Ultrametric Fredholm operators and approximate pseudospectrum

Jawad Ettayb

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Open Access. Article publication date: 31 October 2024

257

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 [] studied the operator I − A where A is a completely continuous linear operator on a free Banach space. On the other hand, Gurson [] lifted this restriction by working on general ultrametric Banach spaces. Recently, Nadathur [] extended and studied some classical results on compact and Fredholm operators on ultrametric Banach spaces over a spherically complete field K. Moreover, Schikhof gave a basic theory for compact and semi-Fredholm operators on ultrametric Banach spaces, for more details, we refer to Ref. []. Furthermore, Perez-Garcia [] studied the Calkin algebras and semi-Fredholm operators on ultrametric Banach spaces. The stability of Fredholm operators and semi-Fredholom operators under smallest perturbation of operators and under compact operators on ultrametric Banach spaces were proved by Araujo, Perez-Garcia and Vega []. There are many studies on ultrametric Fredholm operators, see Refs. [,,,].

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. [].

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 K with a non-trivial valuation |⋅|, L(E,F) denotes the set of all continuous linear operators from E into F, IE is the identity operator on E and IF is the identity operator on F. If E = F, we have L(E,F)=L(E). If AL(E), N(A) and R(A) denote the kernel and the range of A respectively. For more details see Refs. []. Recall that, an unbounded linear operator A: D(A) ⊆ EF is said to be closed if for each (xn) ⊂ D(A) such that ‖xn − x‖ → 0 and ‖Axn − y‖ → 0 as n, for some x ∈ E and y ∈ F, then x ∈ D(A) and y = Ax. A is called densely defined if D(A) is dense in E. The collection of closed linear operators from E into F is denoted by C(E,F). If E = F, we put C(E,F)=C(E).

2. Preliminaries

We continue by recalling some preliminaries.

Definition 2.1.

[] We say that AL(E,F) has an index when both α(A) = dim N(A) and β(A)=dimF/R(A) are finite. In this case, the index of the linear operator A is defined as ind(A) = α(A) − β(A).

Definition 2.2.

[] Let AL(E,F). A is said to be an upper semi-Fredholm operator if α(A) is finite and R(A) is closed. The set of all upper semi-Fredholm operators from E into F is denoted by Φ+(E, F).

Definition 2.3.

[] Let AL(E,F), A is said to be a lower semi-Fredholm operator if β(A) is finite. The set of all semi-Fredholm operators from E into F is denoted by Φ(E, F).

The set of all Fredholm operators from E into F is defined by

Φ(E,F)=Φ+(E,F)Φ(E,F).

Definition 2.4.

[] Let E and F be two ultrametric Banach spaces over K. A linear map A: EF is said to be compact if A(BE) is compactoid in F, where BE = {x ∈ E: ‖x‖ ≤ 1}.

The collection of all compact operators from E into F is denoted by K(E,F).

Definition 2.5.

[] Let AL(E,F). A is called an operator of finite rank if R(A) is a finite dimensional subspace of F.

Theorem 2.6.

[] Let AL(E,F). Then A is compact if and only if for each ɛ > 0, there exists an operator SL(E,F) such that R(S) is finite-dimensional and ‖A − S‖ < ɛ.

Definition 2.7.

[] Let E be an ultrametric Banach space and let SL(E). S is said to be completely continuous if, there exists a sequence of finite rank linear operators (An) such that ‖An − S‖ → 0 as n.

The collection of all completely continuous linear operators on E is denoted by Cc(E).

Example 2.8.

[] Classical examples of completely continuous operators include finite rank operators.

Theorem 2.9.

[] Suppose that K is spherically complete. Then, for each A ∈ Φ(E, F) and KCc(E,F), A + K ∈ Φ(E, F) and ind(A + K) = ind(A).

Theorem 2.10.

[] 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) ⊆ EF. 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

Ax,y=x,Ay
for all x ∈ D(A), y′ ∈ D(A′). As in the classical case, the following property is an immediate consequence of the definition.

Proposition 2.11.

[] Let A be a linear operator with dense domain. Then A′ is a closed linear operator.

Proposition 2.12.

[] Let A be a linear operator with dense domain. Then the following statement holds:

R(A)=N(A)(F/R(A)¯).

For more details on bounded linear operators, see Ref. [].

Theorem 2.13.

[] Let E be an ultrametric Banach space over a spherically complete field K. For each x ∈ E\{0}, there is x′ ∈ E′ such that x′(x) = 1 and ‖x′‖ = ‖x‖−1.

Lemma 2.14.

[] Let E be an ultrametric normed vector space over a spherically complete field K, and suppose that E = N ⊕ E0, where E0 is a closed subspace and N is finite dimensional. If E1 is a subspace of E containing E0, then E1 is closed.

Definition 2.15.

[] Let E and F be two ultrametric Banach spaces and let AL(E,F).

  • (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 F(E,F) the set of Fredholm perturbations and by F+(E,F) (resp. F(E,F)) the set of upper semi-Fredholm (resp. lower semi-Fredholm) perturbations. For E = F, we put F(E,F)=F(E),F+(E,F)=F+(E) and F(E,F)=F(E). The proof of the next proposition is similar to the classical case, see Ref. [].

Proposition 2.16.

[] Let E be an ultrametric Banach space over a spherically complete field K. (i) If A ∈ Φ(E) and FF(E), then A + F ∈ Φ(E) and ind(A + F) = ind(A). (ii) If A ∈ Φ+(E) and FF+(E), then A + F ∈ Φ+(E) and ind(A + F) = ind(A).

The proof of the next theorem is similar to the classical case, see Ref. [].

Theorem 2.17.

[] Let E be an ultrametric Banach space over K. Let A ∈ Φ+(E). Then the following statements are equivalent: (i) ind(A) ≤ 0; (ii) A can be expressed in the form A = S + K where KCc(E), and SC(E) is an operator with closed range with α(S) = 0.

Theorem 2.18.

[] Let E and F be two ultrametric Banach spaces over a spherically complete field K. Let AL(E,F). If there is A0,A1L(F,E) such that A0AIECc(E) and AA1IFCc(F). Then A ∈ Φ(E, F).

Theorem 2.19.

[] Let E and F be two ultrametric Banach spaces over a spherically complete field K. Let A ∈ Φ(E, F), then there is A0L(F,E) such that A0A − IE and AA0 − IF have finite dimensional images.

Theorem 2.20.

[] Let E, F and G be three ultrametric Banach spaces over a spherically complete field K. If A ∈ Φ(E, F) and B ∈ Φ(F, G), then BA ∈ Φ(E, G) and ind(BA) = ind(A) + ind(B).

Theorem 2.21.

[] Let E be an ultrametric Banach space over a spherically complete field K. Let ACc(E) and λK\{0}, then λIE − A ∈ Φ(E) and ind(λIE − A) = 0.

Theorem 2.22.

[] If A,BCc(E) and C,DL(E), then (i) A+BCc(E); (ii) AC,DACc(E).

Lemma 2.23.

[] Let E and F be two ultrametric Banach spaces over a spherically complete field K. Let AL(E,F) and KCc(E,F). Then A + K ∈ Φ(E, F) and ind(A + K) = ind(A) + ind(K).

Corollary 2.24.

[] 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.

Proposition 2.25.

[] Let E and F be two ultrametric Banach spaces over a spherically complete field K. Let A ∈ Φ(E, F), then A′ ∈ Φ(F′, E′) and ind(A′) = −ind(A).

3. Results

As a simple consequence of and , we have:

Corollary 3.1.

Let E and F be two ultrametric Banach spaces over a spherically complete field K. Let A ∈ Φ(E, F) and let A0L(F,E) be such that A0A − IE and AA0 − IF are of finite rank. Then A0 ∈ Φ(F, E) and ind(A0) = −ind(A).

Proof. Since A0A − IE and AA0 − IF are of finite rank, we get A0AIECc(E) and AA0IFCc(F). Using , we have A0 ∈ Φ(F, E). Since A0 ∈ Φ(F, E) and A ∈ Φ(E, F). By , A0A ∈ Φ(E) and ind(A0A) = ind(A) + ind(A0). From , ind(A0A) = ind(A) + ind(A0) = ind(IE + B) = 0, where B = A0A − IE is of finite rank. □

Theorem 3.2.

Let E, F and G be three ultrametric Banach spaces over a spherically complete field K. Let AL(E,F) and BL(F,G) such that BA ∈ Φ(E, G). Then A ∈ Φ(E, F) if and only if B ∈ Φ(F, G).

Proof. Suppose that A ∈ Φ(E, F). By , there is A0L(F,E) such that A0A − IE and AA0 − IF are of finite rank. By , A0 ∈ Φ(F, E). Set C = AA0 − IF, then BC = BAA0 − B. Since BA ∈ Φ(E, G). From , BAA0 ∈ Φ(F, G). From , CCc(F). By , we have BCCc(F,G). From , B ∈ Φ(F, G). Similarly we obtain that if B ∈ Φ(F, G), hence A ∈ Φ(E, F). □

Theorem 3.3.

Let E, F and G be three ultrametric Banach spaces over a spherically complete field K. Let AL(E,F) and BL(F,G) be such that BA ∈ Φ(E, G). If α(B) is finite, then A ∈ Φ(E, F) and B ∈ Φ(F, G).

Proof. Since R(BA) ⊂ R(B), by , 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 , we have A ∈ Φ(E, F). □

Theorem 3.4.

Let E, F and G be three ultrametric Banach spaces over a spherically complete field K. Let AL(E,F) and BL(F,G) be such that BA ∈ Φ(E, G). If β(A) is finite, then A ∈ Φ(E, F) and B ∈ Φ(F, G).

Proof. If BA ∈ Φ(E, G), then by , AB′ ∈ Φ(G′, E′). Since α(A′) = β(A) is finite, from , we get A′ ∈ Φ(F′, E′) and B′ ∈ Φ(G′, F′). Furthermore, α(B′′) is finite. Also

α(B)α(B)is finite.

Using , we get A ∈ Φ(E, F) and B ∈ Φ(F, G). □

Lemma 3.5.

Let E be an ultrametric Banach space over a spherically complete field K. Let A1,,AnL(E) be such that for all i, j ∈ {1, …, n}, AiAj = AjAi. Suppose that A = A1An ∈ Φ(E). Then for all k ∈ {1, …, n}, Ak ∈ Φ(E).

Proof. One can see that for each k ∈ {1, …, n}, N(Ak) ⊂ N(A) and R(A) ⊂ R(Ak). If A = A1An ∈ 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 K is spherically complete, then there is a finite-dimensional subspace M of E such that E = R(A) ⊕ M. Since for any k ∈ {1, …, n}, R(A) ⊂ R(Ak), from , we have R(Ak) is closed for each k ∈ {1, …, n}. □

Theorem 3.6.

Let E and F be two ultrametric Banach spaces over a spherically complete field K. Then A ∈ Φ(E, F), if and only if A ∈ Φ+(E, F) and A′ ∈ Φ+(F′, E′).

Proof. From , 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:

Definition 3.7.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AC(E) be closed and densely defined. The approximate spectrum σap(A) of A on E is defined by

σap(A)={λK:infxD(A),x=1(AλIE)x=0}.

Definition 3.8.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AC(E) be closed and densely defined, and let ɛ > 0. The approximate pseudospectrum σap,ɛ(A) of A on E is defined by

σap,ε(A)=σap(A){λK:infxD(A),x=1(AλIE)x<ε}.

Proposition 3.9.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AC(E) be closed and densely defined. Then, the following statements hold:

  • (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 σap(A)σap,ε1(A)σap,ε2(A);

  • (4).

    For all μK and ɛ > 0, we have σap,ɛ(A + μIE) = σap,ɛ + μ;

  • (5).

    For each μK\{0} and ɛ > 0, we have σap,|μ|ɛ(μA) = μσap,ɛ(A).

Proof.

  • (1).

    Let λσɛ(A). Then (AλIE)1ε1. On the other hand,

1infxD(A),x=1(AλIE)x=supxD(A),x=1x(AλIE)x,mtdmtdmtdmtdmtdmtd

Thus λσap,ɛ(A).

  • (2).

    From , 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 λσap(A), then λ{λK:infxD(A),x=1(AλIE)x<ε}. 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 λσap,ε1(A), then infx  D(A),‖x‖=1‖(A − λIE)x‖ < ɛ1 < ɛ2. Thus λσap,ε2(A).

  • (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

infxD(A),x=1(μAλIE)x=infxD(A),x=1(μAλIE)x,mtdmtdmtdmtd

Thus λμσap,ε(A). Then σap,|μ|ɛ(μA) ⊆ μσap,ɛ(A). Similarly we get μσap,ɛ(A) ⊆ σap,|μ|ɛ(μA).Hence σap,|μ|ɛ(μA) = μσap,ɛ(A). □

Theorem 3.10.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|, and let AC(E) be closed and densely defined, and let ɛ > 0. Then,

σap,ε(A)=SL(E):S<εσap(A+S).

Proof. Let λSL(E):S<εσap(A+S). Then infx  D(A),‖x‖=1‖(A + S − λIE)x‖ = 0. We will prove that λ ∈ σap,ɛ(A). From the estimate

(AλIE)u=(A+SλIE)uSumax{(A+SλIE)u,Su},
We infer that infx  D(A),‖x‖=1‖(A − λIE)x‖ < ɛ. Conversely, suppose that λ ∈ σap,ɛ(A). We discuss two cases. First case: If λ ∈ σap(A), then it suffices to put S = 0. Second case: If λσap(A), then there is y ∈ E\{0} such that ‖y‖ = 1 and ‖(A − λIE)y‖ < ɛ. By , there exists ϕ ∈ E′ such that ϕ(y) = 1 and ‖ϕ‖ = ‖y−1 = 1. Define S on E by
Sx=ϕ(x)(AλIE)yfor allxE.

Then, S is linear and

Sx=ϕ(x)(AλIE)y<εx.

Then, ‖S‖ < ɛ. Furthermore, infx  D(A),‖x‖=1‖(A + S − λIE)x‖ = 0, because

infxD(A),x=1(A+SλIE)x(A+SλIE)ymtdmtd
for y ∈ E.

Theorem 3.11.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AC(E) be closed and densely defined, and let ɛ > 0. Let SL(E) be such thatS‖ < ɛ. Then,

σap,εS(A)σap,ε(A+S)σap,ε+S(A).

Proof. If λ ∈ σap,ɛ−‖S(A), then by , there is BL(E) such that ‖B‖ < ɛ − ‖S‖ and

λσap(A+B)=σap((A+S)+(BS)).

Since ‖B − S‖ ≤ ‖B‖ + ‖S‖ < ɛ, by , we get λ ∈ σap,ɛ(A + S). Similarly, if λ ∈ σap,ɛ(A + S), we obtain that λ ∈ σap,ɛ+‖S(A). □

Definition 3.12.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|, and let AC(E) be closed and densely defined. Then, the essential approximate spectrum σeap(A) of A is defined by

σeap(A)=CCc(E)σap(A+C).

Definition 3.13.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|, and let AC(E) be closed and densely defined, and let ɛ > 0. Then the essential approximate pseudospectrum σeap(A) of A is defined by

σeap,ε(A)=CCc(E)σap,ε(A+C).

Proposition 3.14.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AC(E) be closed and densely defined. Then, the following statements hold:

  • (1).

    σeap(A) = ⋂ɛ>0σeap,ɛ(A);

  • (2).

    For all ɛ1 and ɛ2 such that ɛ1 < ɛ2, we have σeap(A)σeap,ε1(A)σeap,ε2(A);

  • (3).

    σeap,ɛ(A + C) = σeap,ɛ(A), for each CCc(E).

Proof.

  • (1).

    Let λσeap,ɛ(A). Then, there is CCc(E) such that λσap,ɛ(A + C). Hence λσeap(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 CCc(E) 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 λσeap,ε1(A), then for all CCc(E) such that infx  D(A),‖x‖=1‖(A + C − λIE)x‖ < ɛ1 < ɛ2. Thus λσeap,ε2(A).

  • (3).

    Follow from .

In the next theorem, we give a characterization of the essential approximate pseudospectrum by means of ultrametric semi-Fredholm operators.

Theorem 3.15.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AC(E) be closed and densely defined, and let ɛ > 0. Then λσeap,ɛ(A) if and only if for each BL(E) such thatB‖ < ɛ, we have A + B − λIE ∈ Φ+(E) and ind(A + B − λIE) ≤ 0.

Proof. If λσeap,ɛ(A), then there is KCc(E) such that λσap,ɛ(A + K). Using , for each BL(E) such that ‖B‖ < ɛ, we have λσap(A + K + B). Hence for each BL(E) such that ‖B‖ < ɛ, we obtain

A+K+BλIEΦ+(E)
and
ind(A+K+BλIE)0.

From Theorem 2.16, we have

A+BλIEΦ+(E)
and
ind(A+BλIE)0.

Conversely, suppose that for each BL(E) such that ‖B‖ < ɛ, we have A + B − λIE ∈ Φ+(E) and ind(A + B − λIE) ≤ 0. Then from , we get

A+BλIE=(C+K),
where KCc(E) and CC(E) with closed range α(C) = 0. Hence
(3.1) A+K+BλIE=C
and
ind(A+K+BλIE)=α(C)=0.

Since C has a closed range and α(C) = 0, by , there is M > 0 such that

(A+K+BλIE)xMx,for eachxD(A).

Hence infx  D(A),‖x‖=1‖(A + K + B − λIE)x‖ ≥ M > 0. Thus λσap(A + K + B). Consequently, λσeap,ɛ(A). □

Remark 3.16.

From , we get

σeap,ε(A)=BL(E):B<εσeap(A+B).

From and from , we have.

Corollary 3.17.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AC(E) be closed and densely defined, and let ɛ > 0. Then, we have

σeap(A)=limε0CCc(E)σap,ε(A+C)¯=ε>0BL(E):B<εσeap(A+B).

Theorem 3.18.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AC(E) be closed and densely defined, and let ɛ > 0. Then, we have

σeap,ε(A)=FF+(E)σap,ε(A+F).

Proof. If λFF+(E)σap,ε(A+F), then there is FF+(E) such that λσap,ɛ(A + F). By , for each BL(E) with ‖B‖ < ɛ, we have λσap(A + F + B). Hence, for each BL(E) such that ‖B‖ < ɛ, we have

A+F+BλIEΦ+(E),
and
ind(A+F+BλIE)0.

From Theorem 2.16, we have

A+BλIEΦ+(E)
and
ind(A+BλIE)0.

By , we see that λσeap,ɛ(A). Conversely, from Cc(E)F+(E), we infer that

FF+(E)σap,ε(A+F)FCc(E)σap,ε(A+F)=σeap,ε(A).

Remark 3.19.

(i) From , we have σeap,ɛ(A + C) = σeap,ɛ(A), for each CF+(E). (ii) Let J(E) be a subset of L(E). If Cc(E)J(E)F+(E), then we have

σeap,ε(A)=CJ(E)σap,ε(A+C),
and
σeap,ε(A+C)=σeap,ε(A)for eachCJ(E).

Theorem 3.20.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AC(E) be closed and densely defined, and let ɛ > 0. Let SL(E) be such thatS‖ < ɛ. Then, we have (i) σeap,ɛ−‖S(A) ⊆ σeap,ɛ(A + S) ⊆ σeap,ɛ+‖S(A); (ii) For any λ,μK and μ ≠ 0, we have

σeap,ε(λIE+μA)=λ+μσeap,ε|μ|(A).

Proof. Follow from and .

4. Bounded cases

First, we introduce the following definitions.

Definition 4.1.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AL(E). The approximate spectrum σap(A) of A on E is defined by

σap(A)={λK:infxD(A),x=1(AλIE)x=0}.

Definition 4.2.

Let E be an ultrametric Banach space over a spherically complete field K be such that E|K|. Let AL(E) and ɛ > 0. The approximate pseudospectrum σap,ɛ(A) of A on E is defined by

σap,ε(A)=σap(A){λK:infxD(A),x=1(AλIE)x<ε}.

As a particular case of , we have:

Proposition 4.3.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K| and let AL(E). Then, the following statements hold:

  • (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 σap(A)σap,ε1(A)σap,ε2(A);

  • (4).

    For all μK and ɛ > 0, we have σap,ɛ(A + μIE) = σap,ɛ + μ;

  • (5).

    For each μK\{0} and ɛ > 0, we have σap,|μ|ɛ(μA) = μσap,ɛ(A).

As a particular case of , we have:

Theorem 4.4.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|, and let AL(E) and ɛ > 0. Then, we have

σap,ε(A)=SL(E):S<εσap(A+S).

As a particular case of , we have:

Theorem 4.5.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let A,SL(E) be such thatS‖ < ɛ. Then, we have

σap,εS(A)σap,ε(A+S)σap,ε+S(A).

Definition 4.6.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K| and let AL(E). The essential approximate spectrum σeap(A) of A is defined by

σeap(A)=CCc(E)σap(A+C).

As a particular case of , we have:

Definition 4.7.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|, let AL(E) and ɛ > 0. The essential approximate pseudospectrum σeap(A) of A is defined by

σeap,ε(A)=CCc(E)σap,ε(A+C).

As a particular case of , we have:

Proposition 4.8.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AL(E). Then, the following statements hold:

  • (1).

    σeap(A) = ⋂ɛ>0σeap,ɛ(A).

  • (2).

    For all ɛ1 and ɛ2 such that ɛ1 < ɛ2, we have σeap(A)σeap,ε1(A)σeap,ε2(A).

  • (3).

    σeap,ɛ(A + C) = σeap,ɛ(A), for each CCc(E).

In the next theorem, we give a characterization of the essential approximate pseudospectra of bounded linear operators by means of ultrametric semi-Fredholm operators.

Theorem 4.9.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AL(E) and ɛ > 0. Then λσeap,ɛ(A) if and only if for each BL(E) such thatB‖ < ɛ, we have A + B − λIE ∈ Φ+(E) and ind(A + B − λIE) ≤ 0.

Proof. It is a particular case of .

Remark 4.10.

From , we get

σeap,ε(A)=BL(E):B<εσeap(A+B).

From , we have.

Corollary 4.11.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AL(E) and ɛ > 0. Then, we have

σeap(A)=limε0CCc(E)σap,ε(A+C)¯=ε>0BL(E):B<εσeap(A+B).

Theorem 4.12.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let AL(E) and ɛ > 0. Then, we have

σeap,ε(A)=FF+(E)σap,ε(A+F).

Remark 4.13.

(i) From , σeap,ɛ(A + C) = σeap,ɛ(A), for each CF+(E). (ii) Let J(E) be a subset of L(E). If Cc(E)J(E)F+(E), then we have

σeap,ε(A)=CJ(E)σap,ε(A+C)
and
σeap,ε(A+C)=σeap,ε(A)for eachCJ(E).

As a particular case of , we have:

Theorem 4.14.

Let E be an ultrametric Banach space over a spherically complete field K such that E|K|. Let A,SL(E) and ɛ > 0 be such thatS‖ < ɛ. Then, we have (i) σeap,ɛ−‖S(A) ⊆ σeap,ɛ(A + S) ⊆ σeap,ɛ+‖S(A); (ii) For any λ,μK and μ ≠ 0, we have

σeap,ε(λIE+μA)=λ+μσeap,ε|μ|(A).

Availability of data and materials: Data sharing not applicable to this article, as no datasets were generated or analyzed during the current study.

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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.

Corresponding author

Jawad Ettayb can be contacted at: ettayb.j@gmail.com

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