Generalized multipliers for left-invertible analytic operators and their applications to commutant and reflexivity

A Shimorin-type analytic model on an annulus for left-invertible operators and applications

“…By Theorem 3.3 the sequenceφ induces bounded operator Mφ on subspace R(M z ). Changing order of summation we obtain (χ N (k)M k z f + χ Z\N (k)L −k f ) = ∞ k=0ψ (k)M k z f + ∞ k=1ψ (−k)L k f.Let, for n ∈ N , denote by p n : Z → C the coefficients of the n-th Fejer kernel, i.e.,As in the proof of[8, Proposition 15] one can show that Mp nφ SOT Mφ in R(M z ). Wydzia l Matematyki i Informatyki, Uniwersytet Jagielloński, ul.…”

“…Following [20], the reproducing kernel for H is an B(E)-valued function of two variables κ H : Ω × Ω → B(E) that The class of weighted shifts on a directed tree was introduced in [9] and intensively studied since then [7,2,4]. The class is a source of interesting examples (see e.g., [8,13]). In [7] S. Chavan and S. Trivedi showed that a weighted shift S λ on a rooted directed tree with finite branching index is analytic therefore can be modelled as a multiplication operator M z on a reproducing kernel Hilbert space H of E-valued holomorphic functions on a disc centered at the origin, where E := N (S * λ ).…”

Generalized Multipliers for Left-Invertible Operators and Applications

“…[22,Theorem 3.3] this requires considering two-sided sequence of operators. Hence, we have to extend the notion of generalized multipliers for left-invertible and analytic operators introduced in [9]. Since the analytic model for left-invertible operator introduced in the recent paper [22] by the author plays a major role in this paper, we outline it in the following discussion.…”

“…In [9] P. Dymek, A. P laneta and M. Ptak extended the notion of multipliers to left-invertible analytic operators using Shimorin's analytic function theory approach. Namely, they defined generalized multipliers for left-invertible analytic operator T , whose coefficients are bounded operators on N (T * ) and characterized the commutant of such operators in its terms (see [9,Theorem 4]).…”

“…As shown in [22, Theorem 3.2 and 3.8] a left-invertible operator T , which satisfies certain conditions can be modelled as a multiplication operator M z on a reproducing kernel Hilbert space of vector-valued analytic functions on an annulus or a disc. In this paper, we extend the notion of generalized multipliers for leftinvertible analytic operator introduced in [9] to the case of left-invertible operator T , whose model constructed in [22,Section 3] actually represent analytic functions on an annulus or a disc. We show that they are coefficients of analytic functions and characterize the commutant of some left-invertible operators, which satisfies certain conditions in its terms.…”