A new characterization of the bounded operators commuting with Hankel translation
A new characterization for the linear bounded operator \(T\) on Zemanian's space \({\mathcal H}_\mu\), \(\mu>-1/2\), commuting with Hankel translation \[ (\tau_x \psi)(y) = \int_0^\infty D_\mu (x,y,z)\psi(z) dz,\quad x,y\in (0,\infty), \] where \[ D_\mu (x,y,z) = \int_0^\infty t^{-\mu -1/2} (xt)^{1/2} J_\mu(xt) (yt)^{1/2}J_\mu (yt) (zt)^{1/2} J_\mu (zt) dt,\;x,y,z \in (0,\infty) \] and \(J_\mu\) denotes the Bessel function of the first kind and order \(\mu\) is given. Namely, it is shown that the operator \(T\) commutes with the Hankel translation if, and only if, it commutes with the Bessel operator \(S_\mu = x^{-\mu -1/2}D x^{2\mu +1} D x^{-\mu -1/2}\).
scientific article; zbMATH DE number 1145770