Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Recursive approach to the matrix moment problem

R. Curto, A. Ech-charyfy, K. Idrissi, E. H. Zerouali

Abstract

In this paper, we study the truncated matrix moment problem in one variable through recursive matrix extensions. \ We give necessary and sufficient conditions for a recursive matrix extension of finite data to be a matrix moment sequence in the classical cases of Hamburger, Stieltjes, and Hausdorff moment problems. \ We also discuss matricial subnormal completion and matricial $k$--hyponormal completion problems and provide an analog of Stampfli's Theorem on flat propagation for $2$--hyponormal matricial weighted shifts.

A Recursive approach to the matrix moment problem

Abstract

In this paper, we study the truncated matrix moment problem in one variable through recursive matrix extensions. \ We give necessary and sufficient conditions for a recursive matrix extension of finite data to be a matrix moment sequence in the classical cases of Hamburger, Stieltjes, and Hausdorff moment problems. \ We also discuss matricial subnormal completion and matricial --hyponormal completion problems and provide an analog of Stampfli's Theorem on flat propagation for --hyponormal matricial weighted shifts.
Paper Structure (15 sections, 30 theorems, 68 equations)

This paper contains 15 sections, 30 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The truncated matrix moment problem and recursive matrix sequences
  4. Minimal polynomial and finitely atomic representing matrix charge
  5. Minimal polynomial and finitely atomic representing matrix measure
  6. Recursive moment sequences
  7. Hamburger recursive matrix moment sequences
  8. Stieltjes and Hausdorff recursive matrix sequences
  9. Some special cases
  10. Case r=2.
  11. Case r=3.
  12. Propagation phenomena for $k$--hyponormal matricial weighted Shifts
  13. $k$-hyponormal and subnormal matricial weighted Shifts
  14. Subnormal and $k$--hyponormal completion of matricial weighted shifts
  15. Propagation phenomena for matricial weighted shifts

Key Result

Theorem 2.3

The following statements are equivalent: where $\mathcal{P}(K) = \{P \in \mathcal{M}_p(\mathbb{R}[X]) \mid P_{|K} \succeq 0\}$.

Theorems & Definitions (48)

  • Theorem 2.3: $le2019tracial$
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Proposition 2.7: $cimprivc2013moment$
  • Proposition 2.8
  • Proposition 2.9
  • Theorem 2.10
  • Example 3.3
  • Proposition 3.4
  • ...and 38 more