HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR. 1. Introduction

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1 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR CAMILO ARGOTY AND ALEXANDER BERENSTEIN Abstract. We study Hilbert spaces expanded with a unitary operator with a countable spectrum. We show the theory of such a structure is ω-stable and has quantifier elimination. 1. Introduction This paper deals with the expansion of a Hilbert space with a unitary operator with a countable spectrum from the perpective of continuous logic ( for an introduction to this subject see [4]). We study the model-theoretic properties of such a structure in terms of the spectrum of the operator. In other words, our goal is to study the relation between Model Theory and Spectral Theory when the spectrum of the operator involved is countable. The main results are the following: 1.1. Theorem. Let U be a unitary operator with pure point spectrum in a Hilbert space H. Then, the theory of structure (H, U) = (H, +, 0,, U) admits quantifier elimination, is separably categorical (see Definition 3.18), is ω-stable (see Definition 3.20) and the projection into every eigenspace is definable Theorem. Let U be a unitary operator whose spectrum has countably many non-empty accumulation points. Then, the theory of the structure (H, U) = (H, +, 0,, U) admits quantifier elimination and is ω-stable. The projection into an eigenspace is definable if and only if the corresponding eigenvalue is an isolated point in the spectrum. The theory of the structure (H, +, 0,, U) where U is a unitary operator, was first studied by Henson and Iovino in [11], where they observed that it is stable. A geometric characterization of forking in such structures was first done by Berenstein and Buechler [7] after adding to the structure the projections corresponding to the The authors would like to thank Ward Henson for allowing us to include in this paper an unpublished result of his. 1

2 2 CAMILO ARGOTY AND ALEXANDER BERENSTEIN Spectral Decomposition Theorem. Ben Yaacov, Usvyatsov and Zadka characterized the unitary operators corresponding to generic automorphisms of a Hilbert space as those unitary transformations whose spectrum is S 1. In this work we study the definable aspects of the spectral decomposition, we classify the separable models of the expansions and we study orthogonality of types when U has a countable spectrum. We also give a different proof of the characterization of forking given in [7]: we provide an explicit freeness relation and prove some of its properties which show it coincides with non-forking, without adding to the structure the projections corresponding to the spectral decomposition. The framework for this work is continuous logic, we assume the reader is familiar with notions such as definability, definable and algebraic closure, and forking. The background can be found in [4, 5]. This paper is divided as follows. In the second section we give a brief introduction to Spectral Theory. In the third section we study the expansions of a Hilbert space with unitary operators with a pure point (finite) spectrum and with a countably infinite spectrum. 2. Preliminaries: Spectral theory The following section is based on [1] and [12]. Let H = (H, +, 0, ) be an infinite dimensional Hilbert space over C Definition. Let A be a linear operator from H into H. The operator A is called bounded if the set { A(u) : u H, u = 1} is bounded in R. If A is bounded we define the norm of A by: A = sup A(u) u H, u = Definition. A sequence of linear operators {A n } n ω converges uniformly to an operator A, if lim n A A n = Definition. Given a linear operator A : H H, its adjoint operator, denoted A is the linear operator A : H H such that for every u, v H, Au v = u A v Definition. A linear operator A : H H is called self-adjoint if A = A.

3 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR Definition. Let N a linear operator from H to H. N is called normal if N commutes with its adjoint N Definition. Given a normal operator A, a complex number λ is called an eigenvalue or punctual spectral value of A if the operator A λi is not one to one. A complex number λ is called a continuous spectral value if the operator A λi is one to one and the operator (A λi) 1 is densely defined but is unbounded Fact (Theorem 1, section 93 in [1]). A point r belongs to the continuous spectrum of an operator A if and only if there is an orthonormal sequence of elements (f n : n ω) such that lim n (Af n rf n ) = Definition. The spectrum of an operator A, denoted by σ(a), is the set of the punctual and continuous spectral values. It is well known that if A is a bounded normal operator, then σ(a) is a bounded subset of C and that if A is self-adjoint, then σ(a) is a subset of R Definition. A self-adjoint operator A different from the zero operator is called positive and we write A 0, if Au u 0 for all u H. If A and B are selfadjoint operators, we write A B if A B is positive Proposition. For a self-adjoint operator A, A 2 is positive. Proof. Clear Definition. A self-adjoint Q operator is called a square root of a positive self-adjoint operator A if Q 2 = A Fact (Theorem 1, section 36 in [12]). Let A be a self-adjoint operator and let E + be the projection of H onto the null space of the operator A Q where Q is the positive square root of A 2. Then (1) Any bounded operator R that commutes with A, commutes with E +. (2) AE + 0, A(Id E + ) Fact (Theorem 2, section 36 in [12]). Let A be a bounded self-adjoint operator, let r be a real number and let E + (r) the projection operator constructed for A r = A ri according to Fact If we denote by E r the projection operator Id E + (r), then the family {E r } r R satisfies the following conditions:

4 4 CAMILO ARGOTY AND ALEXANDER BERENSTEIN (1) Any bounded operator R that commutes with A commutes with E r. (2) E r E s if r < s. (3) E r is continuous on the left: s<r E s = E r. (4) E r = 0 for < r m and E r = I for M < r <, where m = inf(σ(a)) and M = sup(σ(a)) Definition. The family {E r } r [m,m], where σ(a) [m, M] is called the resolution of the identity generated by A Fact (Spectral Decomposition Theorem, section 35 in [12]). Let A be a bounded self-adjoint operator. For every ɛ > 0, A = M+ɛ rde m r, where the integral is to be interpreted as limit of finite sums in the sense of uniform convergence in the space of operators, and m = inf(σ(a)) and M = sup(σ(a)). Let A as above. The Spectral Decomposition Theorem can be interpreted in the following way: given ɛ > 0 and δ > 0 there is an N such that for any partition m = r 0 < s 0 = r 1 < s 1 = r 2 < < s N = M +ɛ with max{s k r k } < 2(M +ɛ)/n, if we write E( k ) for E sk E rk, we have A N k 1 r k E( k ) < δ Definition. Let A be a normal operator. The real and imaginary parts of A are the operators A r = 1 2 (A + A ) and A i = 1 2 (A A ) Fact. The real and imaginary parts A r and A i of a normal operator A are self-adjoint. Proof. For u, v H, A r u v = 1 2 (A + A )u v = u 1 2 (A + A v = u 1 2 (A + A )v = u A r v. Similarly for A i Corollary. If A is a normal operator, then A r and A i have integral representations A r = b a sde s and A i = b a tdf t, where E s F t = F t E s, and we can write A = (s + it)de s df t Definition. A normal operator U on H is said to be unitary if U is a bijection and for any u, v H, Uu Uv = u v Fact (Section 71 in [1]). Let U be a unitary operator. Then there is a unique resolution of the identity {E r r [0, 2π]} such that U k = π 0 eikt de t for all k Z.

5 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR 5 We want to know if we can recover Theorem (Theorem in [3]). Let N be a normal operator on H, let B(σ(N)) the algebra of Borel functions from σ(n) into the complex numbers, and let B(H) the algebra of linear operators on H. Then there exist a isometric monomorphism π : B(σ(N)) B(H) such that π( f) = (π(f) ), π(1) = Id and if f = i j a ijz i z j, then π(f) = i j a ijn i (N ) j, where by 1 we denote the constant function on σ(n) with value Fact. Let A be a selfadjoint operator on H. Let m = inf{σ(a)} and M = sup(σ(a)). Let f the function on [m, M] such that f(x) = 0 if x 0 and f(x) = 1 if x > 0. Then f is aproximable by polynomials. Proof. This function clearly belongs to L 2 [m, M] and the polynomials are dense in L 2 [m.m], so the conclusion follows Theorem. Let A be a selfadjoint operator. Let E + be the projection of H onto the null space of the operator A Q where Q is the positive square root of A 2. Then E + is approximable by operators of the form g(a) where g is a polynomial. Proof. For any v H we have E + (v) = M m g(r)de r(v) where g is the function defined in Fact But by 2.22 g(x) can approximated by polynomials, E + can also be approximated by polynomials in A. It is important to note that for a selfadjoint operator A on a Hilbert space H, E + is the limit of a sequence of polynomials in A, but that the sequence does not converge uniformly. That is, given a unit vector v H, E + (v) can be approximated by polynomials in A(v), but the rate of convergence depends on v Definition. The essential spectrum σ e (U) of a linear operator U is the set of accumulation points of the spectrum of U together with the set of eigenvalues of infinite multiplicity. The finite spectrum σ fin (U) of a linear operator U, the complement of σ e (U), is the set of isolated points of the spectrum of U of finite multiplicity.

6 6 CAMILO ARGOTY AND ALEXANDER BERENSTEIN 3. Unitary operators We begin this section by characterizing the theory of the expansion of a Hilbert space with a unitary operator in terms of the spectrum of the operator using an instance of a result by C. Ward Henson: 3.1. Theorem. Let A and B be two normal operators on Hilbert spaces H A, H B respectively. Then the structures (H A, +, 0,, A) and (H B, +, 0,, B) are elementarily equivalent if and only if (1) σ e (A) = σ e (B). (2) dim{x H A : Ax = λx} = dim{x H B : Bx = λx} for λ S 1 \ σ e (A). We will provide a proof for the result when the operators involved have a countable spectrum and are unitary. Henson s original proof (unpublished), based on a Theorem by Voiculescu (see [10]), yields stronger information than our proof, in particular it shows that the theory T h(h, N A ) is separably categorical up to perturbations of the automorphism. We follow the standard way of dealing with Hilbert spaces, we write the structure as (H, +, 0, ) and we work inside the unit ball. We denote by L the language of Hilbert spaces and by L U the language of Hilbert spaces with a new unary function, which we denote as U. Whenever we quantify over H we mean we are quantifying over its unit ball Notation. For λ σ(u), we denote by H λ the set {x H : Ux = λx}, by P λ the projection operator onto the space H λ Theorem. Let U A and U B be two unitary operators with countable spectrum on Hilbert space H. Then the structures (H A, +, 0,, U A ) and (H B, +, 0,, U B ) are elementarily equivalent if and only if (1) σ e (U A ) = σ e (U B ). (2) dim{x H A : U A x = λx} = dim{x H B : U B x = λx} for λ S 1 \σ e (U A ). Proof. ) We may assume the structures are separable and we will write H instead of H A, H B. Assume that the structures (H, +, 0,, U A ) and (H, +, 0,, U B ) are elementarily equivalent. Let σ A be the spectrum of U A and let σ B be the spectrum of U B. For every µ S 1 \ σ A, let η = d(µ, σ A ).

7 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR 7 ) Then the statement sup u (η u Uu µu = 0 is true for (H, +, 0,, U A ) and thus true for (H, +, 0,, U B ). Therefore σ B σ A. In a similar way we can show that σ A σ B. For every λ σ A fin whose eigenspace is of dimension m λ, we have that the statement inf max ( u i u j, u i 1, U(u i ) λu i ) = 0 u 1u 2 u mλ is valid in (H, +, 0,, U A ). Thus the same statement is true for (H, +, 0,, U B ). From this condition and the fact that λ is an isolated point in the spectrum it easily follows that dim{x H : U A x = λx} dim{x H : U B x = λx}. In a similar way we can prove that dim{x H : U A x = λx} dim{x H : U B x = λx}. Also, for every λ σe A and every k 1, inf max ( u i u j, u i 1, U(u i ) λu i ) = 0 u 1u 2 u k is true for (H, +, 0, U A ) and thus also true for (H, +, 0,, U B ). This implies that σ A e = σ B e. ) Conversely, assume that σ A e = σ B e and dim{x H : U A x = λx} = dim{x H : U B x = λx} for λ σ A fin. Let σ e = σ A e. We may assume, exchanging (H A, U A ) and (H B, U B ) for elementary superstructures, that (H A, U A ) and (H B, U B ) are ℵ 0 -saturated. Let H fin A H A : U A (x) = λx}. Note that H λ A = λ σfin H λ A, where Hλ A = {x is the domain of a finite dimensional Hilbert subspace of H A. Let H fin B = λ σ fin H λ B, where Hλ B = {x H B : U B (x) = λx}. (H fin B, U B) is a substructure of (H B, U B ). Then for each λ σ fin, (H λ A, U A) = (H λ B, U B), so (H fin A this common substructure as H fin., U A) = (H fin B, U B). We denote Let H C = H A Hfin H B, the free amalgamation of H A and H B over H fin and let U C be the induced unitary map on H C determined by U A and U B. We will prove that (H A, U A ) (H C, U C ) and that (H B, U B ) (H C, U C ). From this we get that the structures (H A, +, 0,, A) and (H B, +, 0,, B) are elementarily equivalent. Claim (H A, A) (H C, C) We use the Tarski-Vaught test. Let ϕ(x 1,..., x n, y) be a formula and let ā = (a 1,..., a n ) H n A. Assume that inf{ϕ(h C,U C ) (a 1,..., a n, b) : b

8 8 CAMILO ARGOTY AND ALEXANDER BERENSTEIN H C } = r and let ɛ > 0. Let b H C be such that ϕ(a 1,..., a n, b) r < ɛ. We may write b = b A + d, where b A = P HA (b), so d H A. There exists k 1 and an L-formula ψ(x 11,..., x n1, y 1 ;... ; x 1k,..., x nk, y k ) such that (H C, C) = sup x1... sup xn sup y ϕ(x 1,..., x n, y) ψ(x 1,..., x n, y;... ; U k (x 1 ),..., U k (x k ), U k (y 0. Clearly d λ σe Hλ B, so d = λ σ e P λ (d). Note that P λ (d) H A so P λ (d) P λ (a i ) for each i n. Since (H A, U A ) is ℵ 0 -saturated, dim(ha λ) = for each λ σ e, so there exists c λ HA λ such that c λ = P λ (d) and c λ P λ (a 1 ),..., P λ (a n ), P λ (b A ). Let c = λ σ e c λ and let b = b A + c. Then tp(c,..., U k (c)/{ā, U(ā),..., U k (ā), b A }) = tp(d,..., U k (d)/{ā, U(ā),..., U k (ā), b A }), so tp(b,..., U k (b)/{ā, U(ā),..., U k (ā)}) = tp(b,..., U k (b )/{ā, U(ā),..., U k (ā)}), so ψ(a 1,..., a n, b,..., U k (a 1 ),..., U k (a k ), U k (b)) = ψ(a 1,..., a n, b,..., U k (a 1 ),..., U k (a k ), U k (b )) and ϕ(a 1,..., a n, b ) r < ɛ. Since ɛ was arbitrary we get inf{ϕ (H C,U C ) (a 1,..., a n, b) : b H C } = r = inf{ϕ (H C,U C ) (a 1,..., a n, b) : b H A } In other words, since Theorem?? this theorem means that the operator theoretic concept of aproximate unitary equivalence and the model theoretic concept of elementary equivalence. The previous Theorem is proved in [6] when σ A = σe A = S 1. Let U be a unitary operator in a separable Hilbert space H and σ be the corresponding spectrum. The previous result says that the theory of the structure (H, U) = (H, +, 0,,, U) determines the dimension of the eigenspace corresponding to the isolated points of the spectrum. We strengthen this observation by showing the definability of these eigenspaces Notation. For λ σ, we denote by B λ the unit ball in the space H λ. Recall that a closed set D H is definable if the function dist(x, D) is a definable predicate. We will use the following characterization of definable sets and functions: 3.5. Fact (Proposition 9.18 of [4]). Let D H be closed. Then D is definable in (H, U) if and only if there is a definable predicate P : H n [0, 1] such that

9 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR 9 P (x) = 0 for all x D and ɛ δ x H n (P (x) δ d(x, D) ɛ) Fact (Proposition 9.22 of [4]). Let (H, U ) be κ-saturated where κ is uncountable and let A H have cardinality < κ. Let f : H n H be any function and let G f its graph. Then the following are equivalent: (1) f is definable in (H, U ) over A. (2) G f is type-definable in (H, U ) over A Lemma. Let λ σ be isolated, let χ = d(λ, σ \ {λ}) and let u H. If U(u) λu < ɛ then u P λ (u) < ɛ χ. Proof. Let λ σ be isolated and let u H. Then U(u) λ u = λ σ,λ λ (λ λ )P λ (u) and U(u) λ u 2 = λ σ,λ λ λ λ 2 P λ (u) 2. We have that λ λ 2 χ 2 for all λ σ such that λ λ. If U(u) λ u 2 < ɛ 2, χ 2 λ σ,λ λ P λ(u) 2 < ɛ 2, so that u P λ (u) 2 = λ σ,λ λ P λ(u) 2 < ɛ2 χ Theorem. For every isolated λ σ the subspace H λ is definable. Proof. By Lemma 3.7, we have that if U(u) λu 0, then d(u, H λ ) 0 uniformly on H. By Fact 3.5 H λ is definable. For general λ σ, we have a weaker result: 3.9. Theorem. For every λ σ the subspace H λ is a zero-set. Proof. H λ is the set of solutions of the statement U(v) λv = 0. We will later show (Corollary 3.27) that for λ σ non-isolated, H λ definable. is not Theorem. For every isolated λ the projection P λ is definable. Proof. We work in (H, U ) (H, U) an ℵ 1 -saturated elementary extension of (H, U). By Fact 3.6, it suffices to show that G Pλ is type-definable. Consider the formula ϕ(u, v) = U(v) λv 2 + sup x Bλ v u x 2. Then ϕ(u, v) = 0 if and only if v = P λ (u).

10 10 CAMILO ARGOTY AND ALEXANDER BERENSTEIN Finally we use Theorem 3.3 to give an explicit axiomatization of the theory of the structure (H, +, 0,,, U). We write σ = σ fin σ e and for each λ σ fin let m λ the dimension of H λ. Let T σ be the theory of Hilbert spaces together with the following statements: (1) sup u sup v u v U(u) U(v) = 0 (2) sup u inf v u U(v) = 0 (3) (Schema) (a) For n N and λ σ e, inf max ( u i u j, u i 1, U(u i ) λu i ) = 0 u 1u 2 u n (b) For λ σ fin, χ = d(λ, σ \ {λ}), and 0 < ɛ < 1: inf sup max ( u i u j, u i 1, Uu i λu i, ɛ U(v) λv, u 1u 2 u mλ v (4) (Schema) For µ S 1 \ σ and η = d(µ, σ), v ( ) sup η u Uu µu = 0 u Proposition. The theory T σ axiomatizes T h(h, +, 0,, U). m v u i u i ) χ) ɛ = 0 k=1 Proof. Clearly the statements (1), (2), (3a), (4) hold for (H, +, 0,, rangle, U). By Lemma 3.7 the statements (3b) are true for (H, +, 0,, rangle, U). Now assume that (H, +, 0,,, A) satisfies the axioms above. The first two axioms say that A is an inner product preserving linear map from H onto H and thus is a unitary operator. The fourth axiom says that the spectrum σ A of A is contained in σ. Finally by the third axiom, σe A = σ e and dim{x H : Ax = λx} = dim{x H : Ux = λx} for λ σ A \ σe A. By Theorem 3.3 we get T h(h, +, 0,, U) = T h(h, +, 0,, A). Our next goal is to study the separable models of T h(h, +, 0,, U) and understand the space of types over. This analysis depends heavily on the properties of the spectrum σ of U and we divide our work accordingly.

11 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR Finite spectrum. Let U be a unitary operator with a pure point spectrum. In this subsection we will study the model theory of the structure (H, 0, +,, U), characterizing the definable closure, the algebraic closure and properties of the space of types. We write σ for the spectrum of U Theorem. All the projections P λ, λ σ can be expressed as polynomials in U. Proof. Let σ = {λ 1,..., λ n }. Then for every u H, u = n k=1 P λ k (u) and for every m N, U m (u) = n k=1 λm k P λ k (u). So, u P λ1 (u) U(u) λ 1 λ 2 λ n P λ2 (u) =,.... U n 1 (u) λ n 1 1 λ n 1 2 λ n 1 n P λn (u) λ 1 λ 2 λ n The matrix is the van der Monde matrix, whose determinant is ( 1) n Π i<j (λ i λ j ) 0. This.. λ n 1 1 λ n 1 2 λ n 1 n implies, P λ1 (u) P λ2 (u) λ 1 λ 2 λ n =... P λn (u) λ1 n 1 λ n 1 2 λ n 1 n u U(u). U n 1 (u) Corollary. For each λ σ, the projection P λ is definable. Proof. Clear. Notice that the previous Corollary was already known by Theorem Theorem. The structure (H, 0, +,, U) admits quantifier elimination. Furthermore, for ā = (a 1,..., a n ) H n, tp(ā) is determined by the values of P λ (a k ), P λ (a l ) for λ σ and k, l n.

12 12 CAMILO ARGOTY AND ALEXANDER BERENSTEIN Proof. Let ā = (a 1,..., a n ), b = (b 1,..., b n ) H n. We write qftp(a 1,..., a n ) for the set of quanfier free statements valid for a 1,..., a n H. We will show that qftp(a 1,..., a n ) determines tp(a 1,..., a n ). Assume that qftp(ā) = qftp( b). Then, by Theorem 3.12 P λ (a k ) P λ (a l ) = P λ (b k ) P λ (b l ) for λ σ and k, l = 1,..., n. Let λ σ. We can assume without loss of generality that there exists 0 < k n such that the set {P λ (a 1 ),..., P λ (a k )} is linearly independent and the elements of the set {P λ (a k+1 ),..., P λ (a n )} can be expressed as linear combinations of the elements of {P λ (a 1 ),..., P λ (a k )}. Because qftp(ā) = qftp( b) we have that the set {P λ (b 1 ),..., P λ (b k )} is linearly independent and the elements {P λ (b k+1 ),..., P λ (b n )} can be expressed in terms of the elements of {P λ (b 1 ),..., P λ (b k )} with exactly the same linear combinations as before. For each λ σ, let B1 λ be an orthonormal basis of H λ {P λ (a 1 ),..., P λ (a k )} and let B2 λ be an orthonormal basis of H λ {P λ (b 1 ),..., P λ (b n )}. Then qftp(p λ (a 1 ),..., P λ (a n ), B1 λ ) = qftp(p λ (b 1 ),..., P λ (b n ), B2 λ ). For each λ σ, let f λ : H λ H λ be the linear transformation generated by the map that sends P λ (a k ) into P λ (b k ) for k = 1,..., n and is a bijection between B1 λ and B2 λ. Recall that H = λ σ H λ. Let f : H H be the automorphism induced by the family (f λ : λ σ) (we can write u H as λ σ u λ, where u λ = P λ (u). Then f( λ σ u λ) = λ σ f λ(u λ )). Clearly f(ā) = b and for any c H, f(u(c)) = U(f(c)) so f is an automorphism of the structure (H, U). This implies tp(ā) = tp( b) Lemma. Let A H. Then dcl(a) is the closed Hilbert space generated by the set {P λ (a) : a A, λ σ}. Proof. Let A be a subset of H and E be the closed vector space generated by the set {P λ (a) a A, λ σ}. We first show that E dcl(a). Let b E. Then there exists a sequence (c n,λ : n ω, λ σ) of complex numbers and a sequence (a n,λ : n ω, λ σ) of elements of A such that b = k=0,λ σ c k,λp λ (a k,λ ). Thus ɛ > 0 N n N, b n k=0,λ σ c k,λp λ (a k,λ ) < ɛ. Let R(x) = x b and φ n (x) := x n k=0,λ σ c kp λ (a k,λ ). Then ɛ > 0 N n N x( R(x) φ n (x) < ɛ) and {b} is definable over A.

13 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR 13 Let b E, then for some λ 0 σ, P λ0 (b) E. Since there are infinitely many vectors in H λ0 E with norm P λ0 (b P E (b)), there are infinitely many realizations of tp(b/a) and therefore b dcl(a) Lemma. Let A H. Then acl(a) is the closed Hilbert space generated by the union of dcl(a) with all the finite dimensional subspaces H λ with λ σ fin. Proof. Every ball B λ for λ σ fin is algebraic over because the closed unit ball of a finite dimensional space is compact. Therefore dcl(a) λ σfin H λ acl(a). Conversely, let E be the closure of the space generated by dcl(a) and λ σfin H λ. If b E, then P λ0 (b) P λ0 (P E (b)) 0 for some λ 0 σ e. Without loss of generality the dimension of H λ0 dcl(a) is infinite and the set {b H : tp(b /A) = tp(b/a)} is unbounded, thus b acl(a) Proposition. Let p, q S 1 ( ) and let a = p, b = q. Then d(p, q) = λ σ ( P λ(a) P λ (b) ) 2. Proof.. It is easy to see that: a b = a 2 2 a b + b 2 = = ( P λ (a) 2 2 P λ (a) P λ (b) + P λ (b) 2 ) = Then, λ σ ( P λ (a) 2 2 P λ (a) P λ (b) + P λ (b) 2 ) λ σ ( P λ (a) P λ (b) ) 2 λ σ d(p, q) = inf{ a b H = p(a ) and H = q(b )} ( P λ (a) P λ (b) ) 2. λ σ. Let a, b be such that tp(a) = tp(a ) and tp(b) = tp(b ). Then P λ (a) = P λ (a ) and P λ (b) = P λ (b ) for λ σ, but the inner products P λ (a ) P λ (b ) depend on the choice of a and b. We may choose a and b such that for each

14 14 CAMILO ARGOTY AND ALEXANDER BERENSTEIN λ σ, P λ (a ) P λ (b ) = P λ (a ) P λ (b ). Then, d(p, q) = inf{ a b H = p(a) and H = q(b)} a b = = a 2 2 a b + b 2 = ( P λ (a ) 2 2 P λ (a ) P λ (b ) + P λ (b ) 2 ) = = = λ σ ( P λ (a ) 2 2 P λ (a ) P λ (b ) + P λ (b ) 2 ) = λ σ ( P λ (a) P λ (b) ) 2. λ σ Definition. Recall that the theory of a metric structure M is called separably categorical if whenever N 1, N 2 = T h(m) are separable we have N 1 = N Lemma. The theory T σ is separably categorical. Proof. Let (H, U ) and (H, U ) be separable models of T σ. Then for each λ σ, dim(h λ ) = dim(h λ ) (which is either finite or ℵ 0). Hence, we have that for every λ σ, H λ = H λ and thus (H, U ) = (H, U ). One could also prove the previous Lemma using Proposition It is clear that the formula presented in Proposition 3.17 is definable and thus the logic topology and the distance topology agree on the space of types and by Theorem 12.4 [4] T σ is separably categorical Definition. The theory of a metric structure Mis called ω-stable if for any N = T h(m) and A N countable, S(A) is separable Theorem. The theory T σ is ω-stable. Proof. Let H = T σ be separable and let A H be a countable set. Let Ā be the algebraic closure of A and write H = Ā + Ā = λ σ H λ. For each λ σ e, let H λ be a separable Hilbert space such that H λ H λ = {0} and let H = H λ σe H λ. We define for each v H λ, U λ (v) = λv and let U be the unitary map on H induced by U and U λ, λ σ e. Then (H, U ) = T σ and dim(h λ Ā ) = for

15 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR 15 each λ σ e. In particular, by Theorem 3.14 H realizes all types over A, so we can work inside the structure H. Since H is separable, so is S(A) Countably infinite spectrum. Our next step is to consider a unitary operator U with countable spectrum σ. We write σ = σ a σ i, where σ i are the isolated points of the spectrum and σ a are the non-isolated points of the spectrum Theorem. The following properties are true: (1) T σ has quantifier elimination. Furtheromore, for any ā H n, tp(ā) is determined by the values P λ (a k ), P λ (a j ) for λ σ, j, l n. (2) T σ is ω-stable. Proof. (1) It follows from Lemma?? that for any v H and λ σ, P λ (v) is in the quantifier free definable closure of v. The rest o the proof follows as in Theorem (2) We can proceed as in Theorem Proposition. Let p, q S 1 ( ) and let a = p, b = q. Then d(p, q) = λ σ ( P λ(a) P λ (b) ) 2. Proof. Similar to Proposition Theorem. The principal types in S 1 (T ) are the ones of elements a H with P λ (a) = 0 for λ σ a. Proof. We can build a structure (H, 0, +,, U ) elementarily equivalent to (H, 0, +,, U) such that H λ = 0 for all λ σ a. In this structure the types of elements a such that P λ (a) 0 for some λ σ a are omitted. Conversely, assume that a is such that P λ (a) = 0 for all λ σ a and let (H, 0, +,,, U ) be a model of T σ. By Theorem 3.10 the projections P λ are definable for every isolated λ σ and thus H λ 0. Let a λ H λ be such that a λ = P λ(a) for each λ σ i and let a = λ σ i a λ. Then a H and tp(a) = tp(a ). One could also use Proposition 3.23 to prove the previous Theorem.

16 16 CAMILO ARGOTY AND ALEXANDER BERENSTEIN Corollary. The atomic models of T σ are the models in which the accumulation points of the spectrum are not eigenvalues Corollary. The ℵ 0 -saturated models of T σ are the ones in which the accumulation points of the spectrum are eigenvalues whose eigenspace is infinite dimensional Corollary. For λ σ a, the set H λ is not definable and the function P λ is not definable. Proof. Assume, in order to get a contradiction, that H λ is definable. In an ℵ 0 - saturated structure the statement inf u Hλ ( u 1) 2 = 0 is true. Thus in the prime model this property also holds and there is a vector of norm one in H λ, a contradiction. Assume now that P λ is definable. Then x P λ (x) measures the distance from x into H λ (x) and thus H λ is definable, a contradiction Theorem. If σ a is finite then T σ has ℵ 0 nonisomorphic separable models. If σ a is infinite then T σ has 2 ℵ0 nonisomorphic separable models. Proof. Let H and H be separable models of T σ. We can write H = λ σ i σ a H λ and H = λ σ i σ a H λ. For λ σ i, H λ = H λ, so the only diference between H and H can come from the spaces H λ and H λ for λ σ a. For such λ, the spaces H λ and H λ are isomorphic if and only if dim(h λ) = dim(h λ ). Assume first that σ a is a finite non-empty set. So there is exactly one model up to isomorphism for every possible dimension of H λ for λ σ a. Thus there are ℵ 0 nonisomorphic separable models of the theory T σ. On the other hand, if σ a is an infinite countable set, there are 2 ℵ0 nonisomorphic separable models of T σ Forking. We fix a countable spectrum σ and a structure (H, U) = T σ which is κ-saturated and strongly κ-homogeneous for some uncountable inaccessible cardinal κ. We say A H is small if A < κ Definition. (1) Let ā = (a 1,..., a m ) H m ; let B, C H be small, let B C = acl(b C) and C = acl(c). We write ā B and say that ā is - C independent from B over C if P B C (P λ (a i )) = P C(P λ (a i )) for i = 1,..., m and λ σ. (2) For A, B, C H small we say A B if and only if for all finite subsets ā C of A, we have that ā B. C

17 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR Lemma. Let B be an algebraically closed small set in (H, U). Then for every λ σ and for every a H, P λ (P B (a)) = P B (P λ (a)). Proof. We first need the following claim: Claim. P B and U commute Proof. Let a H. We can write a = a B + a B, where a B = P B (a) and a B = a a B. Then U(a) = U(a B ) + U(a B ). For any b B, U(b), U (b) B. Thus b U(a B ) = U (b) a B = 0. This implies that U(a B ) B and U(a B ) B ; P B (U(a)) = P B (U(a B ) + U(a B )) = U(a B ) and thus P B (U(a)) = U(P B (a)). The result follows from Fact 2.13 part Notation. For A H small, we write Ā for acl(a) Corollary. Let ā = (a 1,..., a m ) H m ; let B, C H be small. Then ā B C if and only if P B C (a i ) = P C (a i ) for i m. Proof. Assume first that ā B, then P C B C (P λ(a i )) = P C(P λ (a i )) for any i n. By Lemma 3.30 we get P λ (P B C (a i )) = P λ (P C(a i )) for all λ and thus P B C (a i ) = P C(a i ). The converse is proved in a similar way Proposition. Given two tuples ā = (a 1,..., a n ) and b = (b 1,..., b m ) and a small set C, ā b C if and only if Pλ (a k ) P C λ(b j ) for k = 1,..., n; j = 1,..., m and λ σ. Proof. Let ā = (a 1,..., a n ) and b = (b 1,..., b m ). If ā b, C then we have that, P (P {b1,...,b m} C λ(a i )) = P C(P λ (a i )) for every i = 1,..., n and λ σ. For λ σ and j m, we have C C {P λ (b j )} C {b j } C {b 1,..., b m }, so for every i = 1,..., n, P {Pλ (b (P j)} C λ(a i )) = P C(P λ (a i )). Conversely, let us suppose that P {Pλ (b (P j)} C λ(a k )) = P C(P λ (a k )) for all λ σ, k = 1,..., n, j = 1,..., m. We fix λ σ. We can write P λ (a k ) = P C(P λ (a k ))+P C (P λ (a k )). Since P {Pλ (b (P j)} C λ(a k )) = P C(P λ (a k )) for every j = 1,..., m, then P C (P λ (a k )) P λ (b j ) for j = 1,..., m and P C (P λ (a k )) C {P λ (b j )} for j = 1,..., m.

18 18 CAMILO ARGOTY AND ALEXANDER BERENSTEIN Thus P C (P λ (a k )) C {P λ (b 1 ),..., P λ (b m )} and P C {Pλ (b 1),...,P λ (b (P m)} λ(a k )) = P C(P λ (a k )). Since P (P C {b1,...,b n} λ(a k )) belongs to the closed space H λ, we also get P C {Pλ (b 1),...,P λ (b (P m)} λ(a k )) = P (P C {b1,...,b m} λ(a k )). So P (P C {b1,...,b m} λ(a i )) = P C(P λ (a i )) for any λ σ and then ā b. C Lemma. Let C H be algebraically closed and let ā = (a 1,..., a n ) H n, b = (b1..., b m ) H m be tuples in H. Then ā b C if and only if Pλ (a k ) P C(P λ (a k )) P λ (b j ) for k = 1,..., n, j = 1,..., m and λ σ. Proof. Given two tuples ā = (a 1,..., a n ) and b = (b 1,..., b m ), by Proposition 3.34 ā b C if and only if Pλ (a k ) P C λ(b j ) for k = 1,..., n, j = 1,..., m and λ σ. This happens if and only if P C {Pλ (b (P j)} λ(a k )) = P C(P λ (a k )) for k = 1..., n, j = 1,..., m and λ σ. Finally, P C {Pλ (b (P j)} λ(a k )) = P C(P λ (a k )) if and only if P λ (a k ) P C(P λ (a k )) P λ (b j ) Theorem. The relation satisfies the following properties: finite character, local character, transitivity, symmetry, invariance, existence and stationarity. Proof. We prove all the properties: (1) Finite character: we show that ā C B if and only if ā C B 0 for all finite B 0 B. First of all, if ā C B and B B then ā C B. If ā C B, P B C (a k ) P C(a k ) for some 1 k n. Let b = P B C (a k ) P C(a k ). Then there exist b 1,..., b l B, c 1,..., c m C, λ 1,..., λ l+m σ and α 1,..., α n, β 1,..., β m C such that b l k=1 α kp λk (b k ) m j=1 β jp λl+j (c j ) < b /2. Let B 0 = {b 1,..., b l }, then ā C B 0. (2) Local Character: For every a and B there exists a sequence (d n : n ω) such that d n P B(a) and d n = α 1n P λ1n (b 1n ) + + α knnp λknn (b knn) where b ij B, λ ij σ and α ij C. Let B 0 = {b ij : i j, j ω}. B 0 B is countable and a B 0 B. (3) Transitivity of independence: Let A B C H be small and let ā = (a 1,..., a n ). If ā A C, P C(a k ) = P Ā (a k ) for k = 1,..., n. So, P C(a k ) = P B(a k ) = P Ā (a k ) for k = 1,..., n and therefore ā B C and ā B. The converse is proved in a similar way. A (4) Symmetry: It enough to show that for any tuples ā = (a 1,..., a n ) and b = (b1,..., b m ) and small sets C, ā b C if and only if b ā. Let C

19 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR 19 λ σ, k n and j m, we can write P λ (a k ) = P λ (a k C) + P λ (a k C ) and P λ (b j ) = P λ (b j C) + P λ (b j C ), where a k C = P C(a k ), a k C = P C (a k ). If ā C b by Lemma 3.35 Pλ (a k ) P λ (a k C) P λ (b j ) then P λ (a k C ) P λ (b j C) + P λ (b j C ) = 0. So P λ (a k C ) P λ (b j C) + P λ (a k C ) P λ (b j C ) = 0. By Lemma 3.30 P λ (a k C ) P λ (b j C) = 0, so P λ (a k C ) P λ (b j C ) = 0. On the other hand, by Lemma 3.30 P λ (a k C) P λ (b j C ) = 0, so P λ (a k ) P λ (b j C ) = 0. Thus P λ (a k ) P λ (b j ) P λ (b j C) = 0 and therefore P λ (a k ) P λ (b j ) P λ (P C(b j )). By Lemma 3.35, this implies that b ā which completes the proof C (5) Invariance: For every u, v H and f Aut(H), u v = f(u) f(v). Let ā = (a 1,..., a n ) and b = (b 1,..., b m ). Then ā C b if and only if P λ (a k ) P λ (P C(a k )) P λ (b j ) for every k = 1,..., n, j = 1,..., m and λ σ if and only if P λ (f(a k )) P λ (P f( C) (f(a k ))) P λ (f(b j )). (6) Existence: Let ā = (a 1,..., a n ) H n and let A H be small. By quantifier elimination, the type tp(ā/a) is determined by P Ā (P λ (a k )) for λ σ and the inner products P λ (a k ) P λ (a j ) for k, j n. Let b H n and B A be small. tp( b/b) is a free extension of tp(ā/a) if and only if tp( b/a) = tp(ā/a) and P B(P λ (b k )) = P Ā (P λ (a k )) for all λ σ and k = 1,..., n. For each λ σ, let a λ kā = P Ā (P λ(a k )), a λ kā = PĀ (P λ (a k )). Since B is small, for each λ σ e, dim(h λ B ) =, so we can find d λ k H λ B with a λ = d λ kā k. Let b k = λ σ (aλ kā + dλ k ) and let b = (b1,..., b n ). Then tp(ā/a) = tp( b/a) and P B(P λ (b k )) = P Ā (P λ (a k )) for k = 1,... n. (7) Stationarity: Let ā = (a 1,..., a n ), b = (b 1,..., b n ), b = (b 1,..., b n) H n and let A B H be small. Assume that the types of b and b over B are free extensions of tp(ā/a). Then tp( b/a) = tp( b /A) and for every λ σ and i n, P B(P λ (b i )) = P Ā (P λ (b i )) = P Ā (P λ (b i )) = P B(P λ (b i )). Thus tp( b/b) = tp( b /B). Therefore for every p S(A) {tp(ā/b) ā A B, p tp(ā/b)} = Corollary. The theory T σ is stable and -independence coincides with the notion of independence induced by forking.

20 20 CAMILO ARGOTY AND ALEXANDER BERENSTEIN Observation. We had shown in Theorem 3.20 that T σ is ω-stable and thus it has prime models over sets. Let A H be small. For each λ σ i such that dim(h λ ) = and dim(p λ (Ā)) < ℵ 0, let H λ be a subspace of H λ of dimension ℵ 0. Then A λ σi H λ is the prime model over A of T σ. It also follows from ω-stability that for every infinite cardinal µ, T σ has a µ- saturated model of dimension µ. Indeed, for each λ σ e, let H λ be a subspace of H λ of dimension µ. Then H λ = T σ has dimension µ and it is saturated Definition. Let A H be small and p, q S n (A). We say that p is almost orthogonal to q (p a q) if for all ā = p and b = q ā A b Definition. Let A H and p, q S n (A). We say that p is orthogonal to q (p q) if for all B A, p B p a non-forking extension, and q B q a non-forking extension, p B a q B Theorem. Let A H be such that A = acl(a). Let p, q S 1 (A), let a = p and b = q. Let a = P A (a) + a and b = P A (b) + b, let σ p = {λ σ e : P λ (a ) 0} σ q = {λ σ e : P λ (b ) 0}, then, p q if and only if p a q if and only if σ p σ q =. Proof. Assume that p q, then p a q. Let a = p and b = q, let a = a P A (a) and b = b P A (b). The type tp(a/a) is determined by P A (a) and the norms of P λ (a ) for λ σ e. Assume, in order to get a contradiction, that P λ0 (a ) 0 and P λ0 (b ) 0 for some λ 0 σ e. Let a, b such that tp(a /A) = tp(a /A), tp(b /A) = tp(b /A) and P λ0 (a ) is a multiple P λ0 (b ). Then tp(p A (a) + a /A) = tp(a/a) and tp(p A (b) + b /A) = tp(b/a) but (P A (a) + a ) A (P A(b) + b ), a contradiction to p w q. Conversely, assume that σ p σ q =. Let B A and let p B p, q B q be nonforking extensions. Let c and d realizations of p B and q B respectively. We may write c = c B +c, d = d B +d where c B d B are the projections of c and d over acl(b) = B. Then σ pb = σ p and σ qb = σ q. Then c d = ( λ σ p P λ (a ) µ σ q P µ (b ) = λ σ p µ σ q P λ (c ) P µ (d ) = 0. Then c d and c B d. A generalization of the previous result appears in [6] when σ fin = and A =.

21 HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR 21 References [1] N.I. Akhiezer, I.M. Glazman, Theory of linear operators in Hilbert Space vols.i and II. Pitman Advanced Publishing Program, [2] Itaï Ben Yaacov, On perturbations of continuous structures, submitted. [3] W. Arveson, A short course in spectral theory. Springer Verlag, [4] Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson and Alexander Usvyatsov, Model theory for metric structures, to appear in the Proceedings of the Isaac Newton Institute s semester on Model Theory and its applications. [5] Itaï Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability. submitted. [6] Itaï Ben Yaacov, Alexander Usvyatsov and Moshe Zadka, Generic automorphism of a Hilbert space, preprint. [7] Alexander Berenstein, and Steven Buechler, Simple stable homogeneous expansions of Hilbert spaces. Annals of of pure and Applied Logic. Vol. 128 (2004) pag [8] Steven Buechler, Essential stability theory. Springer Verlag, [9] R. Dautray, L. Lions, Mathematical analysis and numerical methods for science and technology, volume 3. [10] Kenneth Davidson, C -Algebras by example, Field Institute Monographs, [11] José Iovino, Stable theories in functional analysis University of Illinois Ph.D. Thesis, [12] L. Liusternik and V. Sobolev, Elements of Functional Analysis. Frederic Ungar Publishing Co., New York, [13] M. Reed, B. Simon, Methods of modern mathematicalphysics volume I:Functional analysis, revised and enlarged edition. Academic Press, [14] Werner Schmeidler, Linear Operators in Hilbert Space, Academic Press, Camilo Argoty, Universidad de los Andes, Departamento de Matemáticas, Cra 1# 18A-10, Bogotá, Colombia. and, Universidad Sergio Arboleda, Departamento de Matemáticas, address: c-argoty@uniandes.edu.co Alexander Berenstein, Universidad de los Andes, Departamento de Matemáticas, Carrera 1 N 18A-10, Bogotá, Colombia. address: aberenst@uniandes.edu.co