Exercise 9, § 7

Let E be a complex Banach space.

a) Let $u\in \mathscr{L}(E)$. Consider the endomorphisms $\boldsymbol{\gamma }_u$ and $\boldsymbol{\delta }_u$ of $\mathscr{L}(E)$ defined by $\boldsymbol{\gamma }_u(v) =u\circ v$ and $\boldsymbol{\delta }_u(v) =v\circ u$. One has Sp($u$) $=$ Sp($\boldsymbol{\gamma }_u$) $=$ Sp($\boldsymbol{\delta }_u$), where the last two spectra are relative to the algebra $\mathscr{L}(\mathscr{L}(E))$.

b) For every function $f$ holomorphic in a neighbourhood of the spectrum of $u$, one has $\boldsymbol{\gamma }_{f(u)}=f(\boldsymbol{\gamma }_u)$ and $\boldsymbol{\delta }_{f(u)}=f(\boldsymbol{\delta }_u)$.

c) Let $u_1$ and $u_2$ be in $\mathscr{L}(E)$. Let $\varrho$ be the endomorphism $v\mapsto u_1\circ v\circ u_2$ of $\mathscr{L}(E)$. Its spectrum is equal to Sp($u_1$) Sp($u_2$)$\subset \mathbf{C}$.

d) Let $u_1$ and $u_2$ be in $\mathscr{L}(E)$. Let $\varrho$ be the endomorphism $v\mapsto u_1\circ v\circ u_2$ of the space $\mathscr{L}_2(E)$ of Hilbert–Schmidt mappings from E into E. Its spectrum is equal to Sp($u_1$) Sp($u_2$)$\subset \mathbf{C}$. (Apply a similar method.)