Proposition 4

Let $u$ be a positive compact endomorphism of E. Let $f\in \mathscr{C}(\mathbf{R}_+)$ be a continuous increasing mapping such that $f(0) = 0$.

The endomorphism $f(u)$ is compact and positive and, for every $n\in I_E$, one has $\lambda_n(f(u)) =f(\lambda_n(u))$.

The endomorphism $f(u)$ is compact and positive by prop. 6, b) of III, p. 91 and prop. 15, a) of I, p. 117. The spectrum of $f(u)$ is the image under $f$ of the spectrum of $u$ (cor. 2 of I, p. 111). If $\lambda \in$ Sp$_s(f(u))$, the spectral multiplicity of $\lambda$ is the sum of the spectral multiplicities of the $\mu\in$ Sp($u$) such that $f(\mu) =\lambda$ (cor. 2 of III, p. 84). Since $f$ is increasing, the sequence $(f(\lambda_n(u)))_{n\in I_E}$ is decreasing. The assertion then results from the definition of the sequence $(\lambda_n(f(u)))_{n\in I_E}$.