Rend. Sem. Mat. Univ. Politec. Torino Vol. 68, 3 (2010), 1 – 17

D. Attimu and T. Diagana REPRESENTATION OF BILINEAR FORMS BY LINEAR OPERATORS IN NON-ARCHIMEDEAN HILBERT SPACE EQUIPPED WITH A KRULL VALUATION Abstract. The paper considers representing bilinear forms by linear operators in the case of a Krull valuation. More precisely, after making some suitable assumptions, we prove that if ϕ is a non-degenerate bilinear form, then ϕ is representable by a unique linear operator A whose adjoint operator A∗ exists.

1. Preliminaries Let N0 = {1, 2, 3, ....} and let N = {0, 1, 2, ...}. Let K be a field and let Γ be a totally ordered Abelian additive group. The total ordering of Γ is denoted ≤ with the property that for γ1 , γ1 , γ3 ∈ Γ, γ1 ≤ γ2 if and only if γ1 + γ3 ≤ γ2 + γ3 . Additionally, we shall write γ1 < γ2 to mean that γ1 ≤ γ2 but γ1 6= γ2 , and we write γ1 ≥ γ2 and γ1 > γ2 to mean γ2 ≤ γ1 and γ2 < γ1 , respectively. Moreover, we write Γ∞ := Γ ∪ {∞} in which γ + {∞} = {∞} for each γ ∈ Γ∞ . Next we extend the total ordering to Γ∞ by declaring that for all γ ∈ Γ, γ < {∞}. Typical examples of ordered Abelian groups include the additive group of integers (Z, +, ≤), or the direct sum Γ=

∞ M

Γ j = Γ1 ⊕ Γ2 ⊕ ... ⊕ Γn ⊕ Γn+1 ⊕ ...,

j=1

where Γ j is an isomorphic copy of the additive group Z of integers, for each j ∈ N0 . Namely, Γ consists of all sequences γ = (γ j ) j∈N0 with γ j ∈ Γ j for which { j ∈ N0 : γ j 6= 0} is finite. The group Γ will be equipped with antilexicographic order, that is, if 0 6= γ = (γ j ) j∈N0 ∈ Γ and j0 = max{ j ∈ N0 : γ j 6= 0}, then γ > 0 in Γ if and only if γ j0 > 0 in Γ j0 . Let us mention that there exist totally ordered Abelian additive groups Γ, which are not subgroups of R. In what follows, we give an example of such groups discussed in the remarkable book by Ribenboim [15]. Indeed, let G and H be nonzero subgroups of R and let Γ = G × H equipped with addition defined componentwise. Now equip Γ with the lexicographic order “≤" as follows: ( f , g) < (h, l) if ( f < h or f = h) and g < l. Clearly, it is not hard to see that the order “≤" defined above is compatible with the operation of addition. Moreover, Γ = G × H defined above is not order-isomorphic to a subgroup of R. 1

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D. Attimu and T. Diagana

A map v : K → Γ∞ = Γ ∪ {∞} is said to be a Krull valuation provided that for all λ, µ ∈ K, the following conditions hold: (P1 ) v(λ) = {∞} if and only if λ = 0; (P2 ) v(λµ) = v(λ) + v(µ); n o (P3 ) v(λ + µ) ≥ min v(λ), v(µ) .

The group Γ is then called a value group for K. The valuation v induces a topology on the field K by considering {Oε : ε ∈ Γ} as a neighborhood base of 0 ∈ K, where Oε = {λ ∈ K : v(λ) > ε}. Consequently, a sequence (λ j ) j∈N ⊂ K converges to 0 for this valuation topology if only if v(λ j ) → ∞ as j → ∞. For the above-mentioned valuation, be have the following properties: (P4 ) v(−ξ) = v(ξ) for each ξ; (P5 ) v(ξ−1 ) = −v(ξ) for each ξ ∈ K − {0}; n o (P6 ) v(µ − λ) ≥ min v(µ), v(λ) ;

n o (P7 ) v(µ + λ) = min v(µ), v(λ) whenever v(µ) 6= v(λ).

Suppose that a valuation v and a value group Γ associated with K are given, as above. Fix once and for all a sequence ω = (ω j ) j∈N ⊂ K of nonzero terms. The space N 2 cK 0 (N, ω, Γ) is defined as the set of all x = (x j ) j∈N ∈ K such that ω j x j → 0 as j → ∞, that is,   lim v(ω j ) + 2v(x j ) = ∞. j→∞

One defines an associated (non-archimedean) norm N : cK 0 (N, ω, Γ) → Γ∞ as follows: for each x = (x j ) j∈N ∈ cK 0 (N, ω, Γ),   N(x) = min 2v(x j ) + v(ω j ) . j∈N

It is not hard to check that N satisfies the following properties: (Q1 ) N(x) = ∞ if and only if x = 0, (Q2 ) N(ξx) = 2v(ξ) + N(x); (Q3 ) N(−x) = N(x);   (Q4 ) N(x + y) ≥ min N(x), N(y) ;

  (Q5 ) N(x − y) ≥ min N(x), N(y) ;

Bilinear forms on non-archimedean Hilbert spaces

3

valid for all ξ ∈ K and x, y ∈ cK 0 (N, ω, Γ). As an immediate consequence of (Q4 ), we have the following: if x, y ∈ cK 0 (N, ω, Γ), then   (Q6 ) N(x + y) = min N(x), N(y) whether N(x) 6= N(y). Indeed, suppose that N(x) < N(y). (This by the way includes the case when N(x) < ∞ and N(y) = ∞.) It follows that   N(x + y) ≥ min N(x), N(y) = N(x). Suppose N(x + y) > N(x). Consequently,

  N(x) = N(x + y − y) ≥ min N(x + y), N(y) > N(x),

which is impossible, and hence

  N(x + y) = N(x) = min N(x), N(y) .

Another important consequence of (Q3 ) is the following: if x j ∈ cK 0 (N, ω, Γ) for all j ∈ N, then   (Q7 ) N ∑ x j ≥ inf N(x j ) whenever the sum exists. j∈N

j∈N

Similarly, from (P3 ) we have the following: if h j ∈ K for all j ∈ N, then   (Q8 ) v ∑ h j ≥ inf v(x j ) whenever the sum exists. j∈N

j∈N

For more on the Krull valuation and related issues, we refer the reader to the remarkable work of Keller and Ochsenius [11], Ochsenius [12] and Ochsenius and Schikhof [14].   (N, ω, Γ), N is complete. MoreOne can check that the “normed” space cK 0

over, each x = (x j ) j∈N ∈ cK 0 (N, ω, Γ) can be written as ∞

x = ∑ xi ei with lim N(xi ei ) = ∞, i=0

i→∞

where ei is the sequence whose j-th term is 0 if i 6= j, and the i-th term is 1. In particular, N(e j ) = v(ω j ) for each j ∈ N. The system (e j ) j∈N will be called an orthogonal basis for cK 0 (N, ω, Γ). Similarly, an inner product (symmetric, non-degenerate, bilinear form) is also K defined on cK 0 (N, ω, Γ) for all x = (x j ) j∈N , y = (y j ) j∈N ∈ c0 (N, ω, Γ) by D E x, y :=

∞

∑ ω j x j y j,

j=0

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D. Attimu and T. Diagana

with corresponding “Cauchy-Schwarz inequality” given by D E (1) 2v x, y ≥ N(x) + N(y) for all x, y ∈ cK 0 (N, ω, Γ).

K The space D E c0 (N, ω, Γ) equipped with the above-mentioned valuation N and in-

ner product ·, · is called a non-archimedean Hilbert space. 2. Introduction

A bilinear form ϕ : D(ϕ) × D(ϕ) → K with domain D(ϕ) is said to be representable (Definition 8) whether there exists a linear operator A : D(A) → cK 0 (N, ω, Γ) (D(A) being the domain of A) such that D E ϕ(x, y) = Ax, y , ∀x ∈ D(A), y ∈ D(ϕ).

An unbounded bilinear form ϕ : D(ϕ)×D(ϕ) → K whose domain D(ϕ) contains all elements of the canonical basis (ei )i∈N will be called stable. The subclass of all these K stable unbounded bilinear forms is denoted ΣS (cK 0 (N, ω, Γ) × c0 (N, ω, Γ)). Similarly, the subclass of all bilinear forms whose domains do not contain the above-mentioned K canonical basis will be called unstable and denoted ΣU (cK 0 (N, ω, Γ) × c0 (N, ω, Γ)). In a recent paper by Attimu and Diagana, that is, in [1], it was shown that if ϕ is a non-degenerate, symmetric bilinear form satisfying     |ϕ(e j , ei )| |ϕ(ei , e j )| (2) = lim = 0, ∀ j ∈ N, lim i→∞ i→∞ kei k kei k then ϕ is uniquely representable. Moreover, if A denotes the (possibly unbounded) linear operator associated with ϕ, then its adjoint A∗ does exist. The main concern in this paper consists of studying representation theorems for (stable) bilinear forms in the case of a Krull valuation. More precisely, it will be shown that a non-degenerate bounded bilinear form ϕ on cK (N, ω, Γ) × cK 0 0 (N, ω, Γ) is representable whenever h i h i (3) lim 2v(ϕ(ei , e j )) − N(ei ) = lim 2v(ϕ(e j , ei )) − N(ei ) = ∞ i→∞

i→∞

for all j ∈ N. Similarly, it will be shown that if in addition to (3), (4)

lim N(ei ) = γ ∈ Γ,

i→∞

K then a possibly non-degenerate bilinear form ϕ on cK 0 (N, ω, Γ) × c0 (N, ω, Γ), not necessarily bounded, is representable. Moreover, if A denotes the linear operator on ∗ cK 0 (N, ω, Γ) associated with the form ϕ, then the adjoint A of A does exist.

Bilinear forms on non-archimedean Hilbert spaces

5

In addition to the above-mentioned representation results for bilinear forms, we also establish a non-archimedean version of the Riesz’s representation theorem for a subclass of linear functionals on cK 0 (N, ω, Γ) in the case of a Krull valuation. Namely, it is shown if F : cK 0 (N, ω, Γ) → K is a linear functional such that   lim 2v(F(ei )) − N(ei ) = ∞,

i→∞

E D then there exists a unique vector x0 ∈ cK for each (N, ω, Γ) such that F(x) = x, x 0 0

x ∈ cK 0 (N, ω, Γ). To deal with the above-mentioned issues we introduce a new formalism of unbounded linear operators on the non-archimedean Hilbert space cK 0 (N, ω, Γ) and that of K (un)bounded bilinear forms on cK 0 (N, ω, Γ) × c0 (N, ω, Γ) in the case of a Krull valuation. Representing (un)bounded bilinear forms by linear operators in the classical setting is a topic that arises in several fields such as quantum mechanics (through the study of form sums associated with the Hamiltonian), mathematical physics, symplectic geometry, and the study of weak solutions to some linear partial differential equations, see, e.g., [4, 9, 10]. In the non-archimedean realm, one may expect some related applications in: (i) the study of weak solutions to some p-adic partial differential equations; and (ii) the study of a non-archimedean version of the square root problem of Kato, which is of a great interest to the second author. 3. Linear Operators on cK 0 (N, ω, Γ) 3.1. Bounded Linear Operators on cK 0 (N, ω, Γ) In contrast with the classical definition of the boundedness of linear operators, we have: K D EFINITION 1. One says that a linear operator A : cK 0 (N, ω, Γ) → c0 (N, ω, Γ) is bounded if there exists γ ∈ Γ such that

N(Ax) ≥ γ + N(x) for each 0 6= x ∈ cK 0 (N, ω, Γ). Equivalently, a linear operator A is bounded if and only if there exists γ ∈ Γ such that N(Ae j ) ≥ γ + N(e j ) for each j ∈ N. K The collection of bounded linear operators from cK 0 (N, ω, Γ) into c0 (N, ω, Γ) K K will be denoted B(c0 (N, ω, Γ)). It can be easily checked that B(c0 (N, ω, Γ)) is an algebra. Note that a bounded linear operator A on cK 0 (N, ω, Γ) is continuous in the norm topology. However, continuous operators need not to be bounded. In contrast with the classical operator theory and except in some special cases, one cannot always assign a

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D. Attimu and T. Diagana

norm to bounded linear operators on cK 0 (N, ω, Γ), as the norm takes its values in Γ, in which bounded sets may fail to have an infimum. K Let A : cK 0 (N, ω, Γ) → c0 (N, ω, Γ) be a bounded linear operator. If n   o sup γ ∈ Γ : N Ax ≥ γ + N(x) for each 0 6= x ∈ cK (N, ω, Γ) 0 exists, it is then called the norm of A and denoted kAk. Clearly, if kAk exists, then   kAk = inf N(Ax) − N(x) . x6=0

′ K ∗ Given the orthogonal basis (ei )i∈N for cK 0 (N, ω, Γ), define ei ∈ c0 (N, ω, Γ) by

x=

∑ xi ei ,

e′i (x) = xi .

i∈N

∗ It turns out that ke′i k = −N(ei ) = −v(ωi ). Moreover, every x∗ ∈ cK 0 (N, ω, Γ) ∗ ∗ ′ can be expressed as x = ∑ hx , ei i ei , and i∈N

h i i kx∗ k = inf v(hx∗ , ei i) − N(ei ) = inf v(hx∗ , ei i) − v(ωi ) . i∈N

h

i∈N

K ∗ Further, for (u, v) ∈ cK 0 (N, ω, Γ) × c0 (N, ω, Γ) , define the operator ⊗ as follows:

(v ⊗ u)(x) = v(x)u, for all x ∈ cK 0 (N, ω, Γ). P ROPOSITION 1. Let A be a bounded linear operator on cK 0 (N, ω, Γ). There exists an infinite matrix (ai j )(i, j)∈N×N with coefficients in K, such that A can be written
as a pointwise convergent sum, namely, A = ∑ ai j e′j ⊗ ei and for all j ∈ N, i, j∈N

lim N(ai j ei ) = ∞.

i→∞

Proof. Clearly for all j, Ae j =

lim N(ai j ei ) = ∞. Now for any ∑ ai j ei where ai j ∈ K , i→∞

i∈N

x=

∑ x j e j ∈ cK0 (N, ω, Γ), we have

j∈N

Ax =

∑ ∑ a i j x j ei = ∑ ∑ a i j

j∈N i∈N

P ROPOSITION 2. Let A =

j∈N i∈N

∑

i, j∈N

kAk exists, then


e′j ⊗ ei x.


ai j e′j ⊗ ei be a bounded operator. If the norm

  h i kAk = inf N(Ae j ) − N(e j ) = inf 2v(ai j ) + N(ei ) − N(e j ) . j∈N

i, j∈N

7

Bilinear forms on non-archimedean Hilbert spaces

Proof. Suppose kAk exists. We first establish the first equality. Indeed, by definition,     kAk := inf N(Ax) − N(x) ≤ inf N(Ae j ) − N(e j ) . x6=0

Now N(Ax) = N

j∈N



∑ xi Aei

i∈N





 ≥ inf 2v(xi ) + N(Aei ) i∈N h i = inf N(Aei ) + 2v(xi ) − N(ei ) + N(ei ) i∈N     ≥ inf N(Aei ) − N(ei ) + inf 2v(xi ) + N(ei ) i∈N i∈N   = inf N(Aei ) − N(ei ) + N(x), i∈N

from which we conclude that     kAk = inf N(Ax) − N(x) ≥ inf N(Ae j ) − N(e j ) . x6=0

j∈N

For the second equality, we have       inf N(Ae j ) − N(e j ) = inf N ∑ ai j ei − N(e j ) j∈N

j∈N

= inf

i, j∈N

i∈N



 2v(ai j ) + N(ei ) − N(e j ) ,

which completes the proof. As in the classical case, if A ∈ B(cK 0 (N, ω, Γ)), an adjoint of A is an operator ∗ satisfying hAu, vi = hu, A∗ vi for any u, v in cK 0 (N, ω, Γ). If it exists, the adjoint A is K also a bounded linear operator on c0 (N, ω, Γ). A∗

∑


ai j e′j ⊗ ei ∈ B(cK 0 (N, ω, Γ)), then the adjoint i, j∈N h i A∗ exists if and only if for all j, lim 2v(a ji ) − N(ei ) = ∞. In this situation, P ROPOSITION 3. Let A =

i→∞

A∗ =

∑

i, j∈N


′ ω−1 i ω j a ji e j ⊗ ei .




 Proof. Write A∗ = ∑ bi j e′j ⊗ ei , then A∗ is the adjoint of A if and only if Aei , e j i, j∈N

 = ei , A∗ e j for all i, j ∈ N, that is, * + + *

∑ ali el , e j

l∈N

= a ji ω j =

ei , ∑ b l j el l∈N

= b i j ωi ,

∀i, j ∈ N.

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D. Attimu and T. Diagana

This is equivalent to bi j = ω−1 i ω j a ji for all i, j ∈ N. Moreover, for all j, lim N(bi j ei ) = ∞.

i→∞

Now N(bi j ei ) = 2v(bi j ) + N(ei ) = 2v(ω−1 i ω j a ji ) + N(ei ) = −2v(ωi ) + 2v(ω j ) + 2v(a ji) + N(ei ) = 2v(ω j ) + 2v(a ji) − N(ei ). Therefore, from v(ω j ) = N(e j ) 6= ∞, lim N(bi j ei ) = ∞ if and only if, for all i→∞

j ∈ N,

i lim 2v(a ji ) − N(ei ) = ∞.

i→∞

h

3.2. Unbounded Linear Operators on cK 0 (N, ω, Γ) D EFINITION 2. A stable unbounded linear operator A from cK 0 (N, ω, Γ) into itself is a pair (D(A), A) consisting of a subspace D(A) ⊂ cK (N, ω, Γ) (called the domain 0 of A) and a (possibly not continuous) linear transformation A : cK (N, ω, Γ) ⊃ D(A) → 0 cK (N, ω, Γ). Namely, the domain D(A) contains the basis (e ) and consists of all i i∈N 0 K (N, ω, Γ), that is, (N, ω, Γ) such Ax = x Ae converges in c x = (xi )i∈N ∈ cK ∑ i i 0 0 i∈N

    K  D(A) := x = (xi )i∈N ∈ c0 (N, ω, Γ) : lim N(ui Aei ) = ∞ ,   i→∞ Ax =

   

∑

ai j e′j (x) ei for each x ∈ D(A).

i, j∈N

Using the proof of Proposition 3 one can easily see that the following holds. P ROPOSITION 4. A stable unbounded linear operator     K  D(A) := x = (xi )i∈N ∈ c0 (N, ω, Γ) : lim N(xi Aei ) = ∞ ,   i→∞    

has an adjoint

Ax =

∑

ai j e′j (x) ei for each x ∈ D(A),

i, j∈N

A∗

if and only if for all j ∈ N, h i lim 2v(a ji ) − N(ei ) = ∞. i→∞

Bilinear forms on non-archimedean Hilbert spaces

9

In this event the adjoint A∗ of A is uniquely expressed by    ∗ ∗ K   D(A ) := y = (yi )i∈N ∈ c0 (N, ω, Γ) : lim N(yi A ei ) = ∞ ,   i→∞    

where a∗i j =

A∗ y =

∑

a∗i j e′j (y)ei for each y ∈ D(A∗ ),

i, j∈N

ω j a ji . ωi

3.3. Bounded Linear Functionals on cK 0 (N, ω, Γ) D EFINITION 3. A linear functional F : cK 0 (N, ω, Γ) → K is said to be bounded if there exists γ ∈ Γ such that v(F(x)) − N(x) ≥ γ for each 0 6= x ∈ cK 0 (N, ω, Γ). n   o If sup γ ∈ Γ : v F(x) − N(x) ≥ γ for each 0 6= x ∈ cK (N, ω, Γ) exists, it is 0 then called the norm of the continuous linear functional F and is defined by |kF|k =

inf

x∈cK 0 (N,ω,Γ),x6=0

h i v(F(x)) − N(x) .

Let us recall that the space of all continuous linear functionals on cK 0 (N, ω, Γ) ∗ K is denoted cK (N, ω, Γ) and called the (topological) dual of c (N, ω, Γ). The space 0 0 ∗ , |k · |k) is a Banach space over K. (cK (N, ω, Γ) 0 ∗ P ROPOSITION 5. Let F ∈ cK 0 (N, ω, Γ) . Then its norm |kF|k, if it exists, can be explicitly expressed as

  |kF|k = inf v(F(ei )) − N(ei ) . i∈N

The next theorem constitutes a non-archimedean version of the well-known Riesz representation theorem [10] in the case of a Krull valuation. T HEOREM 1. Let F : cK 0 (N, ω, Γ) → K be a linear functional such that (5)

  lim 2v(F(ei )) − N(ei ) = ∞.

i→∞

Then there exists a unique x0 ∈ cK 0 (N, ω, Γ) such that F(x) = hx, x0 i,

for all x ∈ cK 0 (N, ω, Γ).

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D. Attimu and T. Diagana

Proof. If x =

∑ xi ei ∈ cK0 (N, ω, Γ),

we then claim that F(x) =

i∈N

∑ xi F(ei ) is well-

i∈N

defined. Indeed, since x ∈ cK 0 (N, ω, Γ), then, lim N(xi ei ) = ∞. Moreover, it is not i→∞

hard to see that   2v(xi F(ei )) = 2v(xi ) + 2v(F(ei )) = N(xi ei ) + 2v(F(ei )) − N(ei ) , and hence lim v(xi F(ei )) = ∞,

i→∞

by using assumption (5).   F(ei ) ei and using (5), one can see that Setting x0 = ∑ ωi i∈N lim N

i→∞

  h  F(e )  i i ei = lim 2v(F(ei )) − N(ei ) → ∞, i→∞ ωi

K and hence x0 ∈ cK 0 (N, ω, Γ). Moreover, F(x) = hx, x0 i for each x ∈ c0 (N, ω, Γ). Suppose that there exists another y0 ∈ cK 0 (N, ω, Γ) such that F(x) = hx, y0 i for K each x ∈ cK 0 (N, ω, Γ). It follows that hx0 − y0 , ui = 0 for each x ∈ c0 (N, ω, Γ), that is, K x0 − y0 ⊥ c0 (N, ω, Γ). In particular, hx0 − y0 , ei i = 0 for each i ∈ N, so all coordinates of x0 − y0 in the basis (ei )i∈N of cK 0 (N, ω, Γ) are zero, and hence x0 = y0 .

K 4. Bilinear Forms on cK 0 (N, ω, Γ) × c0 (N, ω, Γ) K D EFINITION 4. A mapping ϕ : cK 0 (N, ω, Γ) × c0 (N, ω, Γ) → K is said to be a bilinear form whenever x 7→ ϕ(x, y) is linear for each y ∈ cK 0 (N, ω, Γ) and y 7→ ϕ(x, y) linear for each x ∈ cK (N, ω, Γ). 0 K Note that if ϕ : cK 0 (N, ω, Γ)×c0 (N, ω, Γ) → K is a bilinear form over the product K K c0 (N, ω, Γ) × c0 (N, ω, Γ), then the sum

ϕ(x, y) =

(6)

∞

∑

Ω i j xi y j

i, j=0

where Ωi j = ϕ(ei , e j ) for all i, j ∈ N, may or may not convergent. However if both x = (xi )i∈N and y = (yi )i∈N are taken in cK 0 (N, ω, Γ) with (7)

lim

i, j→∞



   v(Ωi j ) + 2v(xi) = ∞ and lim v(Ω ji ) + 2v(yi ) = ∞,

then the sum in (6) converges.

i, j→∞

Bilinear forms on non-archimedean Hilbert spaces

11

4.1. Bounded Bilinear Forms K D EFINITION 5. A non-archimedean bilinear form ϕ : cK 0 (N, ω, Γ) × c0 (N, ω, Γ) → K is said to be bounded if there exists γ ∈ Γ such that

2v(ϕ(x, y)) − N(x) − N(y) ≥ γ for all x, y ∈ cK 0 (N, ω, Γ).

(8)

If sup {γ ∈ Γ : (8) holds} exists, it is then called the norm of the bilinear form ϕ and is defined by   kϕk = inf

x,y6=0

2v(ϕ(x, y)) − N(x) − N(y) .

K P ROPOSITION 6. Let ϕ : cK 0 (N, ω, Γ) × c0 (N, ω, Γ) → K be a bounded bilinear form. If kϕk exists, it can then be explicitly expressed as h i kϕk = inf 2v(ϕ(ei , e j )) − N(ei ) − N(e j ) . i, j∈N

Proof. Suppose kϕk exists. The inequality, h i kϕk ≤ inf 2v(ϕ(ei , e j )) − N(ei ) − N(e j ) , i, j∈N

is a straightforward consequence of the definition of the norm kϕk of ϕ. Now suppose x, y 6= 0. In view of the above, one has 2v(ϕ(x, y))

= 2v



∞

∑

i, j=0

ϕ(ei , e j ) xi y j



  inf 2v ϕ(ei , e j )xi y j i, j∈N h  i = inf 2v(ϕ(ei , e j )) − N(ei ) − N(e j ) + N(xi ei ) + N(y j e j ) i, j∈N h i ≥ inf 2v(ϕ(ei , e j )) − N(ei ) − N(e j ) + N(x) + N(y)

≥

i, j∈N

and hence h i 2v(ϕ(x, y)) − N(x) − N(y) ≥ inf 2v(ϕ(ei , e j )) − N(ei ) − N(e j ) . i, j∈N

One completes the proof by combining the first and the latest inequalities. K D EFINITION 6. A bounded bilinear form ϕ : cK 0 (N, ω, Γ) × c0 (N, ω, Γ) → K is said to be representable whether there exists a bounded linear operator A : cK 0 (N, ω, Γ) → cK 0 (N, ω, Γ) such that D E ϕ(x, y) = Ax, y , ∀x, y ∈ cK 0 (N, ω, Γ).

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D. Attimu and T. Diagana

K T HEOREM 2. Let ϕ : cK 0 (N, ω, Γ) × c0 (N, ω, Γ) → K be a non-degenerate K K bounded bilinear form on c0 (N, ω, Γ) × c0 (N, ω, Γ). Then ϕ is representable whenever (3) holds. In that case, if A denotes the linear operator associated with ϕ, then the adjoint A∗ of A exists.

Proof. Define the linear operator A on cK 0 (N, ω, Γ) associated with ϕ by   ϕ(e j , ei ) ′ e j (x) ei , Ax := ∑ ωi i, j∈N for each x ∈ Eω . We first check that the linear operator A given above is well-defined on the space cK (N, ω, Γ). For that, it suffices to show that, for all j ∈ N, 0   ϕ(e j , ei ) lim N ei = ∞. i→∞ ωi This in fact follows from   ϕ(e j , ei ) ei = 2v(ϕ(e j , ei )) − N(ei ) → ∞, as i → ∞, N ωi D E by using (3). It is also routine to see that ϕ(x, y) = Ax, y for all x, y ∈ cK 0 (N, ω, Γ). Of course, the linear operator A given above is bounded. Moreover, it is unique since ϕ is non-degenerate. It remains to show that A∗ , the adjoint of A exists. Indeed, using Proposition 3 we have     ϕ(ei , e j ) lim 2v − N(ei ) = lim [2v (ϕ(ei , e j )) − 2v(ω j )) − N(ei )] i→∞ i→∞ ωj   = lim 2v (ϕ(ei , e j )) − N(ei ) − 2N(e j ) i→∞   = lim 2v (ϕ(ei , e j )) − N(ei ) i→∞

= ∞ for all j ∈ N,

using assumption (3), and hence the adjoint A∗ of A exists. R EMARK 1. One should mention that in Theorem 2, if we suppose that kϕk exists, then kAk exists and kAk = kϕk. Indeed, using Proposition 2, h  ϕ(e , e )  i j i kAk := inf 2v + N(ei ) − N(e j ) i, j∈N ωi h i = inf 2v(ϕ(e j , ei )) − 2v(ωi) + N(ei ) − N(e j ) i, j∈N   = inf 2v(ϕ(e j , ei )) − N(ei ) − N(e j ) i, j∈N

= kϕk.

13

Bilinear forms on non-archimedean Hilbert spaces

E XAMPLE 1. Let K = Q p (p being a prime number such that p > 2) and let the group Γ be (Z, +, ≤). Define the Krull valuation v : Q p → Z ∪ {∞} by v(0) = ∞ and for 0 6= x ∈ Q p , n o v(x) = max r ∈ Z : pr divides x .

Let ωi = p−i for each i ∈ N. Let N0 ∈ N with N0 ≥ 1 (fixed) and set N

πi j0 = 1 +

1 1 1 + 2 2 + ···+ N N , 0 ω j ωi ω j ωi ω j 0

for all i, j ∈ N. Q

For all x = (xi )i∈N , y = (yi )i∈N ∈ c0 p (N, ω, Z), define the bilinear form ∞

ϕ(x, y) =

∑

πNi j0 xi y j .

i, j=0

Now, h  N kϕk : = inf 2v(πi j0 ) − N(ei ) − N(e j ) i, j∈N h i N = inf 2v(πi j0 ) − v(ωi ) − v(ω j ) i, j∈N

= inf (i + j) i, j∈N

= 0, and hence ϕ is bounded. Therefore, the only bounded linear operator on cK 0 (N, ω, Γ) associated with ϕ is the one defined by " N # π ji0 ′ Au = ∑ e j (u)ei i, j∈N ωi   for each u ∈ cK 0 (N, ω, Γ) with kAk = inf N (Aei ) − N(e j ) = kϕk = 0. i, j∈N

4.2. Stable Unbounded Bilinear Forms In this subsection we provide a representation theorem for some unbounded bilinear forms. More precisely, we consider those unbounded bilinear forms whose domains contain all elements of the canonical basis (ei )i∈N of cK 0 (N, ω, Γ), as such a basis plays a key role in the present setting. The subclass of all those types of unbounded bilinear K forms will be called stable and denoted ΣS (cK 0 (N, ω, Γ) × c0 (N, ω, Γ)). Similarly, the subclass of all unbounded bilinear forms whose domains do not contain elements of the above-mentioned canonical basis will be called unstable and K denoted ΣU (cK 0 (N, ω, Γ) × c0 (N, ω, Γ)). Note that a representation theorem similar K to Theorem 3 for elements of ΣU (cK 0 (N, ω, Γ) × c0 (N, ω, Γ)) will be left as an open question.

14

D. Attimu and T. Diagana

K D EFINITION 7. A mapping ϕ : cK 0 (N, ω, Γ)× c0 (N, ω, Γ) ⊃ D(ϕ) × D(ϕ) → K is said to be a stable unbounded bilinear form if u 7→ ϕ(u, v) is linear for each v ∈ D(ϕ), v 7→ ϕ(u, v) linear for each u ∈ D(ϕ), where D(ϕ) is a vector subspace of cK 0 (N, ω, Γ) that contains the basis (ei )i∈N , and consists of all x = (xi )i∈N ∈ cK 0 (N, ω, Γ) such that     lim v(Ωi j ) + 2v(xi) = lim v(Ω ji ) + 2v(x j ) = ∞ i, j→∞

i, j→∞

and ϕ(x, y) =

∞

∑

Ωi j xi y j , for all x, y ∈ D(ϕ)

i, j=0

where Ωi j = ϕ(ei , e j ). The space D(ϕ) defined above is called the domain of the bilinear form ϕ. D EFINITION 8. A bilinear form ϕ : D(ϕ) × D(ϕ) → K (D(ϕ) being its domain) is said to be representable whenever there exists a (possibly unbounded) linear operator A : D(A) → cK 0 (N, ω, Γ) (D(A) being the domain of A) such that ϕ(x, y) = hAx, yi,

∀x ∈ D(A), v ∈ D(ϕ).

T HEOREM 3. Suppose that ω = (ω j ) j∈N is chosen so that N(e j ) → γ as j → ∞ where γ ∈ Γ. Let ϕ : D(ϕ) × D(ϕ) → K be a non-degenerate stable unbounded bilinear form. Then ϕ is representable whenever assumption (3) holds. In that case, if A denotes the linear operator associated with ϕ, then the adjoint A∗ of A exists. Proof. For all x = (xi )i∈N , y = (y j ) j∈N ∈ D(ϕ), write ϕ(x, y) =

∞

∑

Ωi j xi y j and define

i, j=0

the linear operator A on cK 0 (N, ω, Γ) associated to it as follows:       K  D(A) := x = (xi )i∈N ∈ c0 (N, ω, Γ) : lim N xi Aei = ∞ ,   i→∞     ϕ(e j , ei ) ′    Ax = ∑ e j (x)ei for each x = (xi )i∈N ∈ D(A).  ωi i, j∈N

One can prove as in the proof of Theorem 2 that A is well-defined. Now ! 1 Ax = ∑ ∑ x j ϕ(e j , ei ) ei for each x = (xi )i∈N ∈ D(A), i∈N ωi j∈N and hence

E D Aei , e j = ϕ(ei , e j ) for all i, j ∈ N.

Bilinear forms on non-archimedean Hilbert spaces

15

  Moreover, D(A) ⊂ D(ϕ). That is, lim N xi Aei = ∞ yields i→∞

lim

i, j→∞



   v(Ωi j ) + 2v(xi) = lim v(Ω ji ) + 2v(xi ) = ∞. i, j→∞

Indeed, if x = (xi )i∈N ∈ D(A), then using the Cauchy-Schwarz inequality, it follows that, for all i, j ∈ N, 2v(ϕ(ei , e j )) ≥ N(Aei ) + N(e j ), and hence h i 2 v(ϕ(ei , e j )) + 2v(xi) = 2v(ϕ(ei , e j )) + 4v(xi )

≥ N(Aei ) + N(e j ) + 4v(xi)   = N xi Aei + N(e j ) + 2v(xi)   = N xi Aei + N(xi ei ) + N(e j ) − N(ei ) → ∞ as i, j → ∞.

Similarly, using the fact (e j ) j∈N ⊂ D(A), i.e., N(Ae j ) → ∞ as j → ∞, we obtain h i 2 v(ϕ(e j , ei )) + 2v(xi ) = 2v(ϕ(e j , ei )) + 4v(xi)

≥ N(Ae j ) + N(ei ) + 4v(xi )   = N Ae j + 2N(xi ei ) − N(ei )

→ ∞ as i, j → ∞.   Note that v xi yk ϕ(ei , ek ) → ∞ as i, k → ∞, using the fact that x ∈ D(A) ⊂ D(ϕ)

and y ∈ D(ϕ), as       2v xi yk ϕ(ei , ek ) = 2v(xi ) + v ϕ(ei , ek ) + v ϕ(ei , ek ) + 2v(yk ) → ∞, as i, k → ∞.

Hence

∑ ∑ xi yk ϕ(ei , ek ) = ∑ ∑ xi yk ϕ(ei , ek ).

k∈N i∈N

i∈N k∈N

Consequently, the following sequence of equalities is justified: ! D E 1 Ax, y = ∑ ωk y k ∑ xi ϕ(ei , ek ) ωk i∈N k∈N ! =

∑ yk ∑ xi ϕ(ei , ek )

k∈N

=

∑

i∈N

ϕ(ei , ek )xi yk

i,k∈N

=

ϕ(x, y),

16

D. Attimu and T. Diagana

for all x = (xi )i∈N ∈ D(A) and y = (yi )i∈N ∈ D(ϕ). Furthermore, the uniqueness of A is guaranteed by the fact that the form ϕ is non-degenerate. It remains to show that A∗ , the adjoint of A exists. Though this can be done as in the bounded case. E XAMPLE 2. This is a generalization of Example 1 to the case in which ϕ is possibly unbounded. Consider the bilinear form defined by ϕ(x, y) =

∑

π i j xi y j ,

∀x = (xi )i∈N , y = (yi )i∈N ∈ D(ϕ),

i, j∈N

where (πi j )i, j∈N ⊂ K is an arbitrary sequence, and the domain D(ϕ) of ϕ is defined by all x = (xi )i∈N ∈ cK 0 (N, ω, Γ) such that     lim v(πi j ) + 2v(xi) = lim v(π ji ) + 2v(xi) = ∞. i, j→∞

i, j→∞

Note that ϕ(ei , e j ) = πi j for all i, j ∈ N and therefore an equivalent version of assumption (3) is     (9) lim v(πi j ) − N(ei ) = lim v(π ji ) − N(ei ) = ∞. i→∞

i→∞

Furthermore, if ωi , i ∈ N are chosen such that N(ei ) → γ ∈ Γ as i → ∞ then upon making assumption (9), the unique (possibly unbounded) linear operator associated with ϕ is given by π ji ′ e j (x) ei , ∀x = (xi )i∈N ∈ D(A) Ax = ∑ i, j∈N ωi  o n  where D(A) = x = (xi )i∈N ∈ cK 0 (N, ω, Γ) : lim N xi Aei = ∞ . i→∞

In addition to the above, the adjoint A∗ of A does exist under assumption (9).

Acknowledgments The authors would like to express their thanks to the referee for careful reading of the manuscript and insightful comments.

References [1] ATTIMU , D., AND D IAGANA , T. Representation of bilinear forms in non-archimedian Hilbert spaces by linear operators II. Comment. Math. Univ. Carolin. 48, 3 (2007), 431– 442. [2] D IAGANA , T. Towards a theory of some unbounded linear operators on p-adic Hilbert spaces and applications. Ann. Math. Blaise Pascal 12, 1 (2005), 205–222. [3] D IAGANA , T. Erratum to “Towards a theory of some unbounded linear operators on p-adic Hilbert spaces and applications”. Ann. Math. Blaise Pascal 13 (2006), 105–106.

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[4] D IAGANA , T. Fractional powers of the algebraic sum of normal operators. Proc. Amer. Math. Soc. 134, 6 (2006), 1777–1782. [5] D IAGANA , T. An Introduction to Classical and p-adic Theory of Linear Operators and Applications. Nova Science Publishers, New York, 2006. [6] D IAGANA , T. Representation of bilinear forms in non-archimedian Hilbert spaces by linear operators. Comment. Math. Univ. Carolin. 47, 4 (2006), 695–705. [7] D IARRA , B. An operator on some ultrametric Hilbert spaces. J. Analysis 6 (1998), 55–74. [8] D IARRA , B. Geometry of the p-adic Hilbert spaces. Preprint (1999). [9] J OHNSON , G., AND L APIDUS , M. The Feynman Integral and Feynman Operational Calculus. Oxford Univ. Press, 2000. [10] K ATO , T. Perturbation Theory for Linear Operators. Springer, New York, 1966. [11] K ELLER , H., AND O CHSENIUS , H. Bounded operators on non-classical orthomodular spaces. Math. Slovaca 45 (1995), 413–434. [12] O CHSENIUS , H. Hilbert-like spaces over Krull valued fields. ultrametric functional analysis. In Contemp. Math., 319, Amer. Math. Soc. Nijmegen, 2002, Providence, RI, 2003, pp. 227–238. [13] O CHSENIUS , H., AND S CHIKHOF, H. Banach spaces over fields with an infinite rank valuation. In p-adic Functional Analysis. Dekker, New York, 1999, pp. 233–293. [14] O CHSENIUS , H., AND W, S. Norm Hilbert spaces over Krull valued fields. Indag. Math. (N.S.) 17, 1 (2006), 65–84. [15] R IBENBOIM , P. The Theory of Classical Valuations. Springer-Verlag, 1999. [16]

VAN ROOIJ , A. Non-Archimedean Functional Analysis. Marcel Dekker Inc, New York, 1978.

AMS Subject Classification: 47S10, 46S10. Dodzi ATTIMU, Toka DIAGANA, Department of Mathematics, Howard University, 2441 Sixth Street NW, Washington, DC 20059, USA e-mail: dkattimugmail. om,tdiaganahoward.edu Lavoro pervenuto in redazione il ??.??.???? e, in forma definitiva, il ??.??.????.