**Generalized multipliers for left-invertible analytic operators**

A characterization of the commutant of a given operator is one of the way of investigation of the operator itself. The classical result on unilateral shift of (the multiplication by the independent variable on the Hardy space H^{2}) says that its commutant is the algebra of all multiplications by bounded analytic functions. In the case of unilateral shift of arbitrary multiplicity, its commutant is the algebra of all bounded analytic operator-valued functions. It was shown by Shields in [8], that the commutant of unilateral weighted shift of multiplicity one may be identified with the algebra of its multipliers. On the other hand, the multipliers for weighted shifts on rooted directed trees introduced in [1] are not sufficiently large to determine the whole commutant of the operator.

In this talk we present one possible approach to deal with the mentioned problem. As shown by Shimorin, every left-invertible analytic operator T on a Hilbert space is unitarily equivalent to a multiplication operator by z on a reproducing kernel Hilbert space of analytic functions on a disc with values in the kernel of the adjoint of T. We define generalized multipliers for left-invertible analytic operator T using this unitary equivalence. This enable us to characterize the commutant of T. In particular, we discuss the application of general theory in the case of left-invertible weighted shifts on leafless rooted directed trees.

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