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1. Introduction

Let H be a separable Hilbert space. A countable family of vectors {x_n}_n€I in H is a frame if there are constants O < A ≤ B < ∞ such that for every x € H

In frame theory the most important result is the Decomposition Theorem [8], [9], [10], [11], [12], [18], [13], [15], [21]. It states that if {x_n}_n€I in a frame for H, then every vector x € H can be written as

where the coefficients c_n are computed explicitly by c_n (S^—1x, x_n). Here S^—1 is the inverse of the linear operator Sx=∑_n∊I (x, x_n) x_n. Observe that the elements of the frame are not necessarily orthogonal, therefore frames can be thought of as over-complete bases.

The possible over-completeness of frames makes them more flexible than orthonormal bases, and therefore a powerful tool in several branches of mathematics [8], [9], [11], [12], [18], [13], [15], [21] and physics [2], [3], [4], [12], [13], [20]. For this reason it is natural to consider frame theory in Hilbert spaces with W-metrics and Krein spaces [1], [14], [17], [16], [20].

A Hilbert space with W-metric (or simply W-space) Hw is the completion of (H, (VV. , .), where W* = VV € B(H), and ker W={0} [5], [6]. In [14] frames in Hilbert spaces with W-metrics were introduced and it was showed that the behavior of the frames depends on the boundedness of the linear operator W, and whether or not 0 € spec(W). Independently of this, the Decomposition Theorem for frames in Hilbert spaces with W-metrics still holds. Similar ideas have been developed independently in [17] and [20].

In [9], [19], [18] there are methods to construct frames in a Hilbert space. Our approach follows the results obtained in [9], [19], [18], [14] in order to construct frames in Hilbert spaces with W-metrics. We present some characterization of frames unitarily equivalent in terms of their respective analysis operators. Furthermore we show that if {x_n}_n€N and {Y_n}_n€N are frames for the W-spaces K_G and H_w, respectively, then the coupling of these families is a frame. That is, {X_n}_n€N U {y_n}_n€N is a frame for the Hilbert space with W-metric KW = K_G ⊕ H_W.

We have organized this paper as follows: In Section 2 basic aspects of the Hilbert spaces with W-metrics and frame theory in such spaces are given. In Section 3 we prove the main results of this paper, introduce similar and unitarily equivalent frames and show that two frames are similar if and only if their respective analysis operators have the same range. In addition, we construct frames from a positive and J-adjoint operator (Subsection 3.4). Finally we show that the coupling of two frames is a frame (Subsection 3.3).

2. Preliminares

Let K be a vector space over C. Consider a sesquilinear [. , .] : K X K → C. The vector space (K, [. , .]) is a Krein space if K = K⁺ ⊕ K^- , where (K⁺ , [. , .]), (K^-, - [. , .]) are Hilbert spaces, and K⁺ , K^- are orthogonal with respect to [. , .].

On K, define the following inner product:

This inner product turns (K, (. , .)) into a Hilbert space, which is called the Hilbert space associated to K. Moreover, there exist unique orthogonal projections onto K⁺ and K^-, which are denoted by P⁺ and P^- respectively. The linear bounded operator J = P⁺ — P^- is called Fundamental Symmetry. It satisfies

The Hilbert space (K, [. , .]j) is used to study linear operators acting on Krein space (K, [. , .]). Topological concepts such as continuity, closedness of operators, spectral theory and so on, refer to the topology induced by the J-norm given in (1). Therefore, we may apply the same definitions as in the operator theory of Hilbert spaces. For instance, the adjoint of an operator T in Krein spaces (T^[*]) satisfies [T (x), y] = [x, T ^[*] (y)], but we must take into account that T also has an adjoint operator in the Hilbert space (K, [. , .] J), denoted by T^*J where J is the fundamental symmetry in K. The relation between T^*J and T^[*] is T^[*] = JT^*JJ. Moreover, let K and K be Krein spaces with fundamental symmetries JK and JK, respectively. If T € B(K, K), then T^[*] J_KT^*JKJK. An operator T € B(K) is said to be self-adjoint if T T^[*], and J-self-adjoint if T T^[*] Furthermore, a linear operator T is said to be invertible if its range and domain are the whole space.

Definition 2.1 Let V be a closed subspace of K. The subspace

Is called the J-orthogonal complement of V with respect [. , .] (or simply J-orthogonal complement of V).

Definition 2.2 A closed subspace V of K such that V ∩ V^[⊥] = {0} and V + V^[⊥] = K, where V^[⊥] is given in (2). Is called projectively complete.

Proposition 2.3 ([6, Theorem 7.16]). Let (K, [. , .]) be a Krein space and let V be a closed subspace of K. The following statements are equivalent:

• i) V is projectively complete.

  ii) V is a Krein space.

  iii) Any vector y ε K has at least one J-orthogonal projection onto V.

For more details on Krein spaces, see [6],[7].

2.1. Hilbert spaces with W-metrics

Definition 2.4. Let H be a separable Hilbert space with scalar product (. , .) and induced norm || . || = √(. , .). Consider an operator W = W* є B(H) with ker W = {0}. The sesquilinear form Hw

Defined on H is called W-metric, or W-inner product, and the operator W is called Gram operator.

Remark 2.5 (Consequences induced from the Gram operator), Let W be a Gram operator on H.

• i). If 0 ∉ spec (W), then

  ||w-1||-1||x||2 ≤||x||2J ≤||w|| ||x||2, ⋁x € H (4)

  Therefore,

  Hw:= (H, [. , .]J)||.||J = (H, [. , .]J). (5)

• ii). If 0 ∈spec (W), then

  ||x||J ≤ √||W|| ||x||, ⋁x € H (6)

  In this case,

  Hw := H||.||J ≠ H. (7)

Definition 2.6. Let (H, (. , .)) be a separable Hilbert space. The Krein space HW is said to be regular if the Gram operator W is such that 0 ∈ p(W). Otherwise, it is said to be singular.

In what follows all the Hilbert spaces are separable. For more details about the regular and singular Krein spaces see [5].

Remark 2.7. Consider the polar decomposition of W,

Where the linear operator J : (ker |W|)^⊥ = H Rang |W| = H → Rang W = H is a partial isometry. However, ker J = {0}, thus J = J* is a unitary operator.

Definition 2.8. The space HW is the Krein space (H^||.||J, [. , .]) with the sesquilinear form

And the J-norm ||.||J given by the scalar product

Where J is the symmetry of Hilbert space H defined by W = J|W|.

Proposition 2.9. If HW_i is a family of Hilbert spaces with W-metrics indexed by i є I, then the direct sum of the HW_i, denoted

Is a Hilbert space with W-metric, where the Gram operator W is given by

Proof. It is well known if H_i is a family of Hilbert spaces indexed by i є I, then the direct sum

Is a Hilbert space with scalar product given by (. , .) ҡ = ∑_iєI(. , .) H_i. Thus, if W = ⊕_iЄI W_i, then W-metric

Is well defined and bounded. Therefore, the completion ҡW of (ҡ, (W . , .)) is a Hilbert space with W-metric and ҡw = ⊕iЄI HWi.

2.2 ℓ₂(N) as a Hilbert space with W-metric

On ℓ₂ (N) consider the following sesquilinear form [. , .] ℓ₂(N):

where (a_n)_n€N is a sequence of non zero real numbers belonging to c₀(N). Let {δ_n}_n€N be such that δ_n,m = 1 if m = n, and δ_n,m = 0 for m ≠ n. It is well known that {δ_n}_n€N is a orthonormal basis of ℓ₂ (N), and that the diagonal operator ϐ:ℓ₂(N) → ℓ₂(N) given by

Is bounded, self-adjoint and has the spectrum

Thus, the sesquilinear form [. , .] ℓ₂(N) is a W-metric on ℓ₂(N) and satisfies

Since ϐ = J|ϐ|, where Jδn = sign (a_n)δ_n for each n € N, we have by Definition 2.8 that

[z, w]J = (|ϐ|)z, w, z,w € ℓ₂(N)

Defines a scalar product on ℓ₂(N). Hence, t₂,w (N) :=( ℓ₂(N)), (ϐ . , .) ^||.||J is a Hilbert space with W-metric with Gram operator ϐ. Up to unitary equivalence, we may write

Remark 2.10. Since a_n n → ∞ 0, one gets 0 € spec (ϐ) and ℓ₂(N) ⊊ t_2,w (N) by Theorem 4.2 in [14].

Proposition 2.11. Let t_2.w (N) be the Hilbert space with W-metric with Gram operator ϐ given in [14]. Then U := √|ϐ| is a unitary operator U from the Hilbert space (t_2,w (N), [. , .]_J) onto (ℓ₂ (N), (. , .)).

Proof. Let U = √|ϐ| : t_2,w (N) →ℓ₂ (N), where the operator ϐ is given in (13). Hence U is an isometry, since

Moreover, the inverse operator U^-1 = (√|ϐ|)^-1 : ℓ₂ (N) → t_2,w (N) is well defined, In fact, since {a_n}_N €N is not identically zero and belongs to C₀(N), one gets

2.3. Frames in Hilbert spaces with W-metrics

The next results are based on the frame theory in Krein spaces studied in [14].

Definition 2.12. Let Hw be a Hilbert space with W-metric. A countable sequence {X_n}_n€n ⊂ Hw is called a frame for Hw, if there exist constants 0 < A ≤ B < ∞ such that

Since we are mostly interested in infinite-dimensional spaces, and since one can always fill up a finite frame with zero elements, we assume that n=N whenever the finiteness of n is of no importance

Proposition 2.13 ([4]). Let Hw be a Hilbert space with W-metric. The family {Xn}n€n is a frame for Hw, if and only if {x_n}_n€n is a frame for the associated Hilbert space (Hw, [. , .]J).

Definition 2.14. Let Hw be a Hilbert space with W-metric and t_2,w (N) be the Hilbert space with W-metric (ϐ . , .) given in (14). If {x_n}_n€n is a frame for Hw, the linear map

Is called pre-frame operator.

Remark 2.15. Let {x_n}_n€n be a frame for a Hilbert space Hw_(⋀) with a W-metric.

Then, for every {Y_n}_n€n € t_2,w (N) we have

Thus, ||T|| ≤ √B.

The following result provides a criterion for a family {x_n}_n€n to be a frame for Hw (see [14]) and is an analogue to the Hilbert case.

Proposition 2.16 ([14]). Let Hw be a Hilbert space with W-metric. The family {x_n}_n€n is a frame for HW, if and only if T is well defined (i.e., bounded) and surjective.

Remark 2.17. It is easily seen that for all (y_n)_n€_n € t_2,w(N)

Thus, the adjoint operator T^[*] of T is given by

Definition 2.18. ([14]). Let Hw be a Hilbert space with W-metric, (t_2,w (N), (ϐ . , .)) be a Hilbert space with W-metric given in (14), and {k_n}_n€n ⊂ Hw a frame for Hw. The operator.

Is called frame operator.

Let {x_n}_n€n be a frame for Hw. The linear operator T₀ : ℓ(N) → Hw given by the rule

is well defined, and its adjoint operator is T₀^[*]k = ([k, x_n])_n€n. Observe that the preframe operator T given in (18) is such that

It follows immediately from (21), (22), and J² = UU^-1 = id_ℓ2(N) that

Furthermore, S is self-adjoint and invertible. If we put a_n = 1 for each n € N in (12), then

(t_2,w (N),[. , .]) = (ℓ₂(N), (. , .)), and the frame operator is S — T T* , exactly as in the Hilbert space case.

Proposition 2.19. Let Hw be a Hilbert space with W-metric. If {x_n}_n€n is a frame for Hw, then the image of the analysis operator, denoted by Rang T^[*], is a Krein subspace of t_2,w (N).

Proof. Since {x_n}_n€n is a frame for Hw_(⋀), there is A > such that

Thus, A||k||_J ≤ ||T^[*]k||J for each k € Hw; therefore, Rang T^[*] is a closed subspace of t_2,w (N). On the other hand, if x € Rang T^[*] ⋂ (Rang T^[*])^[⊥], then [x,x]t_2,w(N) =0.

Since x = T^[*] y for some y € Hw, we have that y € ker T[*], and by injectivity of T[*] (Proposition 2.16), one gets that x = 0. Hence, Rang T^[*] is a Krein subspace of t_2,w (N).

3. Main results

3.1. Similar frames Hilbert spaces with W-metrics

Definition 3.1. Let (Hw, [. , .]_H) and (K_G, [. , .]K) be Hilbert spaces with W-metrics and Gram operators VV and G, respectively. Two frames {x_n}_n€n and {y_n}_n€n for Hw and K_G, respectively, are said to be similar if there exists an invertible operator u : Hw → K_G such that U_yn = x_n for n € n. The frames are called unitarily equivalent if U is a unitary operator from Hw onto K_G.

In [14] it was proven that the behavior of frames in Hilbert spaces with W-metrics depends on whether the Gram operator is bounded or not. However, for any unitary operator U: H → Hw, {x_n}_n€n is a frame for H if and only if (Ux_n)n€n is a frame for Hw.

Remark 3.2. Note that H and Hw are unitarily equivalent, since both are separable.

The following result provides a criterion for a couple of frames to be similar. It is an analogue to Deguang and Larson's result [19] for Hilbert spaces.

Theorem 3.3. Let {x_n}_n€n, and {Y_n}_n€n be frames for Hilbert spaces with W-metrics (Hw, [. , .]_H) and (K_G, [. , .]K), respectively. Let T^[*]HW and T_KG^[*]KG be the analysis operators for {xn}n€n and (y_n)_n€m, respectively. The frames {x_n}_n€n and {Y_n}_n€n are similar if and only if T_Hw^[*]HW and T_KG^[*]KG have the same range.

Proof. Let T_Hw^[*]HW and T_KG^[*]KG be the analysis operator for {x_n}_n€n and {Y_n}_n€n, respectively. Suppose that {x_n}_n€n and {Y_n}_n€n are similar; then, there is an invertible operator U : Hw → K_G such that U_xn = Y_n. Hence,

Now, since U^[*] is invertible, we concluded that T_KG^[*]KG (KG) =T_HW^[*]HWU^[*] (KG). Thus, Rang T_KG^[*]KG = Rang T_Hw^[*]HW.

Conversely, supone that Rang T_KG^[*]KG = Rang T_HW^[*]HW =V. Ley U be the unitary operator in (16). Note that T_HW|UJV Y T_KG|UJV are invertible. Thus, the operator

Is well defined and invertible. By Proposition 2.19 the closed subspace V is projectively complete, so there exists a unique J-orthonal projection P from t_2,w(N) on V. If yЄ V^(⊥)t2,w(N), then for every z ∊KG and T_KG^[*]KG Z ∊ V, we got

Since y was arbitrary, we conclude that T_KG (V^[⊥]t2,w(N)) = {0}. Analogously, we have T_HW (V^[⊥]t2,w(N)) = {0}. Now, consider {V_m}_mЄn, where V_m,j =δ_m,j. By the definition of pre-frame operator (18), we get T_HW PV_m =T_HWV_m = ∑_jЄnδ_m,Jx_m. On the other hand, it also satisfies that y_m = T_KGV_m = T_KG P_vm. Hence,

3.2. Construction of frames

Proposition 3.4 (Construction of frames with an operator on finite-dimensional Krein spaces). Let (K, [. , .]) be a N -dimensional Krein space with fundamental symmetry J, and S₀ be a J-self-adjoint positive operator with respect to [. , .]. Let λ₁ ≥ λ₂ ≥ . . . ≥λ_N > 0 be the eigenvalues of S₀. Fix k ≥ N and real numbers a₁ ≥ a₂ ≥ . . . ≥ a_K > 0. The following statements are equivalent:

• (i) For every 1 ≤ j ≤ N,

  ∑^j_n=1 a²n ≤ ∑^j_n=1 λn, ∑^k_n=1 a²n ≤ ∑^N_n=1 λn. (24)

  (ii) There is a frame {x_n}^k_n=1 for the Hilbert space (K, [. , .]J) with frame operator S₀ and ||x_n||_J = a_n, for all n = 1, …, k.

  (iii) There is a frame {x_n}^k_n=1 for the krein space K with frame operator S₀J and ||x_n||_J = a_n, for all n = 1,…, k.

  (iv) There is a frame {Jx_n}^k_n=1 for the Krein space K with frame operator JS₀ and ||X_n||_j = a_n, for all n = 1,…, k.

  (v) There is a frame {Jx_n}^k_n=1 for the Hilbert space (K, [. , .]J) with frame operator JS₀J and ||x_n||_J = a_n, for all n = 1, …, k.

Proof. The equivalence between (i) and (ii) is proved in [9]. By Proposition 2.13 and the remarks following Theorem 3.6 in [14], we have the equivalences (ii) ↔ (iii) ↔ (iv) ↔ (v). Now we only need to find the relationship of the respective frame operators with the operator S₀.

Since S₀ is the frame operator corresponding to the family {x_n}^k_n=1, we have for every x Є K

Therefore, S₁:= S₀J is the frame operator for the frame {x_n}^k_n=1 in the Krein space K.

Next, for every x € K, one gets

Thus, S₂ := JS₀ is the frame operator for the frame {Jx_n}^k_n=1 in the Krein space K.

Finally, for every x € K we have that

Hence, S₃ := JS₀J is the frame operator for the frame {Jx_n}^k_n=1 n in the Hilbert space (K, [. , .]_J).

Consider a Hilbert space (H, (. , .)), with a frame {x_n}n€n and frame bounds 0 < A ≤ B.

The operator frame S : H → H given by

is bounded, invertible and A Id ≤ S ≤ B Id (see for example [10]). Therefore, the Hilbert space with W-metric Hw = (H, ll.llJ) is a regular Krein space with frame {x_n}_n€n, where the norm ll.ll J is given by ll.llJ √(S.,.) and satisfies (4).

Now we pose the following problem: find a Hilbert space with W-metric Hw such that H ⊊ Hw, and construct a frame {y_n}_n€n for Hw unitarily equivalent or similar to (x_n)_n€m. The next lemma solves the problem.

Lemma 3.5. Let (H, (. , .)) be a Hilbert space and {e_n}_n€n be a orthonormal basis for H.

If ϐ is the linear operator defined in (14), then the linear operator Wϐ: H → H given by

is well defined, bounded, Wϐ* = Wϐ, ker Wϐ = {0} and 0 € spec (Wϐ).

Proof. Let x = ∑n€n^xnen. Then x € ker (Wϐ) and only if a e ∑n€n^anxnen =0. Therefore a_nx_n = 0 for all n € N, i.e., x_n = 0 for all n € N. Thus, x 0. Moreover 0 € spec (Wϐ), since spec (Wϐ) = {a_n : n €N} and a_n n→∞ 0.

Proposition 3.6. If (H, (. , .)) is a Hilbert space and {x_n}_n€n is a frame for H, then there exist a Hilbert space with W-metric Hw such that H ⊊ Hw, and a frame {y_n}_n€n for Hw which is unitarily equivalent to {x_n}_n€n.

Proof. Let (H, (. , .)), be a Hilbert space and {x_n}_n€n be a frame for H. If {e_n}_n€n is a orthonormal basis for H, then by Lemma 3.5 the operator W_ϐ given in (25) is well defined, bounded, Wϐ* = {0}. Thus, we may consider Hw, the

Hilbert space with W-metric with Gram operator W_ϐ. Now H ⊊ Hw and this is an equality if and only if 0 ∉spec (Wϐ) (see [14, Theorem 4.2]). On the other hand, the existence of a frame unitarily equivalent to {x_n}_n€n follows from the facts that H and Hw are unitarily equivalent (since both are separable), and that the unitary operators transfer frames into frames with the same frame bounds.

3.3. The coupling of frames

Proposition 3.7. Let K_G and Hw be Hilbert spaces with W-metrics, and {Y_n}_n€n, be frames for K_G, H_w, respectively.

Denote by S_KG, S_Hw their corresponding frame operators. Then {x_n}_n€n ⋃ {Y_n}_n€n is a frame for K_W := K_G ⊕ Hw , and the corresponding frame operator is given by

Proof. Set K_w = K_G ⊕ Hw. By Proposition 2.9, K_G is a Hilbert space with W-metric, where the Gram operator is given by W = G ⊕ W. Now, consider the pre-frame operators

as defined in (18). Let V:t^2,w3 (N) →t₂,w₁(N) ⊕ t₂,w₂ (N) be any unitary isometry, i.e., VV^*J = Id_t2,w3(N) and V^*J V = Id_{t2,w1 (N)⊕ t2,w2 (N)}. Then

is bounded and surjective, hence it is a pre-frame operator for the frame {x_n}_n€n ⋃{Y_n}_n€n for K_w = K_G ⊕ H_w. Then, by Definition 2.18,

Acknowledgements

The second author is grateful to the coauthors, to Ricardo Cedeño (R.I.P.) and to the Universidad Surcolombiana for their hospitality during his stay in the city of Neiva. The authors wish to express his gratitude to the Universidad de Sucre. The work on the paper was partially supported by the project Extensión de Frames en espacios de métrica indefinida. We also thank an anonymous referee for constructive comments, recommendations and useful suggestions.

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