Stability criteria for positive semigroups on ordered Banach spaces
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arXiv
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| Natura: | Preprint |
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2022
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| author | Glück, Jochen Mironchenko, Andrii |
| author_facet | Glück, Jochen Mironchenko, Andrii |
| contents | We consider generators of positive $C_0$-semigroups and, more generally, resolvent positive operators $A$ on ordered Banach spaces and seek for conditions ensuring the negativity of their spectral bound $s(A)$. Our main result characterizes $s(A) < 0$ in terms of so-called \emph{small-gain conditions} that describe the behaviour of $Ax$ for positive vectors $x$. This is new even in case that the underlying space is an $L^p$-space or a space of continuous functions.
We also demonstrate that it becomes considerably easier to characterize the property $s(A) < 0$ if the cone of the underlying Banach space has non-empty interior or if the essential spectral bound of $A$ is negative. To treat the latter case, we discuss a counterpart of a Krein-Rutman theorem for resolvent positive operators. When $A$ is the generator of a positive $C_0$-semigroup, our results can be interpreted as stability results for the semigroup, and as such, they complement similar results recently proved for the discrete-time case.
In the same vein, we prove a Collatz--Wielandt type formula and a logarithmic formula for the spectral bound of generators of positive semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_13566 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stability criteria for positive semigroups on ordered Banach spaces Glück, Jochen Mironchenko, Andrii Functional Analysis Spectral Theory 47B65, 47D06, 47A10, 37L15 We consider generators of positive $C_0$-semigroups and, more generally, resolvent positive operators $A$ on ordered Banach spaces and seek for conditions ensuring the negativity of their spectral bound $s(A)$. Our main result characterizes $s(A) < 0$ in terms of so-called \emph{small-gain conditions} that describe the behaviour of $Ax$ for positive vectors $x$. This is new even in case that the underlying space is an $L^p$-space or a space of continuous functions. We also demonstrate that it becomes considerably easier to characterize the property $s(A) < 0$ if the cone of the underlying Banach space has non-empty interior or if the essential spectral bound of $A$ is negative. To treat the latter case, we discuss a counterpart of a Krein-Rutman theorem for resolvent positive operators. When $A$ is the generator of a positive $C_0$-semigroup, our results can be interpreted as stability results for the semigroup, and as such, they complement similar results recently proved for the discrete-time case. In the same vein, we prove a Collatz--Wielandt type formula and a logarithmic formula for the spectral bound of generators of positive semigroups. |
| title | Stability criteria for positive semigroups on ordered Banach spaces |
| topic | Functional Analysis Spectral Theory 47B65, 47D06, 47A10, 37L15 |
| url | https://arxiv.org/abs/2210.13566 |