Weighted shifts on directed trees: their multiplier algebras, reflexivity and decompositions

Abstract

We study bounded weighted shifts on directed trees. We show that the set of multiplication operators associated with an injective weighted shift on a rooted directed tree coincides with the WOT/SOT closure of the set of polynomials of the weighted shift. From this fact we deduce reflexivity of those weighted shifts on rooted directed trees whose all path-induced spectral-like radii are positive. We show that weighted shifts with positive weights on rooted directed trees admit a Wold-type decomposition. We prove that the pairwise orthogonality of the factors in the decomposition is equivalent to the weighted shift being balanced.2010 Mathematics Subject Classification. Primary: 47B37; Secondary: 47L75. Key words and phrases. Weighted shift on directed tree, multiplication operator, reflexive algebras, Wold-type decomposition. 1 2 P. BUDZYŃSKI, P. DYMEK, A. PŁANETA, AND M. PTAKIt is worth noting that the analytic aspects of the theory of weighted shifts on directed trees were studied also in [9] and [10]. The approach used therein was different than ours and relied on Shimurin's work (see [22]) employing vector-valued analytic functions. PreliminariesLet N, R and C denote the set of all natural numbers, real numbers and complex numbers, respectively. Set N 0 = N ∪ {0} and R + = [0, ∞). Denote by T the unit circle {z ∈ C : |z| = 1} and by D the open unit disc {z ∈stands for the set of all complex polynomials in one variable, whereas T denotes the set of trigonometric polynomials on T. In all what follows we use the convention that i∈∅ x i = 0.Let V be a nonempty set. Then ℓ 2 (V ) denotes the Hilbert space of all functions f :The norm induced by ·, − is denoted by · . For u ∈ V , we define e u ∈ ℓ 2 (V ) to be the characteristic function of the one-point set {u}; clearly, {e u } u∈V is an orthogonal basis of ℓ 2 (V ). We will denote by E the linear span of {e u } u∈V . Given a subset W of V , ℓ 2 (W ) stands for the subspace of ℓ 2 (V ) composed of all functions f such that f (v) = 0 for all W c , and E W denotes the set of all f ∈ ℓ 2 (W ) such that {v ∈ V : f (v) = 0} is finite. By P K we denote the orthogonal projection from ℓ 2 (V ) onto its closed subspace K.Let H be a Hilbert space, J be a nonempty set, and {X j } j∈J ⊆ H be a family of sets. Then j∈J X j stands for the smallest closed linear subspace of H such that X i ⊂ j∈J X j for every i ∈ J. Throughout the paper, unless otherwise stated, every subspace of a Hilbert space is assumed to be closed.Let H be a (complex) Hilbert space. If A is a (linear) operator in H, then D(A), N (A), R(A), and A * denote the domain, the kernel, the range, and the adjoint of A, respectively (in case it exists). We write B(H) for the algebra of all bounded operators on H equipped with the standard operator norm. By F 1 (H) and T(H) we denote the sets of rank one and trace class, respectively, operators on H. Let W be a subalgebra of B(H). Then the preannihilator W ⊥ of W is given by {T ∈ T(H) : tr(AT ) = 0 for all A ∈ W }. The set of all invariant subspaces of all operators A ...

Generalized Multipliers for Left-Invertible Operators and Applications

“…On the other hand if f ∈ R(M n z ), then there exist g ∈ H such that z n g = f . Hence, 1 z n f ∈ H . It turns out that some special generalized multiplication operators can be described with the help of operators M z and L .…”

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

“…Suppose that S λ ∈ B(ℓ 2 (V )) is balanced. Then, [1,Theorem 6.4. ] implies that every f ∈ ℓ 2 (V ) can be decomposed as ∞ n=0 S n λ f n , where f n ∈ N (S * λ ) for n ∈ N 0 .…”

“…If S λ ∈ B(ℓ 2 (V )) is left-invertible, then S λ is reflexive. Now, we are going to prove the second criterion for reflexivity of weighted shift operator, which is a generalization of [1,Theorem 4.3]. Note that we do not assume left-invertibility of the operator.…”

“…Let S λ be a weighted shift on T 2 with weights λ = {λ v } v∈V • 2 given by λ (i,j) = 1 for i = 1 and j ∈ N, 1 2 j for i = 2 and j ∈ N.…”

“…In section 6, we return to general context and obtain a criterion for reflexivity of the left-invertible analytic operator (see Theorem 29). In addition, for the case of weighted shifts on directed trees we prove a second criterion for reflexivity, which is independent from the previous one and it is a generalization of [1, Theorem 4.3].…”