2003
DOI: 10.1007/s00209-002-0473-z
|View full text |Cite
|
Sign up to set email alerts
|

Ordinary differential operators in Hilbert spaces and Fredholm pairs

Abstract: Let A(t) be a path of bounded operators on a real Hilbert space, hyperbolic at ±∞. We study the Fredholm theory of the operator F A = d/dt−A(t). We relate the Fredholm property of F A to the stable and unstable linear spaces of the associated system X = A(t)X. Several examples are included to point out the differences with respect to the finite dimensional case, in particular concerning the role of the spectral flow. We define a general class of paths A for which many properties typical of the finite dimension… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
1
1
1
1

Citation Types

5
89
0

Year Published

2005
2005
2021
2021

Publication Types

Select...
6
2

Relationship

1
7

Authors

Journals

citations
Cited by 47 publications
(94 citation statements)
references
References 13 publications
5
89
0
Order By: Relevance
“…By Theorem 1.6 there exist discrete dichotomies on Z þ and Z À : By Lemma 1.5, there exist dichotomies fP þ t g tX0 and fP À t g tp0 : This proves ði 0 Þ in Theorem 1.2 and, therefore, (i) in Theorem 1.1 for a ¼ 0 ¼ b: By Proposition 5.2 we also have that Nð0; 0Þ is Fredholm, and, using formulas for the defect numbers and index from Theorem 1.4, we derive ðii 0 Þ in Theorem 1.2 and, by Lemma 5.1, (ii) in Theorem 1.1 for a ¼ 0 ¼ b; and the required formulas for the defect numbers and the index. It remains to prove that (i) and (ii) in Theorem 1.1 imply (1.3), see [9,Theorem 4], and also [7,Theorem 8] for the proof in the case when a ¼ À1 and b ¼ 0: We will present a proof, different form [7], as well as from the corresponding proofs in [3,12,28,36,38,46] given in particular cases. Our proof is based on the following abstract fact from [29, p. 23].…”
Section: Article In Pressmentioning
confidence: 97%
“…By Theorem 1.6 there exist discrete dichotomies on Z þ and Z À : By Lemma 1.5, there exist dichotomies fP þ t g tX0 and fP À t g tp0 : This proves ði 0 Þ in Theorem 1.2 and, therefore, (i) in Theorem 1.1 for a ¼ 0 ¼ b: By Proposition 5.2 we also have that Nð0; 0Þ is Fredholm, and, using formulas for the defect numbers and index from Theorem 1.4, we derive ðii 0 Þ in Theorem 1.2 and, by Lemma 5.1, (ii) in Theorem 1.1 for a ¼ 0 ¼ b; and the required formulas for the defect numbers and the index. It remains to prove that (i) and (ii) in Theorem 1.1 imply (1.3), see [9,Theorem 4], and also [7,Theorem 8] for the proof in the case when a ¼ À1 and b ¼ 0: We will present a proof, different form [7], as well as from the corresponding proofs in [3,12,28,36,38,46] given in particular cases. Our proof is based on the following abstract fact from [29, p. 23].…”
Section: Article In Pressmentioning
confidence: 97%
“…in the operator norm as τ → +∞, the operators F The following result will be useful in the proof of Theorem 1.7 (see [3] for a more general presentation): …”
Section: (T)u(t) Is Onto the Same Holds For A : R − → [L] And For Thmentioning
confidence: 97%
“…What is crucial to the proof of Proposition 3.1 is that an asymptotically hyperbolic matrix function has an exponential dichotomy [4] [12,10]. Fredholm properties of differential operators on infinite-dimensional spaces have been recently established in [1,8]. An extension of Proposition 3.1 to this case would open the possibility of applying the general principle described here not only to bifurcation of various types of bounded solutions of ordinary differential equations but also to partial differential equations on unbounded domains, for example of the type considered in [8].…”
Section: T)u(t)mentioning
confidence: 99%