Generalized multipliers for left-invertible operators and applications

We introduce generalized multipliers for left-invertible operators which formal Laurent series $U_x(z)={\sum }_{n=1}^{\mathrm{\infty }}(P_E{T}^{n}x)\frac{1}{z^n}+{\sum }_{n=0}^{\mathrm{\infty }}(P_E{T^{\mathrm{\prime }\ast }}^{n}x)z^n$ actually represent analytic functions on an annulus or a disc.