Generalized Multipliers for Left-Invertible Operators and Applications
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Integral Equations and Operator Theory
Generalized Multipliers for Left-Invertible Operators and Applications Pawel Pietrzycki Abstract. Generalized multipliers T , whose for a left-invertible operator ∞ n ∗n 1 (P T x) + (P T x)z n , formal Laurent series Ux (z) = ∞ E E n n=1 n=0 z x ∈ H actually represent analytic functions on an annulus or a disc are investigated. 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. In addition, we prove that the set of multiplication operators associated with a weighted shift on a rootless directed tree lies in the closure of polynomials in z and z1 of the weighted shift in the topologies of strong and weak operator convergence. Mathematics Subject Classification. Primary 47A45; Secondary 47B33, 47B37. Keywords. Left-invertible operator, (Generalized) multipliers, Commutant, Analytic model, Composition operator, Weighted shift on directed three.
1. Introduction One of the key ideas in operator theory is that of viewing an operator as multiplication by z on a Hilbert space consisting of (vector-valued) holomorphic functions. The point is that this multiplication operator is much easier to analyze than is the case in the original setting because of the richer structure of a space of holomorphic functions. This is its great advantage and one of the reasons why it attracts the attention of researchers. As was mentioned by A.L. Shields in the paper [24] the fact that weighted shift can be viewed as multiplication by z on a Hilbert space of formal power series (in the unilateral case) or formal Laurent series (in the bilateral case) has been long folklore and this point of view was taken by R. Gellar (see [11,12]). Given a standard orthonormal basis {en }n∈Z of 2 (Z), let {λn }n∈Z be a sequence of nonzero scalars such that the bilateral weighted shift Sλ : 2 (Z) → 2 (Z) given by Sλ en = λn+1 en+1 ,
n∈Z
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∞ is bounded. To each vector x = n=−∞ xn en in 2 (Z) Gellar associate the series ∞ 0 ∞ n −1 1 Ux (z) = λi x−n n + λi xn z n . z n=1 i=−n+1 n=0 i=1 This endow 2 (Z) with the structure of a Hilbert space whose elements are formal Laurent series. Moreover operator Sλ is unitarily equivalent to the operator Mz of multiplication byz on H := {Ux : x ∈ 2 (Z)}. He considered ∞ n , such that multiplication by those formal Laurent series ϕ = n=−∞ ϕ(n)z ϕ in H is a bounded operator. Those series are called multipliers. He proved that the commutant of bilateral weighted shift of multiplicity one may be identified with the algebra of its multipliers. In [25] S. Shimorin obtain a weak analog of the Wold decomposition theorem, representing operator close to isometry in some sense as a direct sum of a unitary operator and a shift operator acting in some reproducing kernel Hilbert space of vector-valued holomorphic functions defined on a disc. In particular he constructed an analytic model for a left-invertible analytic
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