Characterization of compact and self-adjoint operators, and study of - PDF document

Characterization of compact and self-adjoint operators, and study of positive operators on a Banach space over a non-Archimedean field Khodr Shamseddine University of Manitoba Canada 1

Collaboration with Jos´ e Aguayo (Universidad de Concepcion) and Miguel Nova (Universidad Cat´ olica de la Sant´ ısima) Concepci´ on, Chile • Characterization of compact and self-adjoint operators on Free Banach spaces of count- able type over the complex Levi-Civita field, Journal of Mathematical Physics, Volume 54 # 2, 2013 . • Inner product on B ∗ -algebras of operators on a free Banach space over the Levi-Civita field, Indagationes Mathematicae , Volume 26 # 1, 2015. • Positive operators on a Banach space over the Levi-Civita field, in preparation .

Contents 1. The Levi-Civita Fields R and C 2. Normal Projections, Compact and Self-adjoint Operators on the Space c 0 ( C ) 3. B ∗ -algebras of Operators on c 0 ( C ) 4. Positive Operators 5. Partial Order on Self-adjoint Operators

1. The Levi-Civita Fields R and C • Let R = { f : Q → R |{ q ∈ Q | f ( q ) ̸ = 0 } is left- finite } . • Notation: An element of R is denoted by x and its function value at q ∈ Q by x [ q ] . • Supp ( x ) = { q ∈ Q | x [ q ] ̸ = 0 } . • For x ∈ R , define { min(supp( x )) if x ̸ = 0 λ ( x ) = if x = 0 . ∞ • Arithmetic on R : Let x, y ∈ R . We define x + y and x · y as follows. For q ∈ Q , let ( x + y )[ q ] = x [ q ] + y [ q ] ∑ ( x · y )[ q ] = x [ q 1 ] · y [ q 2 ] . q 1 + q 2 = q Then, x + y ∈ R and x · y ∈ R . Result: ( R , + , · ) is a field. [Levi-Civita, 1892] Definition: C := R + i R . Then ( C , + , · ) is also a field.

Order in R • Define the relation ≤ on R × R as follows: x ≤ y if x = y or ( x ̸ = y and ( x − y )[ λ ( x − y )] < 0) . • ( R , + , · , ≤ ) is an ordered field. • R is real closed. ⇓ C is algebraically closed. • The map E : R → R , given by { x if q = 0 E ( x )[ q ] = , 0 else is an order preserving embedding. • There are infinitely small and infinitely large elements in R : The number d , given by { 1 if q = 1 d [ q ] = , 0 else is infinitely small; while d − 1 is infinitely large.

For x ∈ R , define { x if x ≥ 0 | x | 0 = if x < 0 ; − x { e − λ ( x ) if x ̸ = 0 | x | = if x = 0 . 0 For z = x + iy ∈ C , define √ x 2 + y 2 ; √ | x | 2 0 + | y | 2 | z | 0 = 0 = { e − λ ( z ) if z ̸ = 0 | z | = 0 if z = 0 = max {| x | , | y |} since λ ( z ) = min { λ ( x ) , λ ( y ) } . Note that |·| o and |·| induce the same topology τ v on R (or C ). Moreover, C is topologically isomorphic to R 2 provided with the product topology induced by |·| in R .

Properties of ( R , τ v ) • ( R , τ v ) is a disconnected topological space. • ( R , τ v ) is Hausdorff. • There are no countable bases. • The topology induced to R is the discrete topology. • ( R , τ v ) is not locally compact. • τ v is zero-dimensional (i.e. it has a base consisting of clopen sets). • τ v is not a vector topology. • For all x ∈ R (or C ): x = ∑ ∞ n =1 x [ q n ] · d q n .

Uniqueness of R and C • R is the smallest complete and real closed non-Archimedean field extension of R . – It is small enough so that the R -numbers can be implemented on a computer, thus allowing for computational applications. • C is the smallest complete and algebraically closed non-Archimedean field extension of R (or C ).

2. Normal Projections, Compact and Self-adjoint Operators on c 0 ( C ) { } c 0 ( C ) = ( x n ) n ∈ N : x n ∈ C ; lim n →∞ x n = 0 ≡ c 0 ; { } c 0 ( R ) = ( x n ) n ∈ N : x n ∈ R ; lim n →∞ x n = 0 ; L ( c 0 ) = { T : c 0 → c 0 : T linear & continuous } . We consider the following form: ∞ ∑ ⟨· , ·⟩ : c 0 × c 0 → C ; ⟨ z, w ⟩ = z n w n . n =1 This is well-defined; and it satisfies: (1) ⟨ z, z ⟩ ≥ 0 and ⟨ z, z ⟩ = 0 if and only if z = 0 ; az 1 + bz 2 , w ⟨ ⟩ ⟨ z 1 , w ⟩ ⟨ z 2 , w ⟩ (2) = a + b for a, b ∈ C and z 1 , z 2 , w ∈ c 0 ; (3) ⟨ z, w ⟩ = ⟨ w, z ⟩ for z, w ∈ c 0 ; (4) |⟨ z, w ⟩| 2 ≤ |⟨ z, z ⟩| |⟨ w, w ⟩| (the Cauchy-Schwarz inequality). √ Let ∥ z ∥ := |⟨ z, z ⟩| . Then ∥·∥ is a non-Archimedean norm on c 0 . Moreover, ∥·∥ = ∥·∥ ∞ .

If M is a subspace of c 0 , then M p Notation: will denote the subspace of all y ∈ c 0 such that ⟨ y, x ⟩ = 0 , for all x ∈ M. A sequence ( z n ) of non-null vec- Definition: tors of c 0 has the Riemann-Lebesgue Property (RLP) if for all z ∈ c 0 , n →∞ ⟨ z n , z ⟩ = 0 . lim Note: Any basis of c 0 has the (RLP) property. Theorem: Let M be an infinite dimensional closed subspace of c 0 . Then, the following state- ments are equivalent: (1) M has a normal complement. That is c 0 = M ⊕ M p . (2) M has an orthonormal base with the Riemann- Lebesgue Property. (3) There exists a normal projection P such that N ( P ) = M.

Any continuous linear operator u ∈ L ( c 0 ) can be identified with a matrix of the form   α 11 α 12 α 13 · · · α 1 j · · · α 21 α 22 α 23 · · · α 2 j · · ·     α 31 α 32 α 33 · · · α 3 j · · ·     . [ u ] = . . .  .  .   α i 1 α i 2 α i 3 · · · α ij · · ·     . . . . .   . ↓ ↓ ↓ · · · ↓ · · ·   0 0 0 0 where (1) lim i →∞ α ij = 0 , for any j ∈ N , (2) sup i,j ∈ N | α ij | < ∞ , (3) ∥ u ∥ = sup i,j ∈ N | α ij | = sup n ∈ N ∥ ue n ∥ . Definition: An operator v : c 0 → c 0 is said to be an adjoint of a given operator u ∈ L ( c 0 ) if ⟨ u ( x ) , y ⟩ = ⟨ x, v ( y ) ⟩ , for all x, y ∈ c 0 . In that case, we will say that u admits an adjoint v. We will also say that u is self-adjoint if v = u.

Lemma: Let u ∈ L ( c 0 ) with associated matrix { α i,j } i,j ≥ 1 . Then, u admits an adjoint operator v if and only if lim j →∞ | α ij | = 0 , for each i ∈ N . So   α 11 α 12 α 13 · · · α 1 j · · · → 0 α 21 α 22 α 23 · · · α 2 j · · · → 0     α 31 α 32 α 33 · · · α 3 j · · · → 0     [ u ] = . . . . .   . .   α i 1 α i 2 α i 3 · · · α ij · · · → 0     . . . . .   .  ↓ ↓ ↓ · · · ↓ · · ·  0 0 0 0 In the classical Hilbert space theory, any con- tinuous linear operator admits an adjoint. This is not true in the non-Archimedean case. For example, the operator u ∈ L ( c 0 ) given by the matrix: b b 2 b 3 · · · b j · · ·   0 0 0 · · · 0 · · ·     0 0 0 · · · 0 · · ·   ,   . . . . .   .   0 0 0 · · · 0 · · ·   . . . . . . with 1 < | b | , doesn’t admit an adjoint.

The following two theorems provide charac- terizations for normal projections. Theorem: Let P ∈ L ( c 0 ) . Then P is a normal projection if and only if P is self-adjoint and P 2 = P . Theorem: If P : c 0 → c 0 is a normal projection with R ( P ) = [ { y 1 , y 2 , · · · } ] , where { y 1 , y 2 , · · · } is an orthonormal finite subset of c 0 or an orthonor- mal sequence with the Riemann-Lebesgue Prop- erty, then Px = ∑ ∞ ⟨ x,y i ⟩ ⟨ y i ,y i ⟩ y i . i =1

Since C is not locally compact, convex com- pact sets of c 0 are trivial. Definition: A subset C of c 0 is called compactoid if for every ϵ > 0 there exists a finite subset S ⊂ c 0 such that C ⊂ B c 0 ( 0 , ϵ ) + co ( S ) , where B c 0 ( 0 , ϵ ) = { x ∈ c 0 : ∥ x ∥ ≤ ϵ } and co ( S ) is the ab- solutely (closed) convex hull of S . Definition: A linear operator T : c 0 → c 0 is said to be compact if T ( B c 0 ) is compactoid, where B c 0 = { x ∈ c 0 : ∥ x ∥ ≤ 1 } is the unit ball of c 0 . Theorem: T ∈ L ( c 0 ) is compact if and only if, for each ϵ > 0 , there exists a linear operator of finite-dimensional range S such that ∥ T − S ∥ ≤ ϵ.

The following theorem provides a way to con- struct compact and self-adjoint operators start- ing from an orthonormal sequence. Theorem: Let { y 1 , y 2 , · · · } be an orthonormal se- quence in c 0 . Then, for any λ = ( λ n ) in c 0 ( R ) , the map T : c 0 → c 0 defined by ∞ ∞ ⟨· , y n ⟩ ∑ ∑ T ( · ) = λ n ⟨ y n , y n ⟩ y n ≡ λ n P n ( · ) n =1 n =1 is a compact and self-adjoint operator. The converse is also true, as the following the- orem shows. Theorem: If the linear operator T : c 0 → c 0 is compact and self-adjoint, then there exist λ = ( λ n ) ∈ c 0 ( R ) and an orthonormal sequence { y n } in c 0 such that ∞ ⟨· , y n ⟩ ∑ T = λ n ⟨ y n , y n ⟩ y n . n =1

3. B ∗ -algebras Let A 0 ≡ A 0 ( c 0 ) := { T ∈ L ( c 0 ) : T has an adjoint } . • A 0 is a non-commutative Banach algebra with unity. • A 0 contains normal projections. • T ∈ A 0 if and only if its associated matrix has the form:   α 11 α 12 α 13 α 1 j → 0 α 21 α 22 α 23 α 2 j → 0     α 31 α 32 α 33 α 3 j → 0   . . . . ... . . . .   . . . . [ T ] =     α i 1 α i 2 α i 3 α ij → 0     ↓ ↓ ↓ ↓ · · ·   0 0 0 0 · · ·