On the equivalence of composite operators to the second-order differentiation operator (Q1177553)

Let \(A_ \infty\) be the space of all entire functions with the topology of compact convergence. Let \(D= d/dz\), \((Pf)(z)= f(-z)\) and \(L=\alpha(D)+ \beta(D)P\), where \(\alpha(\lambda)\) and \(\beta(\lambda)\) are fixed entire functions of class \([1,\infty)\). Theorem 2. Let \(\beta(\lambda)\) be an even function. The operators \(L\) and \(D^ 2\) are equivalent in \(A_ \infty\) if and only if the following conditions are true: 1) \({{\alpha(\lambda)+ \alpha(-\lambda)} \over 2}= a\lambda^ 2+b\) \((a,b\in\mathbb{C})\); 2) \(i {{\alpha(\lambda)- \alpha(-\lambda)}\over 2}+ \beta(\lambda)= (\alpha\lambda^ 2+ b_ 1\lambda+ c_ 1) e^{\sigma\lambda}\) \((\alpha,b_ 1,c_ 1,\sigma\in \mathbb{C})\); 3) \(a^ 2\neq \alpha^ 2\).