Frames associated with an operator in spaces with an indeﬁnite metric

: In the present paper, we study frames associated with an operator ( W -frames) in Krein spaces, and we give the deﬁnition of frames associated with an operator depending on the adjoint of the operator in the Krein space (Deﬁnition 4.1). We prove that the deﬁnition given in [A. Mohammed, K. Samir, N. Bounader, K-frames for Krein spaces, Ann. Funct. Anal. , 14 (2023), 10.], which depends on the adjoint of the operator in the associated Hilbert space, is a consequence of our deﬁnition. We prove that our deﬁnition is independent of the fundamental decomposition (Theorem 4.1) and that having W -frames for the Krein space necessarily gives W -frames for the Hilbert spaces that compose the Krein space (Theorem 4.4). We also prove that orthogonal projectors generate new operators with their respective frames (Theorem 4.2). We prove an equivalence theorem for W -frames (Theorem 4.3), without depending on the fundamental symmetry as usually given in Hilbert spaces.


Introduction
The frame theory for Hilbert spaces has its origin in [7] and was developed by I. Daubechies in [4,5]. Frames can be considered as "overcomplete bases", and their overcompleteness makes them more flexible than orthonormal bases. They have proven to be a powerful tool, for example, in signal processing and wavelet analysis [10].
In [8] a definition of frames for Krein spaces was established by replacing the positive definite inner product in the definition of a frame for a Hilbert space by an indefinite inner product, and it is shown that the theory of frames for Krein spaces and the theory of frames for associated Hilbert spaces are analogous. GȃvruÅ£a in [9] defined K-frames in Hilbert spaces as a generalization of frames, which allows one to precisely reconstruct the images of a bounded linear operator on a Hilbert space. In [11] Mohammed, Samir and Bounader defined K-frames in Krein spaces using the adjoint of the operator on the Hilbert space associated with the Krein space and presented an equivalence result for K-frames depending on the fundamental symmetry ( [11], Proposition 3.14).
In this paper, we give a definition (Definition 4.1) of W-frames in Krein spaces which does not depend directly on the adjoint of the operator in the associated Hilbert space. Instead, it depends on the adjoint of the operator on the Krein space, and we prove that the definition given in [11] is a consequence of ours. Following Wagner, Ferrer and Esmeral in [8], we prove that the definition given in this investigation is independent of the fundamental decomposition and that having W-frames for the Krein space necessarily gives W-frames for the Hilbert spaces that compose this space. We also prove that the orthogonal projectors generate new operators with their respective associated frames.
Note that if we have another fundamental decomposition, then we will have another fundamental symmetry and consequently another J-inner product.
Definition 2.4. [1,2] The fundamental symmetry J associated with Krein space induces a norm in K defined by and this norm is called the J-norm of K. Explicitly, From now on, the topology studied in Krein spaces will be directly related to the J-norm of K.
) be a Krein space and let be two fundamental decompositions. If J 1 and J 2 are the respective fundamental symmetries, it follows that · J 1 and · J 2 are equivalent norms.  Example 2.1.
[8] Now, 2 (N) can also be seen as a Krein space with an inner product whose inner J-product coincides with the usual one. In this sense we define the following mapping: Thus, if {e n } n∈N is the canonical orthonormal basis of 2 (N), then 2 (N) accepts the following fundamental decomposition: where + 2 (N) = span{e 2n : n ∈ N} and − 2 (N) = span{e 2n+1 : n ∈ N} with associated fundamental symmetry From now on whenever we see 2 (N) as Krein space we will understand that it is endowed with a fundamental symmetry J 2 such that [·, ·] J 2 = ·, · 2 . An example of such is the one developed above, and more trivial is the symmetry given by the identity operator on 2 (N). Thus we will write £ 2 (N) instead of 2 (N) when viewed as Krein space with such properties and the fundamental symmetry by J £ 2 , to avoid confusion.

Frames in indefinite metric spaces
The following results were established in [8] for Wagner, Ferrer and Esmeral.
be Krein spaces, such that [·, ·] J £ 2 coincides with the standard inner product ·, · defined in 2 (N). Given a frame {x n } n∈N for K, the linear mapping T : is called a pre-frame operator.
Remark 3.1. The adjoint of T is given by In fact, for all {α n } n∈N ∈ £ 2 (N) and k ∈ K, we have be Krein spaces, so that [·, ·] J £ 2 coincides with the standard inner product ·, · defined in 2 (N), and {x n } n∈N ⊂ K is a frame for K. The operator is called the frame operator. Then, Also, The definition of K-frames given in [11], which is an adaptation of the definition of frames given in [8], was presented apparently depending on the fundamental symmetry. We will show below that the W-frames according to the definition given in this paper are independent of the fundamental decomposition of the Krein space in question.
K − 2 and fundamental symmetries J 1 , J 2 , respectively, and W : K → K is a bounded operator. If {x n } n∈N is a frame for W with respect to J 1 , then {x n } n∈N is a frame for W with respect to J 2 .
The following result was presented in [11] with the restriction on the images of a sequence, under the fundamental symmetry.
In this paper we present and show a result where it is observed that such a restriction is not necessary. The result holds as usual in the Hilbert spaces for any sequence of Krein space. It remains to prove that there exists Since J is an isometry in the Hilbert space (K, [·, ·] J ), we have x, Wy = sup ii) → i) Suppose that {x n } n∈N is a W−frame for (K = K +[ +]K − , [·, ·], J), and then there exist constants A, B > 0 such that From the above inequality we have that In [8] the authors showed that the operator T : £ 2 (N) → K given by T ({a n } n∈N ) = n∈N a n x n , is well defined and bounded, and also T J ≤ √ B. Since W and T are bounded operators, and T is an epimorphism (see [3]), R(W) ⊂ R(T ) = K. By Theorem 2.1 there exists the bounded linear operator M : (K, [·, ·] J ) → 2 (N) such that W = T M.
We consider F n : (K, [·, ·] J ) → C, F n (x) = (Mx) n = a x n . Since Mx ∈ 2 (N), and we write (Mx) n to indicate the terms of the sequence Mx.
We define a x = Mx. We have Therefore, for each n ∈ N, F n : K → C are continuous linear functionals. From the Riesz representation theorem for Krein spaces (see [1]), it follows that there exists {y n } n∈N ⊂ K such that a x n = F n (x) = [x, y n ] for all x ∈ K. It remains to prove that {y n } n∈N ⊂ K is a Bessel sequence. In effect, n∈N x, y n 2 = n∈N |a x n | 2 = a x 2 2 ≤ M 2 x 2 J , and therefore, {y n } n∈N is a Bessel sequence for (K = K +[ +]K − , [·, ·], J).
As an application of the previous theorem, using the fundamental projectors below, we obtain frames associated with these projectors for the subspaces that compose the Krein space.

Conclusions
The W-frames in Krein spaces are well defined, and they are a generalization of the K-frames in Hilbert spaces introduced by GȃvruÅ£a in [9]. The W-frames are independent of the decomposition of the Krein space. By having W-frames for a Krein space one necessarily has W-frames for the Hilbert spaces that compose the Krein space, and the orthogonal projectors project W-frames on WP-frames.