Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A NEW CRITERION FOR MOMENT INFINITELY DIVISIBLE WEIGHTED SHIFTS

  • Hong T. T. Trinh (Department of Mathematics Chungnam National University)
  • Received : 2023.09.13
  • Accepted : 2024.02.29
  • Published : 2024.04.30
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Abstract

In this paper we present the weighted shift operators having the property of moment infinite divisibility. We first review the monotone theory and conditional positive definiteness. Next, we study the infinite divisibility of sequences. A sequence of real numbers γ is said to be infinitely divisible if for any p > 0, the sequence γp = {γpn}∞n=0 is positive definite. For sequences α = {αn}∞n=0 of positive real numbers, we consider the weighted shift operators Wα. It is also known that Wα is moment infinitely divisible if and only if the sequences {γn}∞n=0 and {γn+1}∞n=0 of Wα are infinitely divisible. Here γ is the moment sequence associated with α. We use conditional positive definiteness to establish a new criterion for moment infinite divisibility of Wα, which only requires infinite divisibility of the sequence {γn}∞n=0. Finally, we consider some examples and properties of weighted shift operators having the property of (k, 0)-CPD; that is, the moment matrix Mγ(n, k) is CPD for any n ≥ 0.

Keywords

Acknowledgement

The author is extremely thankful to the supervisor, Professor Sang Hoon Lee, for his great guidance and support throughout my PhD research studies at Chungnam National University

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