Embeddability of real and positive operators

Abstract

Embedding discrete Markov chains into continuous ones is a famous open problem in probability theory with many applications. Inspired by recent progress, we study the closely related questions of embeddability of real and positive operators into real or positive C₀-semigroups, respectively, on finite and infinite-dimensional separable sequence spaces. For the real case we give both sufficient and necessary conditions for embeddability. For positive operators we present necessary conditions for positive embeddability including a full description for the 2 × 2-case. Moreover, we show that real embeddability is topologically typical for real contractions on ℓ².

Keywords:

• Embedding problem
• real and positive operators
• finite and infinite matrices
• strongly continuous semigroups

2010 Mathematics Subject Classifications:

• 15B48
• 47B65
• 47B37
• 47B99
• 47D06

Acknowledgments

Our paper was motivated by the recent work of Baake and Sumner [4] on the Markov embedding problem. We are very grateful to Michael Baake for the inspiration and helpful comments, to Rainer Nagel for interesting discussions, and to Jochen Glück for valuable remarks and references. We sincerely thank the referees for their comments and suggestions which improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).