Chapter IV

Hilbertian spectral theory

For every vector space E, we denote by $1_E$ the identical mapping of E. If E, F and G are vector spaces and if $v: E\rightarrow F$ and $u: F\rightarrow G$ are linear mappings, the linear mapping $u\circ v$ from E into G will sometimes be denoted by $uv$.

Let E be a prehilbertian space. We denote by $\langle x|y\rangle_E$, or simply by $\langle x|y\rangle$, the scalar product on E. The orthogonal of a subset A of E is denoted by $A^{\circ}$. It is a closed subspace of E; it is reduced to 0 if and only if A is total in E (EVT, V, p. 16, cor. 1). If $F\subset E$ is a vector subspace of E, one has $F^{\circ \circ}= F ($cf. EVT, V, p. 13). The spectrum of an endomorphism of E is always relative to the Banach algebra $\mathscr{L}(E)$.

Let F be a hilbertian space. We shall denote by $\mathscr{L}_1(E)$ the space of endomorphisms of finite trace of E (EVT, V, p. 50, déf. 8) and by $\mathscr{L}_2(E; F)$ the space of Hilbert–Schmidt mappings from E into F (EVT, V, p. 51, déf. 9). The mapping $u\mapsto \|u\|_2=$ (Tr($u^*u$))$^{1/2}$ is a hilbertian norm on $\mathscr{L}_2(E; F)$ (EVT, V, p. 52, th. 1, (i)).

The space $\mathscr{L}^f(E; F)$ of continuous linear mappings of finite rank is dense in $\mathscr{L}^c(E; F)$ (prop. 10 de III, p. 15 et cor. de la prop. 11 de III, p. 16). In particular, the space $\mathscr{L}_2(E; F)$ is dense in $\mathscr{L}^c(E; F)$.

The Hilbertian tensor product of E and F (EVT, V, p. 28, déf. 1) is denoted by $E\widehat{\otimes}_2F$.

If X is a locally compact topological space and $\mu$ a measure on X, we shall denote $\mathscr{L}^p(X, \mu) =\mathscr{L}_{\mathbf{C}}^p(X, \mu)$ and $L^p(X, \mu) = L^p_{\mathbf{C}}(X, \mu)$ for $p\in [1,+\infty ]$.