Sign patterns that require eventual exponential nonnegativity

Erickson, Craig

doi:10.13001/1081-3810.3027

Cited by 4 publications

(4 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These observations suggest that a general theory of eventually positive C 0semigroups would be useful. While such a theory has been developed in finite dimensions during the last decade (see, for instance, [19,20], [6,Theorem 2.9] and [8]), a systematic study of this phenomenon in infinite dimensions was initiated only recently in [3,4]. Several spectral results for infinite-dimensional operators with eventually positive powers were recently proved by the second author in [13], after eventually positive matrix powers had been intensively studied for at least two decades; see the introduction of [13] for references and additional details.…”

Section: Introductionmentioning

confidence: 99%

The Role of Domination and Smoothing Conditions in the Theory of Eventually Positive Semigroups

Daners¹,

Glück²

2017

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

We perform an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron-Frobenius type spectral theorems. We furthermore prove a Kreȋn-Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions. * Partially supported by a scholarship within the scope of the LGFG Baden-Württemberg, Germany.Mathematics Subject Classification (2010): 47D06, 47B65, 34G10, 47A10

show abstract

Section: Introductionmentioning

confidence: 99%

The Role of Domination and Smoothing Conditions in the Theory of Eventually Positive Semigroups

Daners¹,

Glück²

2017

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…by a diagonal matrix whose entries are all ≥ 0). Our example is a manifestation of the fact that certain sign patterns may or may not lead to eventual positivity as extensively discussed in [2,11] and references therein.…”

Section: Large Perturbationsmentioning

confidence: 93%

“…Motivated by applications to partial differential equations (see e.g. [12,13,3]; see also [27] for an overview over related elliptic problems) and by the rapid development of a corresponding theory in finite dimensions (see for instance [21,22,8,11]), a study of eventually positive semigroups on Banach lattices was initiated in a series of recent papers [6,5,4]. In particular, these papers clarified that there are several distinct notions of eventual positivity such as an individual and a uniform one which are worthwhile studying.…”

Section: Introductionmentioning

confidence: 99%

Towards a perturbation theory for eventually positive semigroups

Daners¹,

Glück²

2017

Preprint

View full text Add to dashboard Cite

We consider eventually positive operator semigroups and study the question whether their eventual positivity is preserved by bounded perturbations of the generator or not. We demonstrate that eventual positivity is not stable with respect to large positive perturbation and that certain versions of eventual positivity react quite sensitively to small positive perturbations. In particular we show that if eventual positivity is preserved under arbitrary positive perturbations of the generator, then the semigroup is positive. We then provide sufficient conditions for a positive perturbation to preserve the eventual positivity. Some of these theorems are qualitative in nature while others are quantitative with explicit bounds.

show abstract

“…Note that it is shown in [17] that the n-by-n sign pattern requires the eventual exponential positivity, then every cycle in its signed digraph of length 2 or more is arc-positive.…”

mentioning

confidence: 99%

Eventual Positivity of a Class of Double Star-like Sign Patterns

Mou

et al. 2022

Symmetry

View full text Add to dashboard Cite

Identifying and classifying the potentially eventually positive sign patterns and the potentially eventually exponentially positive sign patterns of orders greater than 3 have been raised as two open problems since 2010. In this article, we investigate the potential eventual positivity of the class of double star-like sign patterns S(n,m,1) whose underlying graph G(S(n,m,1)) is obtained from the underlying graph G(S(n,m)) of the (n+m)-by-(n+m) double star sign patterns S(n,m) by adding an additional vertex adjacent to the two center vertices and removing the edge between the center vertices. We firstly establish some necessary conditions for a double star-like sign pattern to be potentially eventually positive, and then identify all the minimal potentially eventually positive double star-like sign patterns. Secondly, we classify all the potentially eventually positive sign patterns in the class of double star-like sign patterns S(n,m,1). Finally, as an application of our results about the potentially eventually positive double star-like sign patterns, we identify all the minimal potentially eventually exponentially positive sign patterns and characterize all the potentially eventually exponentially positive sign patterns in the class of double star-like sign patterns S(n,m,1).

show abstract

Sign patterns that require eventual exponential nonnegativity

Cited by 4 publications

References 49 publications

The Role of Domination and Smoothing Conditions in the Theory of Eventually Positive Semigroups

The Role of Domination and Smoothing Conditions in the Theory of Eventually Positive Semigroups

Towards a perturbation theory for eventually positive semigroups

Eventual Positivity of a Class of Double Star-like Sign Patterns

Contact Info

Product

Resources

About