• An operator-theoretic proof that the classical Hamburger moment problem admits a solution (see e.g. Methods of modern mathematical physics by Reed and Simon, volume 2, Theorem X.4).

• Weyl's proof of the Bohr analogue of Parseval's identity for almost periodic functions. More precisely, let $f$ be a uniformly almost periodic $\mathbb C$-valued function on $\mathbb R$ and let $c_k=a(\lambda_k)$ be the corresponding sequence of the nontrivial Fourier constants of the function $f$ $$a(\lambda)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t)e^{-i\lambda t}dt\neq 0 $$ for some $\lambda=\lambda_k\in\mathbb R$ (it is known that the set $\{\lambda\in \mathbb R:\ a(\lambda)\neq 0\}$ is at most countable). Then $$\sum_{k}|c_k|^2=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|f(t)|^2dt.$$ The proof is based on the spectral analysis of the operator $$Au=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t-s)u(s)dt.$$ Weyl shows that $A$ is a normal compact operator on the space of uniformly almost periodic functions (endowed with the scalar product $(u,v)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}u(t-s)\overline{v(s)}dt$). The result then follows from a clever application of the spectral theorem.

A detailed exposition of the proof can be found in Theory of linear operators in Hilbert space by Akhiezer and Glazman (see Section 57).

• An operator-theoretic proof that the classical Hamburger moment problem admits a solution (see e.g. Methods of modern mathematical physics by Reed and Simon, volume 2, Theorem X.4).

• Weyl's proof of the Bohr analogue of Parseval's identity for almost periodic functions. More precisely, let $f$ be a uniformly almost periodic $\mathbb C$-valued function on $\mathbb R$ and let $$a(\lambda)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t)e^{-i\lambda t}dt.$$ It is known that the set $\{\lambda\in \mathbb R:\ a(\lambda)\neq 0\}$ is at most countable for any uniformly almost periodic function. Let $c_k=a(\lambda_k)\neq 0$ be the sequence of the nontrivial Fourier constants of the function $f$. Then $$\sum_{k}|c_k|^2=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|f(t)|^2dt.$$ The proof is based on the spectral analysis of the operator $$Au=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t-s)u(s)dt.$$ Weyl shows that $A$ is a normal compact operator on the space of uniformly almost periodic functions (endowed with the scalar product $(u,v)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}u(t-s)\overline{v(s)}dt$). The result then follows from a clever application of the spectral theorem.

A detailed exposition of the proof can be found in Theory of linear operators in Hilbert space by Akhiezer and Glazman (see Section 57).

One example is the proof that Hamburger's moment problem admits a solution.

• An operator-theoretic proof that the classical Hamburger moment problem admits a solution (see e.g. Methods of modern mathematical physics by Reed and Simon, volume 2, Theorem X.4).

• Weyl's proof of the Bohr analogue of Parseval's identity for almost periodic functions. More precisely, let $f$ be a uniformly almost periodic $\mathbb C$-valued function on $\mathbb R$ and let $c_k=a(\lambda_k)$ be the corresponding sequence of the nontrivial Fourier constants of the function $f$ $$a(\lambda)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t)e^{-i\lambda t}dt\neq 0 $$ for some $\lambda=\lambda_k\in\mathbb R$ (it is known that the set $\{\lambda\in \mathbb R:\ a(\lambda)\neq 0\}$ is at most countable). Then $$\sum_{k}|c_k|^2=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|f(t)|^2dt.$$ The proof is based on the spectral analysis of the operator $$Au=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t-s)u(s)dt.$$ Weyl shows that $A$ is a normal compact operator on the space of uniformly almost periodic functions (endowed with the scalar product $(u,v)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}u(t-s)\overline{v(s)}dt$). The result then follows from a clever application of the spectral theorem.

A detailed exposition of the proof can be found in Theory of linear operators in Hilbert space by Akhiezer and Glazman (see Section 57).