Abstract:

In this paper, we study the Hardy operator $H_{\mathbb{R}} $ defined on the space $ L^2(\mathbb{R})$. By introducing an orthogonal basis associated with Laguerre polynomials, we prove that the operator $ I - H_{\mathbb{R}} $ acts as an isometric shift on $ L^2(\mathbb{R})$. We further analyze the properties of its adjoint and establish that $ I - H_{\mathbb{R}}^*$ is also an isometric shift, satisfying $(I - H_{\mathbb{R}})(I - H_{\mathbb{R}}^*) = I,$ which implies that the operator norm of $ H_{\mathbb{R}}$ is 2. Moreover, we investigate the spectral structure of $H_{\mathbb{R}} $ in detail, proving that its spectrum lies on the circle in the complex plane centered at $ (1,0)$ with radius 1, and that $ H_{\mathbb{R}}$ has an empty point spectrum. Finally, we generalize the definition of the Hardy operator by considering a weighted version and demonstrate its application to solving certain first-order differential equations.

Key words: Hardy operator, Schur test, equidistant shift operator