A Subnormal Completion Problem for Weighted Shifts on Directed Trees, II

Abstract

The subnormal completion problem on a directed tree is to determine, given a collection of weights on a subtree, whether the weights may be completed to the weights of a subnormal weighted shift on the directed tree. We study this problem on a directed tree with a single branching point, η branches and the trunk of length 1 and its subtree which is the "truncation" of the full tree to vertices of generation not exceeding 2. We provide necessary and sufficient conditions written in terms of two parameter sequences for the existence of a subnormal completion in which the resulting measures are 2-atomic. As a consequence, we obtain a solution of the subnormal completion problem for this pair of directed trees when η < ∞. If η = 2, we present a solution written explicitly in terms of initial data.

Subnormal and completely hyperexpansive completion problem of weighted shifts on directed trees

“…In this section we will exploit theory developed in the previous section to obtain a solution of subnormal completion problem for weighted shifts on directed trees with one branching point. Our results generelizes the solution obtained in [7] For reader's convenience we state below the general result on subnormal completions on T η,κ . For more details see [7].…”

“…Our results generelizes the solution obtained in [7] For reader's convenience we state below the general result on subnormal completions on T η,κ . For more details see [7].…”

“…In the hypothesis of [7,Problem 4.10] there is no assumption of uniform boundedness as in Theorem 6.7.(ii). At this point we do not know whether this assumption is superfluous or not, but it leads us to another natural problem.…”

“…1-generation completions and flat completions). In [7] (which is a continuation of [6]) the solution of 2-generation completion problem was obtained for directed trees with trunk of length 1.…”

“…These all allow us to obtain the crucial characterizations of truncated moment sequences in terms of extremal values of the integral of reciprocal function. In Theorem 6.2using theory developed in Section 5 -we give a solution (in the kind of [7,Theorem 4.1]) of subnormal completion problem for directed tress with finite trunk. The next part of Section 6 is devoted to trees with infinite trunk -exploiting compactness of certain sets we link these two cases.…”