Abstract
Purpose
The paper deals with ultrametric bounded Fredholm operators and approximate pseudospectra of closed and densely defined (resp. bounded) linear operators on ultrametric Banach spaces.
Design/methodology/approach
The author used the notions of ultrametric bounded Fredholm operators and approximate pseudospectra of operators.
Findings
The author established some results on ultrametric bounded Fredholm operators and approximate pseudospectra of closed and densely defined (resp. bounded) linear operators on ultrametric Banach spaces.
Originality/value
The results of the present manuscript are original.
Keywords
Citation
Ettayb, J. (2024), "Ultrametric Fredholm operators and approximate pseudospectrum", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-09-2023-0007
Publisher
:Emerald Publishing Limited
Copyright © 2024, Jawad Ettayb
License
Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
In ultrametric operator theory, Serre [1] studied the operator I − A where A is a completely continuous linear operator on a free Banach space. On the other hand, Gurson [2] lifted this restriction by working on general ultrametric Banach spaces. Recently, Nadathur [3] extended and studied some classical results on compact and Fredholm operators on ultrametric Banach spaces over a spherically complete field
The pseudospectra of bounded linear operators and the pseudospectra of bounded linear operator pencils and the condition pseudospectra of matrices and bounded linear operators were extended and studied by several authors, see Refs. [11–14].
In this paper, we demonstrate some results on Fredholm operators on ultrametric Banach spaces. On the other hand, we introduce and study the approximate pseudospectra of closed and densely defined linear operators on ultrametric Banach spaces. In particular, we prove that the approximate pseudospectra associated with various ɛ are nested sets and that the intersection of all the approximate pseudospectra is the approximate spectrum. On the other hand, we introduce the essential approximate pseudospectra and we study some of its properties.
Throughout this paper, E, F and G are infinite-dimensional ultrametric Banach spaces over a complete ultrametric valued field
2. Preliminaries
We continue by recalling some preliminaries.
[7] We say that
[7] Let
[7] Let
The set of all Fredholm operators from E into F is defined by
[15] Let E and F be two ultrametric Banach spaces over
The collection of all compact operators from E into F is denoted by
[9] Let
[15] Let
[9] Let E be an ultrametric Banach space and let
The collection of all completely continuous linear operators on E is denoted by
[9] Classical examples of completely continuous operators include finite rank operators.
[12] Suppose that
[16] Assume that E, F are ultrametric Banach spaces. Let A: D(A) ⊆ E → F be a surjective closed linear operator. Then A is an open map.
Let A: D(A) ⊆ E → F. When the domain of A is dense in E, the adjoint operator A′ of A is defined as usual. Specifically, the operator A′: D(A′) ⊆ F′ → E′ satisfies
[16] Let A be a linear operator with dense domain. Then A′ is a closed linear operator.
[16] Let A be a linear operator with dense domain. Then the following statement holds:
For more details on bounded linear operators, see Ref. [17].
[18] Let E be an ultrametric Banach space over a spherically complete field
[19] Let E be an ultrametric normed vector space over a spherically complete field
[20] Let E and F be two ultrametric Banach spaces and let
- (1).
The operator A is called Fredholm perturbation if A + B ∈ Φ(E, F) whenever B ∈ Φ(E, F).
- (2).
A is called an upper (resp. lower) semi-Fredholm perturbation A + B ∈ Φ+(E, F) (resp. A + B ∈ Φ−(E, F)) whenever B ∈ Φ+(E, F) (resp. B ∈ Φ−(E, F)).
We denote by
[20] Let E be an ultrametric Banach space over a spherically complete field
The proof of the next theorem is similar to the classical case, see Ref. [20].
[20] Let E be an ultrametric Banach space over
[3] Let E and F be two ultrametric Banach spaces over a spherically complete field
[3] Let E and F be two ultrametric Banach spaces over a spherically complete field
[3] Let E, F and G be three ultrametric Banach spaces over a spherically complete field
[3] Let E be an ultrametric Banach space over a spherically complete field
[9] If
[12] Let E and F be two ultrametric Banach spaces over a spherically complete field
[16] Suppose that E, F are ultramtric Banach spaces. Let A be a closed linear operator with dense domain. If R(A) is a closed subspace which has the weak extension property in F, then R(A′) = N(A)⊥.
In the next proposition, we assume that A′ exists.
[3] Let E and F be two ultrametric Banach spaces over a spherically complete field
3. Results
As a simple consequence of Theorems 2.18 and 2.20, we have:
Let E and F be two ultrametric Banach spaces over a spherically complete field
Proof. Since A0A − IE and AA0 − IF are of finite rank, we get
Let E, F and G be three ultrametric Banach spaces over a spherically complete field
Proof. Suppose that A ∈ Φ(E, F). By Theorem 2.19, there is
Let E, F and G be three ultrametric Banach spaces over a spherically complete field
Proof. Since R(BA) ⊂ R(B), by Lemma 2.14, we get that R(B) is closed. Since R(BA) ⊂ R(B) and α(B) is finite, we have β(B) ≤ β(BA). Using the fact that α(B) is finite, we get B ∈ Φ(F, G). By Theorem 3.2, we have A ∈ Φ(E, F). □
Let E, F and G be three ultrametric Banach spaces over a spherically complete field
Proof. If BA ∈ Φ(E, G), then by Proposition 2.25, A′B′ ∈ Φ(G′, E′). Since α(A′) = β(A) is finite, from Theorem 3.3, we get A′ ∈ Φ(F′, E′) and B′ ∈ Φ(G′, F′). Furthermore, α(B′′) is finite. Also
Using Theorem 3.3, we get A ∈ Φ(E, F) and B ∈ Φ(F, G). □
Let E be an ultrametric Banach space over a spherically complete field
Proof. One can see that for each k ∈ {1, …, n}, N(Ak) ⊂ N(A) and R(A) ⊂ R(Ak). If A = A1⋯An ∈ Φ(E), then for all k ∈ {1, …, n}, α(Ak) and β(Ak) are finite. Since R(A) is closed and β(B) is finite and
Let E and F be two ultrametric Banach spaces over a spherically complete field
Proof. From Proposition 2.25, if A ∈ Φ(E, F), then A′ ∈ Φ(F′, E′). Thus A ∈ Φ+(E, F) and A′ ∈ Φ+(F′, E′). Conversely, if A ∈ Φ+(E, F) and A′ ∈ Φ+(F′, E′), then α(A) is finite and R(A) is closed. From the fact that A′ ∈ Φ+(F′, E′), we get α(A′) = β(A) is finite. Thus A ∈ Φ(E, F). □
We introduce the following definitions:
Let E be an ultrametric Banach space over a spherically complete field
Let E be an ultrametric Banach space over a spherically complete field
Let E be an ultrametric Banach space over a spherically complete field
- (1).
For any ɛ > 0, we have σap,ɛ(A) ⊆ σɛ(A);
- (2).
σap(A) = ⋂ɛ>0σap,ɛ(A);
- (3).
For all ɛ1 and ɛ2 such that 0 < ɛ1 < ɛ2, we have
; - (4).
For all
and ɛ > 0, we have σap,ɛ(A + μIE) = σap,ɛ + μ; - (5).
For each
and ɛ > 0, we have σap,|μ|ɛ(μA) = μσap,ɛ(A).
Proof.
- (1).
Let λ∉σɛ(A). Then
. On the other hand,
Thus λ∉σap,ɛ(A).
- (2).
From Definition 3.8, for each ɛ > 0, we see that σap(A) ⊆ σap,ɛ(A). Thus σap(A) ⊆⋂ɛ>0σap,ɛ(A). Conversely, if λ ∈⋂ɛ>0σap,ɛ(A), then for each ɛ > 0, λ ∈ σap,ɛ(A). If λ ∈ σap(A), there is nothing to prove. If λ∉σap(A), then
. Taking the limit as ɛ → 0+, we get infx ∈ D(A),‖x‖=1‖(A − λIE)x‖ = 0, which is a contradiction. Hence λ ∈ σap(A). - (3).
Let ɛ1 and ɛ2 be such that 0 < ɛ1 < ɛ2. If
, then infx ∈ D(A),‖x‖=1‖(A − λIE)x‖ < ɛ1 < ɛ2. Thus . - (4).
If λ ∈ σap,ɛ(A + μIE), then either λ ∈ σap(A + μIE) or infx ∈ D(A),‖x‖=1‖(A − (λ − μ)IE)x‖ < ɛ. Thus λ ∈ μ + σap,ɛ(A). Similarly, if λ ∈ μ + σap,ɛ(A), then λ ∈ σap,ɛ(A + μIE).
- (5).
If λ ∈ σap,|μ|ɛ(μA), then
Thus
Let E be an ultrametric Banach space over a spherically complete field
Proof. Let
Then, S is linear and
Then, ‖S‖ < ɛ. Furthermore, infx ∈ D(A),‖x‖=1‖(A + S − λIE)x‖ = 0, because
Let E be an ultrametric Banach space over a spherically complete field
Proof. If λ ∈ σap,ɛ−‖S‖(A), then by Theorem 3.10, there is
Since ‖B − S‖ ≤ ‖B‖ + ‖S‖ < ɛ, by Theorem 3.10, we get λ ∈ σap,ɛ(A + S). Similarly, if λ ∈ σap,ɛ(A + S), we obtain that λ ∈ σap,ɛ+‖S‖(A). □
Let E be an ultrametric Banach space over a spherically complete field
Let E be an ultrametric Banach space over a spherically complete field
Let E be an ultrametric Banach space over a spherically complete field
- (1).
σeap(A) = ⋂ɛ>0σeap,ɛ(A);
- (2).
For all ɛ1 and ɛ2 such that ɛ1 < ɛ2, we have
; - (3).
σeap,ɛ(A + C) = σeap,ɛ(A), for each
.
Proof.
- (1).
Let λ∉σeap,ɛ(A). Then, there is
such that λ∉σap,ɛ(A + C). Hence λ∉σeap(A). Thus σeap(A) ⊂⋂ɛ>0σeap,ɛ(A). Conversely, if λ ∈⋂ɛ>0σeap,ɛ(A), then for each ɛ > 0, λ ∈ σeap,ɛ(A). Hence, for all such that λ ∈ σap,ɛ(A + C). Thus, infx ∈ D(A),‖x‖=1‖(A + C − λIE)x‖ < ɛ. Taking the limit as ɛ → 0, we get infx ∈ D(A),‖x‖=1‖(A + C − λIE)x‖ = 0. Hence λ ∈ σeap(A). - (2).
If
, then for all such that infx ∈ D(A),‖x‖=1‖(A + C − λIE)x‖ < ɛ1 < ɛ2. Thus . - (3).
Follow from Definition 3.13.
In the next theorem, we give a characterization of the essential approximate pseudospectrum by means of ultrametric semi-Fredholm operators.
Let E be an ultrametric Banach space over a spherically complete field
Proof. If λ∉σeap,ɛ(A), then there is
From Theorem 2.16, we have
Conversely, suppose that for each
Since C has a closed range and α(C) = 0, by (3.1), there is M > 0 such that
Hence infx ∈ D(A),‖x‖=1‖(A + K + B − λIE)x‖ ≥ M > 0. Thus λ∉σap(A + K + B). Consequently, λ∉σeap,ɛ(A). □
Let E be an ultrametric Banach space over a spherically complete field
Let E be an ultrametric Banach space over a spherically complete field
Proof. If
From Theorem 2.16, we have
By Theorem 3.15, we see that λ∉σeap,ɛ(A). Conversely, from
(i) From Theorem 3.18, we have σeap,ɛ(A + C) = σeap,ɛ(A), for each
Let E be an ultrametric Banach space over a spherically complete field
Proof. Follow from Theorem 3.11 and Proposition 3.9. □
4. Bounded cases
First, we introduce the following definitions.
Let E be an ultrametric Banach space over a spherically complete field
Let E be an ultrametric Banach space over a spherically complete field
As a particular case of Proposition 3.9, we have:
Let E be an ultrametric Banach space over a spherically complete field
- (1).
For any ɛ > 0, we have σap,ɛ(A) ⊆ σɛ(A).
- (2).
σap(A) = ⋂ɛ>0σap,ɛ(A);
- (3).
For all ɛ1 and ɛ2 such that 0 < ɛ1 < ɛ2, we have
; - (4).
For all
and ɛ > 0, we have σap,ɛ(A + μIE) = σap,ɛ + μ; - (5).
For each
and ɛ > 0, we have σap,|μ|ɛ(μA) = μσap,ɛ(A).
As a particular case of Theorem 3.10, we have:
Let E be an ultrametric Banach space over a spherically complete field
As a particular case of Theorem 3.11, we have:
Let E be an ultrametric Banach space over a spherically complete field
Let E be an ultrametric Banach space over a spherically complete field
As a particular case of Definition 3.13, we have:
Let E be an ultrametric Banach space over a spherically complete field
As a particular case of Proposition 3.14, we have:
Let E be an ultrametric Banach space over a spherically complete field
- (1).
σeap(A) = ⋂ɛ>0σeap,ɛ(A).
- (2).
For all ɛ1 and ɛ2 such that ɛ1 < ɛ2, we have
. - (3).
σeap,ɛ(A + C) = σeap,ɛ(A), for each
.
In the next theorem, we give a characterization of the essential approximate pseudospectra of bounded linear operators by means of ultrametric semi-Fredholm operators.
Let E be an ultrametric Banach space over a spherically complete field
Proof. It is a particular case of Theorem 3.15. □
Let E be an ultrametric Banach space over a spherically complete field
Let E be an ultrametric Banach space over a spherically complete field
(i) From Theorem 4.12, σeap,ɛ(A + C) = σeap,ɛ(A), for each
As a particular case of Theorem 3.20, we have:
Let E be an ultrametric Banach space over a spherically complete field
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Further reading
21Blali A, El Amrani A, Ettayb J. A note on pencil of bounded linear operators on non-archimedean Banach spaces. Methods Funct Anal Topology. 2022; 28(2): 105-9. doi: 10.31392/mfat-npu26_2.2022.02.