Dissertation
Recursiveness for commutative and noncommutative moment problems on directed trees
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2026
DOI: 10.25820/etd.008397
Abstract
We begin by studying the notion of \textit{recursively generated} for 2-variable weighted shift. We then outline a method to construct all polynomials that induce recursive relations for a 2-variable weighted shift (2VWS) form the atoms of the representing measure. We then give the construction of the noncommutative moment matrix and use it to define recursively generated for a weighted shift on a full binary tree with root by needing to have the \textit{left ideal-like} property. We then show that if a noncommutative moment matrix has this property and rank $n$, then there exists a triple $(X,Y,v)$ consisting of $n\times n$ matrices $X$ and $Y$ and an $n$-dimensional vector $v$ that can be used to construct all moments and subsequently all weights of the weighted shift on the full binary trees. We also show how this construction can be extended to weighted shifts on full $m$-ary trees with root. Further, we show that this construction also gives us a characterization for subnormality from if the matrices in this triple. We then recall the procedure for turning a 2VWS into weighted shifts on a full binary tree with root and prove that subnormality and recursive relations are maintained under this construction. Further, we recall the procedure for turning a weighted shift on a directed tree with root into a 2VWS and show that hyponormality and subnormality may not be maintained. We conclude by giving several possible future directions that come from these findings.
Details
- Title: Subtitle
- Recursiveness for commutative and noncommutative moment problems on directed trees
- Creators
- Edward L White
- Contributors
- Raul Curto (Advisor)Ionut Chifan (Committee Member)Daniel Drimbe (Committee Member)Palle Jorgensen (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2026
- DOI
- 10.25820/etd.008397
- Publisher
- University of Iowa
- Number of pages
- viii, 63 pages
- Copyright
- Copyright 2026 Edward L. White
- Language
- English
- Date submitted
- 04/22/2026
- Description illustrations
- illustrations, graphs
- Description bibliographic
- Includes bibliographical references (pages 61-63).
- Public Abstract (ETD)
Patterns appear everywhere in not only mathematics, but in nature as well. Perhaps the most famous pattern in mathematics is the Fibonacci sequence {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, · · · }. This pattern is constructed by adding the previous two terms to get the next term (an = an−1 + an−2). However, what if we were only interested in the 100th term of the sequence? One might think that we need to compute all terms that come before it. Actually, we can use the pattern of the sequence and in this case the first three terms of the sequence to get a kind of encoding of the sequence. This encoding will contain all information about the terms of the sequence and will allow us to generate the n-th term without the terms immediately preceding it. In this work, we will study these encodings for sequences that have two indices. In this case, we will have a rectangle of numbers instead of a string shown above. We outline a method to find all the patterns for a given sequence. We also study sequences that are indexed by strings of letters instead of numbers and give a construction for an encoding of these sequences as well.
- Academic Unit
- Mathematics
- Record Identifier
- 9985177175402771
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