Linear combinations of gammaoperators in \(L_p\) -- spaces (Q1266944)

Gamma operators \(G_n\), \(n\in\mathbb{N}\), \(n\geq 2\) are defined for functions \(f\in L_{1,\text{loc}}(I)\), \(I= (0,\infty)\) by \[ G_n(f; x)= (G_n f)(x)= \int^\infty_0 g_n(x, t)f\Biggl({n\over t}\Biggr)dt \] with the kernel \(g_n(x, t)= {x^{n+1}\over n!} e^{-xt} t^n\), \(x\in I\). These operators were introduced by \textit{M. Müller} [``The sequence of gamma functions'', Dissertation, Stuttgart (1967)]. In this paper, the authors define special linear combinations of gamma operators and prove a global direct result and an equivalence theorem. Their proofs are based on a remarkable connection between the derivatives of the kernel of gamma operators with respect to \(x\) and Laguerre polynomials.