Abstract:
This paper is devoted to the spectral theory for the adjoint of the operator of multiplication by the independent variable in weight spaces of entire functions of one complex variable, and is closely connected with the theory of constant coefficient ordinary differential equations of infinite order $\displaystyle\sum^\infty_{k=0}c_k\frac{d^kf(z)}{dz^k_k}=0$ and of convolution type equations $\langle\varphi,f(z+\zeta)\rangle<0$, with the theory of mean-periodic functions, and with the general theory of subspaces which are invariant under differentiation.
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