Generalized multipliers for left-invertible operators and applications
Pawel Pietrzycki
TL;DR
The paper develops a generalized analytic framework for left-invertible operators by extending Shimorin’s model through vector-Laurent multipliers. It defines generalized multipliers $\hat{\varphi}:\mathbb{Z}\to\mathbf{B}(E)$ and the associated bounded operators $M_{\hat{\varphi}}$ on a reproducing kernel Hilbert space $\mathscr{H}$ that encodes the $T$- and $T'$-dynamics via a Cauchy-type convolution. The resulting multiplier algebra $\mathcal{GM}(T)$ is a Banach algebra under the convolution, and operators commuting with $T$ correspond to multipliers, i.e., $A=U^*M_{\hat{\varphi}_A}U$, establishing a functional-calculus-like structure for left-invertible operators. The framework is instantiated for weighted shifts on directed trees, where $M^{\lambda}_{\hat{\varphi}}$ acts on $\ell^2(V)$ with a tree-adapted convolution, and where explicit decompositions involve both forward and backward shift components; approximation results via Fejér kernels demonstrate convergence in this discrete setting. Overall, the work unifies operator-model theory with a multiplier-algebra perspective, enabling analytic control of left-invertible and analytic operators beyond classical shifts.
Abstract
We introduce generalized multipliers for left-invertible operators which formal Laurent series $U_x(z)=\sum_{n=1}^\infty(P_ET^{n}x) \frac{1}{z^n}+\sum_{n=0}^\infty(P_E{T^{\prime*}}^{n}x)z^n$ actually represent analytic functions on an annulus or a disc.