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8). If T has a root and u = root, then we write Sλ→(u) := Sλ . We show that if at least one weight of the weighted shift Sλ on a directed tree vanishes, then Sλ is an orthogonal sum of two weighted shifts on directed trees. 6. Let Sλ be a weighted shift on a directed tree T with weights λ = {λv }v∈V ◦ . Assume that λu = 0 for some u ∈ V ◦ . Then Sλ = Sλ→(u) ⊕ Sλ←(u) . ◦ Proof. 8 that the graphs TDes(u) and TV \Des(u) are directed trees. , Pu f = χDes(u) f for f ∈ 2 (V ). We show that Pu Sλ ⊆ Sλ Pu .

Fredholm weighted shifts on directed trees and their indexes can be characterized as follows (below we adopt the convention that inf ∅ = ∞). 2. Let Sλ be a densely deﬁned weighted shift on a directed tree T with weights λ = {λv }v∈V ◦ . 2) for the deﬁnition of V≺ ) and card(V \ Vλ+ ) < ∞. 1) ind(Sλ ) = a(Sλ ) − b(Sλ ) − 1 a(Sλ ) − b(Sλ ) if T has a root, otherwise. Proof. First we prove that if b(Sλ ) < ∞, then a(Sλ ) < ∞, card(Chi(u)) < ∞ for all u ∈ V and card(V≺ ) < ∞. We begin by recalling that Vλ+ ⊆ V ⊆ V .

This means that supu∈V Mu < ∞ implies ´ ZENON JAN JABLONSKI, IL BONG JUNG AND JAN STOCHEL 44 supu∈V Tu < ∞. 1), the inequality supu∈V Mu < ∞ is equivalent to supu∈V ◦ Sλ eu < ∞. 8, is equivalent to Sλ ∈ B( 2 (V )). Summarizing, we have shown that if the operator Sλ is densely deﬁned, then the inequality supu∈V Mu < ∞ is equivalent to the boundedness of Sλ . 2, the question of when D(Sλ∗ ) ⊆ D(Sλ ) reduces to estimating the norms of rank one perturbations of positive diagonal operators. The class of such operators is still not very well understood, despite the structural simplicity of diagonal operators (cf.