Ultrametric Fredholm operators and approximate pseudospectrum Open Access

[7] We say that $A∈L(E,F)$ 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).

[7] Let $A∈L(E,F)$⁠. 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).

[7] Let $A∈L(E,F)$⁠, 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

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

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

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

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

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

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

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

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

[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

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

[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 $K$⁠. For each x ∈ E\{0}, there is x′ ∈ E′ such that x′(x) = 1 and ‖x′‖ = ‖x‖⁻¹.

[19] 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.

[20] Let E and F be two ultrametric Banach spaces and let $A∈L(E,F)$⁠.

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. [20].

[20] Let E be an ultrametric Banach space over a spherically complete field $K$⁠. (i) If A ∈ Φ(E) and $F∈F(E)$, then A + F ∈ Φ(E) and ind(A + F) = ind(A). (ii) If A ∈ Φ₊(E) and $F∈F+(E)$, 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. [20].

[20] 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 $K∈Cc(E)$⁠, and $S∈C(E)$ is an operator with closed range with α(S) = 0.

[3] Let E and F be two ultrametric Banach spaces over a spherically complete field $K$⁠. Let $A∈L(E,F)$⁠. If there is $A0,A1∈L(F,E)$ such that $A0A−IE∈Cc(E)$ and $AA1−IF∈Cc(F)$⁠. Then A ∈ Φ(E, F).

[3] Let E and F be two ultrametric Banach spaces over a spherically complete field $K$⁠. Let A ∈ Φ(E, F), then there is $A0∈L(F,E)$ such that A₀A − I_E and AA₀ − I_F have finite dimensional images.

[3] 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).

[3] Let E be an ultrametric Banach space over a spherically complete field $K$⁠. Let $A∈Cc(E)$ and $λ∈K\{0}$, then λI_E − A ∈ Φ(E) and ind(λI_E − A) = 0.

[9] If $A,B∈Cc(E)$ and $C,D∈L(E)$, then (i) $A+B∈Cc(E)$; (ii) $AC,DA∈Cc(E)$⁠.

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

[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 $K$⁠. Let A ∈ Φ(E, F), then A′ ∈ Φ(F′, E′) and ind(A′) = −ind(A).

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

Proof. Since A₀A − I_E and AA₀ − I_F are of finite rank, we get $A0A−IE∈Cc(E)$ and $AA0−IF∈Cc(F)$⁠. Using Theorem 2.18, we have A₀ ∈ Φ(F, E). Since A₀ ∈ Φ(F, E) and A ∈ Φ(E, F). By Theorem 2.20, A₀A ∈ Φ(E) and ind(A₀A) = ind(A) + ind(A₀). From Theorem 2.21, ind(A₀A) = ind(A) + ind(A₀) = ind(I_E + B) = 0, where B = A₀A − I_E is of finite rank. □

Let E, F and G be three ultrametric Banach spaces over a spherically complete field $K$⁠. Let $A∈L(E,F)$ and $B∈L(F,G)$ 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 Theorem 2.19, there is $A0∈L(F,E)$ such that A₀A − I_E and AA₀ − I_F are of finite rank. By Theorem 2.18, A₀ ∈ Φ(F, E). Set C = AA₀ − I_F, then BC = BAA₀ − B. Since BA ∈ Φ(E, G). From Theorem 2.20, BAA₀ ∈ Φ(F, G). From Example 2.8, $C∈Cc(F)$⁠. By Theorem 2.22, we have $BC∈Cc(F,G)$⁠. From Lemma 2.23, B ∈ Φ(F, G). Similarly we obtain that if B ∈ Φ(F, G), hence A ∈ Φ(E, F). □

Let E, F and G be three ultrametric Banach spaces over a spherically complete field $K$⁠. Let $A∈L(E,F)$ and $B∈L(F,G)$ 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 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 $K$⁠. Let $A∈L(E,F)$ and $B∈L(F,G)$ 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 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 $K$⁠. Let $A1,…,An∈L(E)$ be such that for all i, j ∈ {1, …, n}, A_iA_j = A_jA_i. Suppose that A = A₁⋯A_n ∈ Φ(E). Then for all k ∈ {1, …, n}, A_k ∈ Φ(E).

Proof. One can see that for each k ∈ {1, …, n}, N(A_k) ⊂ N(A) and R(A) ⊂ R(A_k). If A = A₁⋯A_n ∈ Φ(E), then for all k ∈ {1, …, n}, α(A_k) and β(A_k) 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(A_k), from Lemma 2.14, we have R(A_k) is closed for each k ∈ {1, …, n}. □

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 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 $K$ such that $‖E‖⊆|K|$⁠. Let $A∈C(E)$ be closed and densely defined. The approximate spectrum σ_ap(A) of A on E is defined by

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

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

Proof.

Thus λ∉σ_ap,ɛ(A).

Thus $λμ∈σap,ε(A)$⁠. Then σ_ap,|μ|ɛ(μA) ⊆ μσ_ap,ɛ(A). Similarly we get μσ_ap,ɛ(A) ⊆ σ_ap,|μ|ɛ(μA).Hence σ_ap,|μ|ɛ(μA) = μσ_ap,ɛ(A). □

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

Proof. Let $λ∈⋃S∈L(E):‖S‖<εσap(A+S)$⁠. Then inf_{x ∈ D(A),‖x‖=1}‖(A + S − λI_E)x‖ = 0. We will prove that λ ∈ σ_ap,ɛ(A). From the estimate

We infer that inf_{x ∈ D(A),‖x‖=1}‖(A − λI_E)x‖ < ɛ. Conversely, suppose that λ ∈ σ_ap,ɛ(A). We discuss two cases. First case: If λ ∈ σ_ap(A), then it suffices to put S = 0. Second case: If λ∉σ_ap(A), then there is y ∈ E\{0} such that ‖y‖ = 1 and ‖(A − λI_E)y‖ < ɛ. By Theorem 2.13, there exists ϕ ∈ E′ such that ϕ(y) = 1 and ‖ϕ‖ = ‖y‖⁻¹ = 1. Define S on E by

Then, S is linear and

Then, ‖S‖ < ɛ. Furthermore, inf_{x ∈ D(A),‖x‖=1}‖(A + S − λI_E)x‖ = 0, because

for y ∈ E.

Let E be an ultrametric Banach space over a spherically complete field $K$ such that $‖E‖⊆|K|$⁠. Let $A∈C(E)$ be closed and densely defined, and let ɛ > 0. Let $S∈L(E)$ be such that ‖S‖ < ɛ. Then,

Proof. If λ ∈ σ_{ap,ɛ−‖S‖}(A), then by Theorem 3.10, there is $B∈L(E)$ such that ‖B‖ < ɛ − ‖S‖ and

Since ‖B − S‖ ≤ ‖B‖ + ‖S‖ < ɛ, by Theorem 3.10, we get λ ∈ σ_ap,ɛ(A + S). Similarly, if λ ∈ σ_ap,ɛ(A + S), we obtain that λ ∈ σ_{ap,ɛ+‖S‖}(A). □

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

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

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

Proof.

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 $K$ such that $‖E‖⊆|K|$⁠. Let $A∈C(E)$ be closed and densely defined, and let ɛ > 0. Then λ∉σ_eap,ɛ(A) if and only if for each $B∈L(E)$ such that ‖B‖ < ɛ, we have A + B − λI_E ∈ Φ₊(E) and ind(A + B − λI_E) ≤ 0.

Proof. If λ∉σ_eap,ɛ(A), then there is $K∈Cc(E)$ such that λ∉σ_ap,ɛ(A + K). Using Theorem 3.10, for each $B∈L(E)$ such that ‖B‖ < ɛ, we have λ∉σ_ap(A + K + B). Hence for each $B∈L(E)$ such that ‖B‖ < ɛ, we obtain

and

From Theorem 2.16, we have

and

Conversely, suppose that for each $B∈L(E)$ such that ‖B‖ < ɛ, we have A + B − λI_E ∈ Φ₊(E) and ind(A + B − λI_E) ≤ 0. Then from Theorem 2.17, we get

where $K∈Cc(E)$ and $C∈C(E)$ with closed range α(C) = 0. Hence

and

Since C has a closed range and α(C) = 0, by (3.1), there is M > 0 such that

Hence inf_{x ∈ D(A),‖x‖=1}‖(A + K + B − λI_E)x‖ ≥ M > 0. Thus λ∉σ_ap(A + K + B). Consequently, λ∉σ_eap,ɛ(A). □

From Theorem 3.15, we get

From Definition 3.13 and from Proposition 3.14, we have.

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

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

Proof. If $λ∉⋂F∈F+(E)σap,ε(A+F)$⁠, then there is $F∈F+(E)$ such that λ∉σ_ap,ɛ(A + F). By Theorem 3.10, for each $B∈L(E)$ with ‖B‖ < ɛ, we have λ∉σ_ap(A + F + B). Hence, for each $B∈L(E)$ such that ‖B‖ < ɛ, we have

and

From Theorem 2.16, we have

and

By Theorem 3.15, we see that λ∉σ_eap,ɛ(A). Conversely, from $Cc(E)⊂F+(E)$⁠, we infer that

(i) From Theorem 3.18, we have σ_eap,ɛ(A + C) = σ_eap,ɛ(A), for each $C∈F+(E)$⁠. (ii) Let J(E) be a subset of $L(E)$⁠. If $Cc(E)⊂J(E)⊂F+(E)$⁠, then we have

and

Let E be an ultrametric Banach space over a spherically complete field $K$ such that $‖E‖⊆|K|$⁠. Let $A∈C(E)$ be closed and densely defined, and let ɛ > 0. Let $S∈L(E)$ be such that ‖S‖ < ɛ. Then, we have (i) σ_{eap,ɛ−‖S‖}(A) ⊆ σ_eap,ɛ(A + S) ⊆ σ_{eap,ɛ+‖S‖}(A); (ii) For any $λ,μ∈K$ and μ ≠ 0, we have

Proof. Follow from Theorem 3.11 and Proposition 3.9. □

Let E be an ultrametric Banach space over a spherically complete field $K$ such that $‖E‖⊆|K|$⁠. Let $A∈L(E)$⁠. The approximate spectrum σ_ap(A) of A on E is defined by

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

As a particular case of Proposition 3.9, we have:

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

As a particular case of Theorem 3.10, we have:

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

As a particular case of Theorem 3.11, we have:

Let E be an ultrametric Banach space over a spherically complete field $K$ such that $‖E‖⊆|K|$⁠. Let $A,S∈L(E)$ be such that ‖S‖ < ɛ. Then, we have

Let E be an ultrametric Banach space over a spherically complete field $K$ such that $‖E‖⊆|K|$ and let $A∈L(E)$⁠. The essential approximate spectrum σ_eap(A) of A is defined by

As a particular case of Definition 3.13, we have:

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

As a particular case of Proposition 3.14, we have:

Let E be an ultrametric Banach space over a spherically complete field $K$ such that $‖E‖⊆|K|$⁠. Let $A∈L(E)$⁠. Then, the following statements hold:

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 $K$ such that $‖E‖⊆|K|$⁠. Let $A∈L(E)$ and ɛ > 0. Then λ∉σ_eap,ɛ(A) if and only if for each $B∈L(E)$ such that ‖B‖ < ɛ, we have A + B − λI_E ∈ Φ₊(E) and ind(A + B − λI_E) ≤ 0.

Proof. It is a particular case of Theorem 3.15. □

From Theorem 4.9, we get

From Proposition 4.8, we have.

Let E be an ultrametric Banach space over a spherically complete field $K$ such that $‖E‖⊆|K|$⁠. Let $A∈L(E)$ and ɛ > 0. Then, we have

Let E be an ultrametric Banach space over a spherically complete field $K$ such that $‖E‖⊆|K|$⁠. Let $A∈L(E)$ and ɛ > 0. Then, we have

(i) From Theorem 4.12, σ_eap,ɛ(A + C) = σ_eap,ɛ(A), for each $C∈F+(E)$⁠. (ii) Let J(E) be a subset of $L(E)$⁠. If $Cc(E)⊂J(E)⊂F+(E)$⁠, then we have

and

As a particular case of Theorem 3.20, we have:

Let E be an ultrametric Banach space over a spherically complete field $K$ such that $‖E‖⊆|K|$⁠. Let $A,S∈L(E)$ and ɛ > 0 be such that ‖S‖ < ɛ. Then, we have (i) σ_{eap,ɛ−‖S‖}(A) ⊆ σ_eap,ɛ(A + S) ⊆ σ_{eap,ɛ+‖S‖}(A); (ii) For any $λ,μ∈K$ and μ ≠ 0, we have