Generalized multipliers for left-invertible operators and applications

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


Introduction
In [20] 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. The construction of the Shimorin's model for a left-invertible analytic operator T ∈ B(H) is as follows. Let E := N (T * ) and define a vector-valued holomorphic functions U x as where T ′ is the Cauchy dual of T . Then we equip the obtained space of analytic functions H := {U x : x ∈ H} with the norm induced by H. The operator U : H ∋ x → U x ∈ H becomes a unitary operator. Moreover, Shimorin proved that H is a reproducing kernel Hilbert space and the operator T is unitary equivalent to the operator M z of multiplication by z on H and T ′ * is unitary equivalent to the operator L given by the Following [20], the reproducing kernel for H is an B(E)-valued function of two variables κ H : Ω × Ω → B(E) that (i) for any e ∈ E and λ ∈ Ω κ H (·, λ)e ∈ H (ii) for any e ∈ E, f ∈ H and λ ∈ Ω f (λ), e E = f, κ H (·, λ)e H 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 ). Moreover, they proved that the reproducing kernel associated with H is multi-diagonal.
In [4] P. Budzyński, P. Dymek and M. Ptak introduced the notion of multiplier algebra induced by a weighted shift. In [8] P. Dymek, A. P laneta and M. Ptak extended this notion to the case of left-invertible analytic operators.

Preliminaries
In this paper, we use the following notation. The fields of rational, real and complex numbers are denoted by Q, R and C, respectively. The symbols Z, Z + , N and R + stand for the sets of integers, positive integers, nonnegative integers, and nonnegative real numbers, respectively. Set D(r) = {z ∈ C : |z| r} and A(r − , r + ) = {z ∈ C : r − |z| r + } for r, r − , r + ∈ R + . The expression "a countable set" means a finite set or a countably infinite set.
All Hilbert spaces considered in this paper are assumed to be complex. Let T be a linear operator in a complex Hilbert space H. Denote by T * the adjoint of T . We write B(H) for the C * -algebra of all bounded operators and the cone of all positive operators in H, respectively. The spectrum and spectral radius of T ∈ B(H) is denoted by σ(T ) and r(T ) respectively. Let T ∈ B(H). We say that T is left-invertible if there exists S ∈ B(H) such that ST = I. The Cauchy dual operator T ′ of a left-invertible operator T ∈ B(H) is defined by The notion of the Cauchy dual operator has been introduced and studied by Shimorin in the context of the wandering subspace problem for Bergman-type operators [20]. We call T analytic if Let X be a set and φ : X → X. If n ∈ N then the n-th iterate of φ is given by is called the orbit of f containing x. If x ∈ X and φ (i) (x) = x for some i ∈ Z + then the cycle of φ containing x is the set  Let (X, A , µ) be a µ-finite measure space, φ : X → X and w : X → C be measurable transformations. By a weighted composition operator C ϕ,w in L 2 (µ) we mean a mapping Let us recall some useful properties of composition operator we need in this paper: Lemma 2.1. Let X be a countable set, φ : X → X and w : X → C be measurable transformations. If C ϕ,w ∈ B(ℓ 2 (X)) then for any x ∈ X and n ∈ N We now describe Cauchy dual of weighted composition operator Lemma 2.2. Let X be a countable set, φ : X → X and w : X → C be measurable transformations. If C ϕ,w ∈ B(ℓ 2 (X)) is left-invertible operator then the Cauchy dual C ′ ϕ,w of C ϕ,w is also a weighted composition operator and is given by: Let T = (V ; E) be a directed tree (V and E are the sets of vertices and edges of T , respectively). For any vertex u ∈ V we put Chi(u) = {v ∈ V : (u, v) ∈ E}. Denote by par the partial function from V to V which assigns to a vertex u a unique v ∈ V such that (v, u) ∈ E. A vertex u ∈ V is called a root of T if u has no parent. If T has a root, we denote it by root. Put V • = V \ {root} if T has a root and V • = V otherwise. The Hilbert space of square summable complex functions on V equipped with the standard inner product is denoted by ℓ 2 (V ). For u ∈ V , we define e u ∈ ℓ 2 (V ) to be the characteristic function of the set {u}. It turns out that the set {e v } v∈V is an orthonormal basis of ℓ 2 (V ). We put V ≺ := {v ∈ V : card(Chi(V )) ≥ 2} and call the a member of this set a branching vertex of T Given a system λ = {λ v } v∈V • of complex numbers, we define the operator S λ in ℓ 2 (V ), which is called a weighted shift on T with weights λ, as follows A subgraph of a directed tree T which itself is a directed tree will be called a subtree of T . We refer the reader to [9] for more details on weighted shifts on directed trees.

Generalized multipliers
In the recent paper [15] we introduced a new analytic model for left-invertible operators. Now, we recall this model. Let T ∈ B(H) be a left-invertible operator and E be a subspace of H denote by [E] T * ,T ′ the direct sum of the smallest T ′invariant subspace containing E and the smallest T * -invariant subspace containing E: where T ′ is the Cauchy dual of T .
To avoid the repetition, we state the following assumption which will be used frequently in this section.
Suppose (LI) holds. In this case we may construct a Hilbert H associated with T , of formal Laurent series with vector coefficients. We proceed as follows.
For each x ∈ H, define a formal Laurent series U x with vector coefficients as Let H denote the vector space of formal Laurent series with vector coefficients of the form U x , x ∈ H. Consider the map U : H → H defined by U x = U x . As shown in [15] U is injective. In particular, we may equip the space H with the norm induced from H, so that U is unitary.
By [15] the operator T is unitary equivalent to the operator M z : H → H of multiplication by z on H given by (3.2) (M z f )(z) = zf (z), f ∈ H and operator T ′ * is unitary equivalent to the operator L : H → H given by For left-invertible operator T ∈ B(H), among all subspaces satisfying condition (LI) we will distinguish those subspaces E which satisfy the following condition Observe that every f ∈ H can be represented as follows Proof. It follows directly from (3.5).
In We introduce generalized multipliers for left-invertible operators which formal Laurent series (3.1) actually represent analytic functions on an annulus or a disc. Define the Cauchy-type multiplication * : We  (i) for every e ∈ E and n ∈ Z, (φ * (U e))(n) =φ(n)e (ii) Proof.
We callφ a generalized multiplier of T and Mφ a generalized multiplication operator if Mφ ∈ B(H ). The set of all generalized multipliers of the operator T we denote by GM(T ). One can easily verify that the set GM(T ) is a linear subspace of B(E) Z . Consider the map V : GM(T ) ∋φ → Mφ ∈ B(H ). By Lemma 3.2, the kernel of V is trivial. In particular, we may equip the space GM(T ) with the norm · : GM(T ) → [0, ∞) induced from B(H ), so that V is isometry: is a Banach algebra.

Proof. (i)
Consider first the case when n ∈ N. Fix f ∈ H and set g = M n z f and ϕ = χ {n} I E . Thenφ * f =ĝ. If n ∈ Z \ N and f ∈ R(M z ) then by (3.3) we have L f = 1 z f . Define g = L −n f . As in the previous case we obtainφ * f =ĝ. If Hence,φ * f (0) = 0 and there exist some k ∈ Z such thatφ * f (k) = 0 which contradicts (3.5).  Proof. Let A = U AU * . All we need to prove is the following equality Fix n ∈ Z. Consider first the case when f = U T m e, e ∈ E, m ∈ N. By (3.4) This altogether implies that where f = U T m e for e ∈ E, m ∈ N.
In turn, if f = U T ′ * m e m ∈ Z + . It is plain that It follows from (3.4) and inclusion (3.8) that Let g e = T m T ′ * m e then As a consequence, we have where f = U T ′ * m e, for e ∈ E, m ∈ N. We extend the previous equality by linearity to the following space lin{U T n x : x ∈ E, n ∈ N} ⊕ lin{U T ′ * n x : x ∈ E, n ∈ N}.
An application of Lemma 3.1 gives (3.9) which completes the proof.
It is interesting to observe that the class of left-invertible and analytic operators and the class of weighted shift on leafless directed trees satisfy the assumptions of the previous theorem.

Weighted shifts on directed trees
In [4] P. Budzyński, P. Dymek, M. Ptak. introduced a notion of a multiplier algebra induced by a weighted shift, which is defined via related multiplication operators. Assume that T = (V, E) is a countably infinite rooted and leafless directed tree, For u ∈ V and v ∈ Des(u) we set The multiplication operator M λ ϕ : ℓ 2 (V ) ⊇ D(M λ ϕ ) → ℓ 2 (V ) is given by We can write the above definition of M λ ϕ in the following form