmidt endomorphisms such that $R^2 = T*T$ and $L^2 = TT^*$ (V, p. 57, cor. 3); let $R = \operatorname{abs}(T)$, and call this the « absolute value » of T (cf. V, p. 75, exerc. 4); we have $L = \operatorname{abs}(T^*)$. Show that $\operatorname{Ker}(T) = \operatorname{Ker}(R)$ and $\overline{L(E)} = \overline{T(E)}$. There exists one, and only one isometry V from $R(E)$ onto $T(E)$ such that $T = VR$; if we extend V by continuity to $\overline{R(E)}$, then to an operator $U \in \mathscr{L}(E)$ by taking $U.x = 0$ on the orthogonal complement of $R(E)$, we also have $T = UR$ (polar decomposition of T). Then $R = U*T = U^*UR = RU^*U$ and $L = URU^*, T = LU^*$. If T belongs to $\mathscr{L}^1(E)$, then so does $R = \operatorname{abs}(T)$, and T is the product of two Hilbert-Schmidt endomorphisms.

b) If $T$ belongs to $\mathscr{L}^1(E)$, show that $\operatorname{Tr}(\operatorname{abs}(T)) = \sup (\sum_i | \langle a_i|T.b_i \rangle |)$ where, on the right hand side, $(a_i)$ and $(b_i)$ range over the set of orthonormal bases of E (use the polar decomposition of T). Show that if we put $\| T \|_1 = \operatorname{Tr}(\operatorname{abs}(T))$, then $\| T \|_1$ 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, then $T \in \mathscr{L}^1(E)$ (first observe that T is a Hilbert-Schmidt endomorphism, then 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 $(\langle x|T_v.y \rangle)$ converges to $\langle x|T.y \rangle$, where T is a linear mapping from E into itself; in addition, assume that the sequence of norms $\| T_v \|_2$ (resp. $\| T_v \|_1$) is bounded. Show 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), note that $|\langle e_i|R.e_i \rangle| \leq \| T.e_i \|$. g) In the space $\ell_c^2$, 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 1-dimensional subspace $\mathbf{C}.a$, the orthoprojector $p_F$ has finite trace, but the series $\sum_n \| p_F.e_n \|$ is not convergent.