A characterization, in operator theory, of exactly which bounded left-invertible weighted shift operators on a Hilbert space indexed by the vertices of a rootless directed tree admit a Wold-type decomposition, meaning the space splits orthogonally into the operator's hyper-range (on which it acts as a unitary) and the subspace generated by its wandering subspace (the analytic part). Existence of the decomposition is reduced to the convergence of an explicit series built from the operator's moments (products of edge weights across successive generations of descendants), extending the classical Wold decomposition of isometries to this tree-indexed setting and connecting to Shimorin's question on m-concave norm-increasing operators.
A functional-analysis / operator-theory paper about weighted shift operators acting on a Hilbert space indexed by the vertices of a rootless directed tree. A left-invertible operator is said to admit…