Towards a theory of some unbounded linear operators on <span class="mathjax-formula">$p$</span>-adic Hilbert spaces and applications

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BLAISE PASCAL

Toka Diagana

Towards a theory of some unbounded linear operators on p-adic Hilbert spaces and applications

Volume 12, n^o1 (2005), p. 205-222.

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Towards a theory of some unbounded linear operators on p-adic Hilbert spaces and

applications

Toka Diagana

Abstract

We are concerned with some unbounded linear operators on the so-called p-adic Hilbert space Eω. Both the Closedness and the self- adjointness of those unbounded linear operators are investigated. As applications, we shall consider the diagonal operator onEω, and the solvability of the equationAu=vwhereAis a linear operator onEω.

AMS subject classification. 47S10; 46S10; 47A05; 47B25.

Keywords and phrases: p-adic Hilbert space, free Banach space, unbounded linear operator, closed linear operator, self-adjoint operator, diagonal operator.

1 Introduction

The primary goal of this paper is to investigate upon some unbounded linear operators on the so-called p-adic Hilbert space Eω. For that, we first introduce and give the required background on thep-adic Hilbert space Eω. Next, we shall be dealing with natural issues such as the closedness and the self-adjointness of those unbounded linear operators. To do so, one equips the so-called direct sumEω×Eω ofEω with itself, with both an ultrametric norm and a hilbertian structure. Afterwards, it goes back to introduce an unitary operatorΓonEω×Eω which yields a remarkable description of A^∗, the adjoint of a linear operatorAdefined onEω which does have an adjoint, in terms ofA.

Let us mention that the p-adic Hilbert space Eω will play a key role throughout the paper. Apart from their intrinsic interests, p-adic Hilbert spaces have found extensive applications in theoretical physics.

For more on these and related issues we refer the reader to([1], [2], [3], [4], [5], [6], [7], [8], [9], [10], and [11]) and the references therein.

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LetKbe a complete ultrametric valued field. Classical examples of such a field includeQpthe field ofp-adic numbers wherep≥2is a prime,Cp the field ofp-adic complex numbers, and the field of formal Laurent series([4]).

An ultrametric Banach spaceEoverKis said to be a free Banach space (see [2], [3], and [4]) if there exists a family (e_i)i∈I (I being an index set) of elements of Esuch that each element x∈Ecan be written in an unique fashion as:

x=X

i∈I

x_ie_i, lim

i∈I x_ie_i= 0, and kxk= sup

i∈I

|x_i|ke_ik.

The family(e_i)i∈I is then called anorthogonal basisforE, and ifke_ik= 1, for alli∈I, the family (e_i)_i∈I is called an orthonormal basis. For a detailed description and properties of these spaces, we refer the reader to ([2], [3], [4], and [11]) and the references therein.

Up to now, we shall suppose that the index set I is N, the set of all natural integers.

For a free Banach spaceE, letE^∗denote its (topological) dual and B(E) the Banach space of all bounded linear operators onE(see [2], [3], and [4]).

BothE^∗ andB(E) are equipped with their respective natural norms.

For(u, v)∈E×E^∗we define the linear operator (v⊗u)by setting:

∀x∈E, (v⊗u) (x) :=v(x)u=hv, xiu.

It follows that(v⊗u)∈B(E)and kv⊗uk=kvk.kuk.

Let(e_i)_i∈

Nbe an orthogonal basis for E. We then definee⁰_i∈E^∗ by x=X

i∈N

x_ie_i, e⁰_i(x) =x_i.

It turns out that ke⁰_ik = 1

ke_ik. Furthermore, every x⁰ ∈ E^∗ can be expressed as a pointwise convergent series: x⁰ = X

i∈N

hx⁰, e_iie⁰_i. In addition to that, we have that:

kx⁰k:= sup

i∈N

|hx⁰, e_ii|

ke_ik

.

Now let us recall that every bounded linear operator A on E can be expressed as a pointwise convergent series, that is, there exists an infinite

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matrix(a_ij)_(i,j)∈N×N with coefficients inKsuch that A=X

ij

a_ij(e⁰_j⊗e_i), and for any j∈N, lim

i→∞|a_ij| ke_ik= 0. (1.1) Moreover, for eachj∈N, Ae_j=X

i∈N

a_ij e_i and its norm is defined by:

kAk:= sup

i,j

|a_ij| ke_ik ke_jk

.

In this paper we shall make extensive use of the p-adic Hilbert spaceEω

whose definition is given below. Again, for details, we refer the reader to ([2], [3], and [4]) and the references therein.

Let ω = (ω_i)_i∈_N be a sequence of non-zero elements in a complete non- Archimedean fieldK. We define the space Eω by

Eω :=n

u= (u_i)_i∈

N| ∀i, u_i∈Kand lim

i→∞|u_i| |ω_i|^1/2= 0o . Clearly, u= (u_i)_i∈

N ∈Eω if and only if lim

i→∞u²_iω_i = 0. Actually Eω is an ultrametric Banach space overKwith the norm given by

u= (u_i)_i∈

N∈Eω, kuk= sup

i∈N

|u_i| |ω_i|^1/2.

Let us also notice that Eω is a free Banach space (see [3]) and it has a canonical orthogonal basis. Namely, (e_i)_i∈_N, where e_i is the sequence all of whose terms are0except thei-th term which is1,in other words,e_i= (δ_ij)_j∈

N

whereδ_ijis the usual Kronecker symbol. We shall make extensive use of such a canonical orthogonal basis throughout the paper. It should be mentioned that for eachi,ke_ik=|ω_i|^1/2. Now if|ω_i|= 1we shall refer to (e_i)_i∈_Nas the canonical orthonormal basis.

Let h,i:Eω×Eω →Kbe the K-bilinear form defined by

∀u, v∈Eω, u= (u_i)_i∈

N, v= (v_i)_i∈

N, hu, vi:=X

i∈N

ω_i u_iv_i.

Then, h,i is a symmetric, non-degenerate form onEω×Eω with value in K, and it satisfies the Cauchy-Schwarz inequality:

|hu, vi| ≤ kuk.kvk, ∀u, v∈Eω.

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Let us also mention that elements(e_i)i∈Nof the canonical orthogonal basis forEω satisfy

he_i, e_ji=ω_iδ_ij=







0 if i6=j ω_i if i=j.

The space Eω endowed with the bilinear formh,i defined above is called a p-adic Hilbert space.

It should also be observed that for every bounded linear operator A on Eω, the domain D(A) (D(A) :={u= (u_i)∈Eω: lim

i |u_i|kAe_ik= 0}) ofAis the whole of Eω. This is actually due to the fact that for every u= X

i∈N

u_ie_i inEω: |u_i| kAe_ik ≤ kAk |u_i| |ω_i|^1/2.

It is well-known ([2] and [3]) that one can find bounded linear operators onEω which do not have adjoints. Similarly, we shall see that the latter is still true for unbounded linear operators we shall deal with. Consequently, we denote by B₀(Eω) the space of all bounded linear operators which do have adjoints with respect to the non-degenerate formh,i defined above. It is known ([2], [3] and [4]) that a bounded linear operatorA=X

ij

a_ij(e⁰_j⊗e_i) is inB₀(Eω)if and only if: ∀i, lim

j→∞|a_ij| |ω_j|^−1/2= 0.

Let us recall thatB₀(Eω)is stable under the operation of taking an adjoint and for any A∈B₀(Eω) : (A^∗)^∗=Aand kAk=kA^∗k.

Letω = (ω_i)i∈N be a sequence of nonzero elements in a (complete) non- Archimedean fieldK. Consider the correspondingp-adic Hilbert space given by(Eω,h,i).

In this paper, we initiate a theory for some unbounded linear operators within thep-adic framework, that is, on thep-adic Hilbert spacesEω. As for bounded linear operators on Eω, some of the results go along the classical line and others deviate from it. For the most part, the statements of the results are inspired by their classical settings. However their proofs may depend heavily on the ultrametric nature of Eω and the ground field K. We especially emphasis on both the closedness and the self-adjointness of those unbounded linear operators on Eω (Theorems 3.3 and 3.4). To deal with those issues, we equip the direct sum Eω ×Eω with both a complete ultrametric norm as well as a hilbertian structure ( it will be shown that this can be even done for a finite direct sumEω1×Eω2×...×Eωn ofEω1,Eω2, ...,

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andEωn where the sequencesω_k = (ω_i^k)i∈Nfork= 1,2, ..., nare respectively nonzero elements inK). Next we consider an unitary operatorΓonEω×Eω

which enables to describe the adjointA^∗of a given operatorAin terms ofA.

To illustrate our abstract results, a few examples are discussed. In particular, a special attention is paid to the diagonal operator onEω as well as the solvability of the equation Au = v where A is (possibly unbounded) a linear operator onEω (Theorems 5.1 and 5.4).

2 Unbounded linear operators on E

^ω

Let ω = (ω_i)_i∈_N, $ = ($_i)_i∈_N be sequences of non-zero elements in a complete non-Archimedean fieldK, and letEω, E$ be their correspondingp-adic Hilbert spaces. We suppose that (e_i)_i∈_N and (h_j)_j∈_N are respectively the canonical orthogonal bases associated to the p-adic Hilbert spaces Eω and E$.

Let D ⊂ Eω be a subspace and let A : D ⊂ Eω 7→ E$ be a linear transformation. As for bounded linear operators one can decomposeA as a pointwise convergent series defined by:

A=X

i,j

a_i,j e⁰_j⊗h_i and, ∀j∈N, lim

i→∞|a_i,j| kh_ik= 0.

Definition 2.1: An unbounded linear operatorAfromEω intoE$ is a pair (D(A), A)consisting of a subspaceD(A)⊂Eω (called the domain ofA) and a (possibly not continuous) linear transformationA:D(A)⊂Eω7→E$.

Throughout the paper, we consider unbounded linear operators Awhose domainD(A) consists of allu∈Eω such thatAu∈E$, that is,











D(A) :={u= (u_i)i∈N∈Eω : lim

i→∞|u_i| kAe_ik= 0}, A= X

i,j∈N

a_i,j e⁰_j⊗h_i, ∀j∈N, lim

i→∞|a_i,j| kh_ik= 0.

(2.1)

We denote the collection of those unbounded linear operators byU(Eω,E$).

Clearly,B(Eω,E$)⊂U(Eω,E$).

We have seen that if A∈B(Eω,E$)then its domain is the whole of Eω. We shall see that one can find elements of U(Eω,E$) whose domains differ

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fromEω (see Remark 4.2). As for bounded operators, there are elements of U(Eω,E$)which do not have adjoints (see Example 2.3 below). Actually in the next definition we state conditions which do guarantee the existence of the adjoint. Without lost of generality we shall suppose that Eω =E$. As usual we denoteU(Eω,Eω)byU(Eω).

In what follows,(K,|.|) denotes a complete non-Archimedean field.

Definition 2.2: Let ω = (ω_i)i∈N ⊂ K be a sequence of nonzero terms and let (Eω,h,i) be the corresponding p-adic Hilbert space. An operator A= X

i,j

a_i,j e⁰_j⊗e_i in U(Eω) is said to have an adjoint A^∗ ∈U(Eω) if and only if

j→∞lim

|a_i,j|

|ω_j|^1/2

= 0, ∀i∈N. (2.2)

Furthermore, the adjoint A^∗ ofAis uniquely expressed by











D(A^∗) :={u= (u_i)i∈N∈Eω: lim

i→∞|u_i| kA^∗e_ik= 0}, A^∗= X

i,j∈N

a^∗_i,j e⁰_j⊗e_i, ∀j∈N, lim

i→∞|a^∗_i,j| |ω_i|¹² = 0, where a^∗_i,j=w_i⁻¹w_ja_j,i.

We denote byU₀(Eω)the collection of operators inU(Eω)which do have adjoint operators. Clearly,B₀(Eω)⊂U₀(Eω).

Example 2.3:(Unbounded operator with no adjoint). Set K=Qp the field ofp-adic numbers endowed with thep-adic norm|.|_pand letω_i=p³ⁱ(in the appropriateQp) so that |ω_i|¹² =p⁻³²ⁱ. Now let A=X

i,j

a_i,j (e⁰_j⊗e_i) be the linear operator defined by its coefficients:

a_ij=







p^−j if i < j 1 if i=j p⁻ⁱ if i > j.

and |a_ij| =







p^j if i < j 1 if i=j pⁱ if i > j.

We have

Proposition 2.4: The linear operator A= X

i,j

a_i,j (e⁰_j⊗e_i) defined above is inU(Eω) and it does not have an adjoint.

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Proof: Since ∀j, lim

i |a_ij||ω_i|¹² = lim

i>jpⁱp⁻³²ⁱ = 0 it is clear that A is well-defined. Furthermore,∀i, j ∈N,

λ_i,j:= |a_ij||ω_i|¹²

|ω_j|¹² =







p³²^(j−i)+j if i < j

1 if i=j

pⁱ⁺³²^j−³²ⁱ if i > j.

Consequently, ∀i, j, λ_i,j := |a_ij||ω_i|^1/2

|ω_j|^1/2 ≥







p^j if i < j 1 if i=j p⁻³²ⁱ if i > j.

Clearly,kAk := sup

i,j

λ_i,j=∞, hence A∈ U(Eω). To complete the proof we have to show that∀i, lim

j

"

|a_ij|

|ω_j|¹²

#

6= 0. Indeed, ∀i,lim

j

"

|a_ij|

|ω_j|¹²

#

= lim

j>ip^jp³²^j

=∞,hence the adjoint of Adoes not exist.

3 Closed linear operators on E

^ω

LetA∈U(Eω). As in the classical setting we define the graph of the linear operatorAby

G(A) :={(x, Ax)∈Eω×Eω: x∈D(A)}.

Definition 3.1: An operatorA∈U(Eω)is said to be closed if its graph is a closed subspace inEω×Eω. The operatorAis said to be closable if it has a closed extension.

As in the classical setting we characterize the closedness of an operator A∈U(Eω)as follows: ∀u_n∈D(A)such thatku−u_nk 7→0andkAu_n−vk 7→

0 (v∈Eω) asn7→+∞, then u∈D(A)and Au=v.

It is now clear that ifA∈B(Eω), it is closed. Indeed sinceAis bounded, D(A) = Eω. Moreover if x_n ∈ Eω such that x_n 7→ x on Eω, then by the boundedness ofAit follows thatAx_n7→Ax, that is,(x_n, Ax_n)7→(x, Ax)on Eω×Eω, hence G(A) is closed.

We denote the collection of closed linear operatorsA∈U(Eω)byC(Eω).

In view of the above,B(Eω)⊂C(Eω).

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Let us notice that in the classical context, the collection of closed operators is not a vector space. However if A is an unbounded linear operator acting in a Banach space X and if B ∈ B(X) then A+B their algebraic sum is closed. In the p-adic setting, there is no reason that the situation differs from the classical one. Similarly, ifA∈C(Eω)and ifB ∈B(Eω)then A+B ∈C(Eω).

Definition 3.2: An operatorA∈U₀(Eω)is said to be self-adjoint ifD(A) = D(A^∗)and Au=A^∗u for eachu∈D(A).

We shall prove the following important theorem:

Theorem 3.3: Let A∈U₀(Eω), then its adjoint A^∗ is a closed linear oper- ator. In particular ifA is self-adjoint, then it is closed.

As previously mentioned, to deal with the closedness of an operatorA∈ U(Eω) we first equip the direct sumEω×Eω with both an ultrametric norm and a p-adic hilbertian structure. Actually, we shall show that this can be even done for a finite direct sum of Hilbert spaces Eω.

Throughout the rest of this section, we suppose thatchar(K) the char- acteristic of the fieldK is zero. Notice that examples of such fields include Qp(p≥2being prime) the field ofp-adic numbers.

Let ω_k = (ω_i^k)_i∈N with k = 1,2, ..., n be (n being fixed) n sequences of nonzero terms inK. For eachk (k= 1,2, ...n) we consider the corresponding p-adic Hilbert spaceEωk. Now let us consider the so-called finite direct sum En := Eω1 ×Eω2×...×Eωn of Eω1, Eω2, ..., and Eωn, respectively. Let us mention thatEn is also denoted by: Eω1⊕Eω2⊕...⊕Eωn.

We now equipEn with the ultrametric norm defined by

||(u₁, u₂, ..., u_n)| |_n:= max (ku₁k,ku₂k, ...,ku_nk), (3.1)

∀(u₁, u₂, ..., u_n)∈En, wherek.kis the norm on eachEωk (k= 1,2, ...n).

Now arguing that each (Eωk,k.k) for k = 1,2, ...n is an ultrametric Ba- nach space it follows that(En,||.||_n) is also an ultrametric Banach space. It also clear that En (n fixed) is a free Banach space with orthogonal basis as {(L^k_i)_i∈N, k = 1,2, ..., n} where L^k_i = (0,0, ...e^k_i, ..0,0,0) whose terms are 0 except thei-th term which ise^k_i ((e^k_i)_i∈_Nbeing the canonical orthogonal basis for Eωk withk = 1,2, .., n). Actually, kL^k_ik_n = ke^k_ik = |ω^k_i|^1/2 for all i ∈ N and for each k = 1,2, ..., n. We confer to {(L^k_i)_i∈_N, k = 1,2, ..., n} as the canonical orthogonal basis forEn.

Leth,i_n:En×En 7→Kbe the K-bilinear form defined by

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h(u₁, u₂, ..., u_n),(v₁, v₂, ..., v_n)i_n:=

n

X

k=1

hu_k, v_ki, (3.2) for all(u₁, u₂, ..., u_n),(v₁, v₂, ..., v_n)∈Enwhereh,iis the corresponding inner product of eachEωk fork= 1,2, ..., n.

Then, h,i_n is a symmetric, non-degenerate form onEn×En.

The space En:=Eω1×Eω2×...×Eωn endowed with the K-bilinear form h,i_n defined in (3.2) will be called ap-adic Hilbert space.

One can easily see that if n = 1 we retrieve the p-adic Hilbert space (Eω,h,i) that we had introduced in Section 1.

Clearly, the norm (3.1) is not deduced from the K-bilinear form (3.2).

However we have the Cauchy-Schwarz inequality:

|h(u₁, u₂, ..., u_n),(v₁, v₂, ..., v_n)i_n| ≤ ||(u₁, u₂, ..., u_n)||_n .||(v₁, v₂, ..., v_n)||_n, for all (u₁, u₂, ..., u_n),(v₁, v₂, ..., v_n)∈En.

We shall prove the Cauchy-Schwarz inequality above only forn= 2since the proof of the general case follows along the same lines.

Proof: For all (u, v), (x, y)∈Eω1×Eω2,

|h(u, v),(x, y)i₂| = |hu, xi+hv, yi|

≤ max(|hu, xi|,|hv, yi|)

≤ max(kuk kxk,kvk kyk)

≤ max(||(u, v)||₂ kxk,||(u, v)||₂kyk)

≤ max(||(u, v)||₂ ||(x, y)||₂,||(u, v)||₂||(x, y)||₂)

= ||(u, v)||₂||(x, y)||₂.

In view of the above, the direct sumEω×Eω is ap-adic Hilbert space.

Let us mention that if M ⊂Eω×Eω is a subspace, then its orthogonal complementM^⊥ with respect theK-bilinear formh,i₂ is defined by

M^⊥:={(u, v)∈Eω×Eω: h(u, v),(x, y)i₂= 0, ∀(x, y)∈M}.

One can easily check that M^⊥ is closed. This is actually a consequence of the continuity of the bilinear form defined byΦ_(x,y) :Eω×Eω 7→Kand

Φ_(x,y)(u, v) =h(u, v),(x, y)i₂, ∀(u, v)∈Eω×Eω,

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where (x, y)∈Eω×Eω is fixed.

LetΓbe the bounded linear operator which goes fromEω×Eω into itself defined by

Γ(u, v) := (−v, u), ∀(u, v)∈Eω×Eω. Clearly, the operatorΓdefined above satisfies:

(i) Γ²(u, v) = Γ(−v, u) = (−u,−v) =−(u, v), ∀(u, v)∈Eω×Eω; (ii) Γ²(M) =M for any subspace M ofEω×Eω.

From (i) it follows thatΓ²=−I(Γis an unitary operator) whereIis the identity operator ofEω×Eω.

As we mentioned in Section 1, the unitary operator Γ enables us to describe yieldA^∗ (if A∈U₀(Eω)) in terms ofA.

We have

Theorem 3.4: If A ∈U₀(Eω), then G(A^∗) = [ΓG(A)]^⊥ where [ΓG(A)]^⊥ is the orthogonal complement of [ΓG(A)].

Proof: (Theorem 3.4). Since the adjoint A^∗ofAdoes exist one defines its graph by:

G(A^∗) :={(x, A^∗x)∈Eω×Eω : x∈D(A^∗)}.

Thus for(x, y)∈D(A^∗)×D(A)one has

h(x, A^∗x),Γ(y, Ay)i₂ = h(x, A^∗x),(−Ay, y)i₂

= −hx, Ayi+hA^∗x, yi

= 0, henceG(A^∗)⊂[ΓG(A)]^⊥.

Conversely, if(x, y)∈[ΓG(A)]^⊥, ∀z ∈D(A), then, 0 = h(x, y),(−Az, z)i₂

= −hx, Azi+hy, zi.

It follows that hAz, xi = hz, yi. Now by uniqueness of the adjoint, we obtain that: x∈D(A^∗) andA^∗x=y,hence(x, y)∈ G(A^∗).

Proof: (Theorem 3.3). Since the adjoint of A^∗ of A does exist and that ΓG(A) is a subspace of Eω ×Eω, then using Theorem 3.4 it follows that G(A^∗) = [ΓG(A)]^⊥is closed, henceA^∗ is closed.

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Let A ∈ U(Eω). We define the resolvent set ρ(A) of A as the set of all λ ∈ K such that the operator A_λ := λI −A is one-to-one and that A⁻¹_λ ∈ B(Eω). In this event, the spectrum σ(A) of A is defined as the complement ofρ(A)in K.

4 The diagonal operator on E

^ω

Letω= (ω_i)i∈Nbe a sequence of nonzero terms inKand let(Eω,h,i)be the correspondingp-adic Hilbert space. Define the diagonal operatorA∈U(Eω) by its coefficients: a_i,j =λ_iδ_i,j where δ_i,j is the usual Kronecker symbols and (λ_i)_i∈_N⊂Kis a sequence of nonzero elements satisfying

i→∞lim|λ_i|=∞. (4.1)

Observe that: D(A) ={x= (x_i)i∈N⊂K: lim

i |λ_i||x_i|ke_ik= 0}.

Proposition 4.1: Under previous assumptions, suppose that (4.1) holds.

ThenAis self-adjoint. Furthermore, ρ(A) =K− {λ_i}_i∈_N.

Proof: First of all, let us make sure that the operatorA is well-defined.

For that note that |a_i,i| = |λ_i| and that |a_i,j| = 0 if i 6= j. We have, limi |a_i,j| |ω_i|^1/2= lim

i>j |a_i,j| |ω_i|^1/2= 0, hence A is well-defined. Next, it is clear that |a_i,j| |ω_i|^1/2

|ω_j| = |λ_i| if i = j and 0 if i 6= j. It follows that kAk:= sup

i,j

|a_i,j| |ω_i|^1/2

|ω_j|

= sup

i

|λ_i|=∞,henceA∈U(Eω).

Let us show that the adjointA^∗ ofAdoes exist. This is actually obvious since ∀i ∈ N, lim

j |a_i,j| |ω_j|^−1/2= lim

j>i |a_i,j| |ω_j|^−1/2= 0. Now the adjoint A^∗ is defined by A^∗ = X

i,j

b_i,j e⁰_j⊗e_i, where b_i,j = w⁻¹_i w_ja_j,i = a_i,j for all i, j∈N. The latter yieldsA=A^∗.

To complete the proof one needs to compute ρ(A). For that, we have to solve the equation (A−λI)x = y, where x = X

i

x_ie_i ∈ D(A) and y = X

i

y_ie_i∈Eω. Considering the previous equation on(e_i)i∈Nand using the fact Ais self-adjoint it follows that: ∀i∈N, (λ_i−λ).he_i, xi=he_i, yi.Equivalently,

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∀i ∈N, (λ_i−λ).ω_ix_i = ω_iy_i. Now if ∀i ∈N, λ_i 6= λ the previous equation has a unique solutionx. Moreover, x= (A−λ)⁻¹y=X

i

y_i λ_i−λe_i.

Let us show that (A−λ)⁻¹y given above is well-defined. For that it is sufficient to prove that lim

i

|y_i|

|λ_i−λ| ke_ik

= 0. According to (4.1) the sequence

1

|λ_i−λ|

i∈N

is bounded, hence lim

i

|y_i|

|λ_i−λ| ke_ik

= 0.

It remains to find conditions on λ so that x defined above belongs to D(A). For that it is sufficient to find conditions so that: lim

i

|y_i|

|λ_i−λ|kAe_ik= limi

|λ_i|

|λ_i−λ| |y_i| ke_ik= 0. Actually, sincelim

i |y_i| ke_ik= 0one has:

0≤lim

i

|y_i|

|λ_i−λ|kAe_ik ≤ lim

i

|λ_i|

| |λ_i| − |λ| | . lim

i |y_i| ke_ik= 0.

From (4.1) it follows that (A−λ)⁻¹ ∈ B(Eω) for all λ 6= λ_i, i ∈ N. Actually, k(A−λ)⁻¹k ≤sup

i∈N

1

|λ_i−λ|

<∞,henceρ(A) =K− {λ_i}i∈N. Remark 4.2: Let us notice that the domainD(A)of the diagonal operator Amay not be equal to the whole of Eω. To see it, suppose that the ground fieldKcontains a square of each of its elementsand choosex˜= (˜x_i)i∈Nwhere

˜

x_i is given by: x˜²_i = 1

λ²_i ω_i for all i ∈ N. According to the assumption on the field K it is clear that for all i ∈ N, x˜_i lies in K. Now, x˜ ∈ Eω since

i→∞lim|˜x_i|ke_ik = lim

i→∞

1

|λ_i| = 0, by (4.1). Meanwhile, one can easily see that

˜

x6∈D(A) sincelim

i |x˜_i| |λ_i|ke_ik= 16= 0.

5 The equation Ax = y on E

^ω

5.1 The bounded case

LetA∈B(Eω) be a bounded linear operator onEω. Consider the existence and uniqueness of a solutionx=X

i∈N

x_ie_i∈Eω to the equation:

Ax=y, (5.1)

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wherey=X

i∈N

y_ie_i∈Eω is an arbitrary element.

We give sufficient conditions ensuring the existence and uniqueness of a solution to (5.1). Next this will be applied to the perturbations of bases.

In the last part of this section, we consider the existence and uniqueness of solutions to(5.1) whenAis a (particular) unbounded linear operator.

Theorem 5.1: Let A : Eω 7→ Eω be a bounded linear operator on Eω. Suppose that kI−Ak < 1, then (5.1) has a unique solution given by x = X

n∈N

(I−A)ⁿy.Moreover, the solutionx satisfies: kxk=kyk.

Proof: The first part of the proof follows along the same lines as in the classical setting. Indeed, sinceAbelongs toB(Eω) which is Banach algebra with unit and sincekI−Ak<1it follows thatAis invertible. Thus, setting B=I−Ait follows that the only solution to (5.1) is given byx= (I−B)⁻¹y, that is,x=X

n∈N

Bⁿy=X

n∈N

(I−A)ⁿy.Now from the assumptionkI−Ak<1 it is clear thatkAk= 1, hencekxk=kAxk=kyk.

In view of the above it clear that 0 ∈ρ(A). In addition to that, Theo- rem 5.1 is still valid in any arbitrary non-Archimedean Banach spaceE. Example 5.2: Let us illustrate Theorem 5.1 by an example. Consider thep- adic space(Eω,h,i)overQpcorresponding toω_k =p^kfor allk∈ {1,2,3, ...}.

Now define the operator AonEω by its coefficientsa_ij:

a_ij=







p^i−j if i > j 1 if i=j 0 if i < j.

and |a_ij|=







p^−(i−j) if i > j

1 if i=j

0 if i < j.

First of all, let us make sure that A= X

i,j≥1

a_ij e⁰_j⊗e_i is well-defined.

Indeed, one has: lim

i |a_ij| ke_ik= lim

i>j p^−(i−j) . p^−i/2= 0.Next, one shows that Ais bounded. For that, we note that

|a_ij| ke_ik ke_jk =







p⁻³² ^(i−j) if i > j

1 if i=j

0 if i < j,

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and |a_ij| ke_ik

ke_jk ≤ 1 for all i, j = 1,2,3, ... Thus, kAk := sup

i,j

|a_ij| ke_ik ke_jk ≤1, henceAis bounded. Actually, one can easily show that the adjoint A^∗ does exist. Nevertheless,Ais not self-adjoint.

ConsiderB = I−A, where I is the identity operator ofEω. It is then clear thatB= X

i,j≥1

b_ij e⁰_j⊗e_i, where

b_ij=

−p^i−j if i > j

0 if i≤j. and |b_ij|=

p^−(i−j) if i > j

0 if i≤j.

Similarly,

|b_ij| ke_ik ke_jk =

p⁻³² ^(i−j) if i > j

0 if i≤j.

Thus it is clear that |b_ij| ke_ik

ke_jk ≤ p⁻³², that is, kBk ≤ p⁻³² < 1, hence for eachy∈Eω, (5.1) has a unique solutionxgiven byx=X

n∈N

Bⁿy. Moreover, kxk=kyk,by Theorem 5.1.

Another application to Theorem 5.1 concerns the perturbation of bases forEω. This is discussed in the next subsection.

5.2 Application to the perturbation of bases on E

ω

We now apply Theorem 5.1 to the perturbation of bases. Let(e_i)i∈N be the canonical basis forEω and let (f_i)_i∈_N⊂Eω be a family of vectors.

We require the following assumption:

sup

n∈N

ke_n−f_nk ke_nk

<1. (5.2)

Corollary 5.3: Let(e_i)_i∈_Nbe the canonical basis forEω and let(f_i)_i∈_N⊂Eω

be a family of vectors. Suppose that (5.2) holds true. Then (f_i)i∈N is also a basis for Eω.

Proof: One defines a linear operator A : Eω 7→ Eω by: ∀x = X

k∈N

x_ke_k, A(X

k∈N

x_ke_k) =X

k∈N

x_kf_k.In particular,Ae_k=f_k, ∀k ∈N. Now from (5.2) one haske_n−f_nk<ke_nk, ∀n∈N, that iske_nk=kf_nk, ∀n∈N.

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In view of the above, we have:

(i) The decompositionAx=X

k∈N

x_kf_k is well-defined;

(ii) kAk:= sup

n∈N

kAe_nk ke_nk = sup

n∈N

kf_nk

kf_nk = 1, in particularA is a bounded linear operator on Eω. Furthermore,kI−Ak:= sup

n∈N

ke_n−Ae_nk ke_nk <1.

According to Theorem 5.1 the operatorAis a bĳective and isometric. Since in thep-adic context, an isometric linear bĳection transforms an orthogonal basis into an orthogonal basis, then(f_n)n∈N is an orthogonal basis forEω.

5.3 The unbounded case

Let($_n)⊂Kbe sequence of nonzero terms and let(f_i)_i∈_Nbe another orthogonal basis for Eω. We shall suppose that there exists a nontrivial isometric linear bĳectionT (see Subsection 5.2) such that T e_i=f_ifor alli∈N, where (e_i)i∈Nis the canonical orthogonal basis forEω. As a consequence,|ω_i|=|$_i| for alli∈N. However, let us notice that the fact |ω_i|=|$_i| for eachi∈N, the sequences(ω_i)i∈N and($_i)i∈Nneed not be equal.

Since(f_i)_i∈N is an orthogonal basis forEω, for eachx∈Eω,x=X

n∈N

x_nf_n with lim

n→∞|x_n|kf_nk= 0,wherekf_ik=:|$_i|^1/2=|ω_i|^1/2, ∀i∈N, andhf_i, f_ji=

$_iδ_i,j (δ_i,j being the usual Kronecker symbol).

We shall study the existence and uniqueness of solutions to (5.1) in the particular case whereAis the linear operator defined by

D(A) ={x= (x_i)i∈N∈Eω : lim

i→∞|x_i| |µ_i| kf_ik= 0}, and

Ax:=X

i∈N

µ_i x_i f_i, for each x= (x_i)i∈N∈D(A), where(µ_i)_i∈_N⊂Kis a given sequence of nonzero terms.

It is not hard to see that A is well-defined on D(A) and that it is an element ofU(Eω). Now sincef_m∈Eω it can be written as: ∀ m∈N,

f_m=X

i

a_ime_i with lim

i |a_im| ke_ik= 0. (5.3)

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We require the following assumption:

m→∞lim





 sup

n∈N

|a_nm| |ω_n|^1/2

|$_m|^1/2





= 0. (5.4)

Theorem 5.4: Under assumptions (5.3)-(5.4), the equation (5.1) has a unique solution x= X

m∈N

x_me_m ∈D(A) where

x_m= 1 µ_m $_m

"

X

n∈N

ω_n y_na_nm

#

, ∀m∈N.

Proof: Let y = X

n∈N

y_ne_n ∈Eω (lim

n→∞|y_n| |ω_n|^1/2 = 0). Now from (5.3) it is clear thathAx, f_mi=hX

n∈N

y_ne_n, f_mi,henceµ_m$_mx_m=X

n∈N

ω_n y_n a_nm. It follows that a solutionxto (5.1) is given by its coefficients:

x_m = 1 µ_m $_m

"

X

n∈N

ω_n y_n a_nm

#

, ∀m∈N.

Since(f_s)s∈N is an orthogonal basis forEω it is then clear thatN(A) ={0}

(N(A)being the kernel ofA), hence the above solution is unique. Now from (5.3) it is clear that x_m is well-defined. The remaining task now is to prove that the solutionx∈D(A). Indeed from the expression ofx_m one has:

|x_m| |µ_m| kf_mk = |x_m| |µ_m| |$_m|^1/2

=







|X

n∈N

ω_n y_n a_nm|

|$_m|^1/2







≤





 sup

n∈N

|a_nm| |ω_n|^1/2

|$_m|^1/2





 .kyk.

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The previous inequality and (5.4) yield: lim

m→∞|x_m||µ_m|kf_mk = 0, hence x∈D(A).

Acknowledgement. The author wants express his deepest thanks to the anonymous referee for comments and suggestions on the paper.

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Toka Diagana Howard University

Department of Mathematics 2441 6th Street N.W.

Washington, D.C. 20059 U.S.A.

[email protected]