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Continuous frames in Krein spaces

The purpose of this paper is to propose a definition of continuous frames of rank n for Krein spaces and to study their basic properties. Similarly to the Hilbert space case, continuous frames are characterized by the analysis, the pre-frame and the frame operator, where the latter gives rise to a frame decomposition theorem. The paper includes a discussion of similar, dual and Parseval frames and of reproducing kernels. In addition, the importance of the fundamental symmetry in the formula for the frame operator in a Krein space is clarified. As prime examples, it is shown how to transfer continuous frames for Hilbert spaces to Krein spaces arising from a possibly non-regular Gram operator.


Introduction
A generalization of the concept of frame in a Hilbert space to a family indexed by a locally compact space with a Radon measure was introduced by G. Kaiser [29] and, independently, by Ali, Antoine and Gazeau [2]. Such frames are known as continuous frames in distinction from usual frames that are given by a countable spanning set. Continuous frames appear under different names in the literature. For example, Gabardo and Han [25] call them frames associated to measure spaces, and Askari-Hemmat, Dehghan, and Radjabalipour [5] call them generalized frames. Moreover, most of the (vector) coherent states in mathematical physics are actually described by the continuous frames [1,35]. For more details, see e.g. [2,5,9,12,13,15,23,25,38].
The possibility of being over-complete makes frames more flexible than orthonormal bases and, for this reason, a powerful tool in signal processing, electrical engineering and several branches of mathematics [10,11,14,16,17,18,19,24,26,36], as well as in physics [1,2,3,4,17,19,34]. On the other hand, Krein spaces arise naturally in mathematics, for instance in relation to signed measure spaces, and in mathematical physics, for instance in quantum field theory. It is therefore natural to extend frame theory for Hilbert spaces to Krein spaces [21,27,31,34].
Discrete frames in a Krein space were introduced simultaneously by Giribet, Maestripieri, Martínez Pería and Massey [27], and Esmeral, Ferrer and Wagner [21], albeit with slightly different definitions. The definition in [27] manifests directly the independence of a J-norm, whereas in [21] frames in a Krein space are essentially the same objects as frames for the associated Hilbert space although the property of being a frame also does not depend on the J-norm. Other approaches can be found in [30,31,33,34].
The present paper introduces continuous frames of rank n for Krein spaces by weak integrals (Definition 3.4) following the Hilbert space approach developed in [2]. As in [21], continuous frames in Krein spaces are essentially the same objects as continuous frames in the associated Hilbert space (Theorem 3.5), and are characterized by the analysis, the pre-frame and the frame operator (see Section 3.2). The most important result in frame theory is the decomposition theorem [10,11,16,17,18,19,26,32,36], so it is clearly desirable to extend this result to the continuous case. This will be achieved in Theorem 3.6. In Section 3.3, we discuss the importance of the fundamental symmetry for the invertibility of the frame operator. A treatment of reproducing kernels will be presented in Section 3.4. Unfortunately, we cannot speak about reproducing kernel Krein spaces unless the image of the analysis operator is ortho-complemented. Sections 3.5 and 3.6 are devoted to the study of similar, dual and Parseval frames, where we also comment briefly on the notion of coherent state. In the final section, it is shown how to transfer continuous frames in a Hilbert space to an associated Krein space arising from a possibly non-regular and unbounded Gram operator W. This will be realized by extending the square root √ |W| to a J-unitary operator. The motivation for using √ |W| stems from the fact that a frame in the Hilbert space can never be a frame for the associated Krein space if the Gram operator is unbounded or if 0 belongs to its spectrum.

Preliminaries
The purpose of this section is to fix notations and to recall the basic elements of frame theory. For more details on Krein spaces, we refer the reader to [7] and [8]. A comprehensive introduction to frame theory can be found in [12].
2.1. Krein spaces. Throughout this paper, (K, [·, ·]) denotes a Krein space with fundamental decomposition K + [∔] K − and fundamental symmetry J given by such that the J-inner product turns (K, [·, ·] J ) into a Hilbert space. The positive definite inner product [·, ·] J defines a topology on K by the J-norm and K + [∔] K − becomes the orthogonal sum of Hilbert spaces. Note that J 2 = 1 by (2.1). 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 (2.3). Therefore, we may apply the same definitions as in operator theory of Hilbert spaces. The unique adjoint is always taken with respect to the specified inner products, i.e., Furthermore, the linear operator A similar distinction will be made between unitary and J-unitary operators. By a J-orthogonal projection we mean a J-self-adjoint idempotent.
The bounded operator A is called weak integral of F and will be written as The next proposition is an immediate consequence of Definition 3.2. Note that the inte- ii) For all linear operators T ∈ B(K 0 , K 1 ) and S ∈ B(K 2 , K 3 ), 3.2. Continuous frames of rank n. From now on, we will work with a signed measure space (M, B, ν) as in Example 1. Our paper is based on the following fundamental definition.
x are weakly measurable for each i = 1, 2, . . . , n and if there exist positive constants 0 < a ≤ b, called frame bounds, such that In case a = b, the frame is said to be tight, and it is called a Parseval frame, if a = b = 1.

Remark 3.
Clearly, the J-norm of a Krein space depends on the fundamental decomposition. At the first glance, it seems that our definition of continuous frames of rank n ∈ N also depends on the chosen fundamental decomposition. However, two J-norms corresponding to different fundamental decompositions of a Krein space are equivalent, see e.g. [7, §7, Theorem 7.19]. Therefore, only the frame bounds may change, but the property of being a continuous frame in a Krein space is independent from the fundamental decomposition.
Next we state the analogue of Theorem 3.3 in [21] for continuous frames of rank n. This result shows that continuous frames for a Krein space are essentially the same objects as continuous frames for the associated Hilbert space. The proof differs from that in [21] only in notation and will be omitted. More information on the proof can be found in the remarks below Proposition 3.11. x , η 2 x , . . . , η n x } x∈M ⊂ K, the following statements are equivalent: Example 2 (Wavelets). Let (L 2 (R), ·, · ) be the Hilbert space consisting of all measurable functions that are square integrable with respect to the Lebesgue measure of R, where ·, · denotes the usual inner product given by f, g = R f (t) g(t) dt. On L 2 (R), we consider the following sesquilinear form and fundamental symmetry: .
Our next aim is to define the analogue of the frame operator for continuous frames. Let {η 1 x , η 2 x , . . . , η n x } x∈M be a continuous frame of rank n ∈ N for the Krein space (K, [·, ·]) with frame bounds 0 < a ≤ b and with respect to the measure space (M, B, ν). Combining Definition 3.1, Remark 1 and Definition 3.4 shows that the function of rank 1 operators Consider the the sesquilinear form Ψ : K × K −→ C given by By the Cauchy-Schwarz inequality, we have |Ψ( Thus, the Riesz representation theorem ensures that there exists a unique bounded linear operator S : where j L 2 := χ M + − χ M − denotes the Radon-Nikodym derivative of ν with respect to |ν|, it follows from Definition 3.2 that is the weak integral of the weakly measurable function with respect to ν. We call the weak integral S in (3.3) the frame operator and Sk the frame transform of k ∈ K. From (3.2) and (3.3), it follows that As a consequence, S is uniformly positive and therefore invertible. Moreover, the inverse operator S −1 , regarded as an operator on the Hilbert space (K, [·, ·] J ), satisfies where the second relation follows from the first by applying J 2 = 1.
To describe the frame operator in terms of an analogue of the so-called analysis operator, let where J L 2 denotes the fundamental symmetry of the Krein space (L 2 (M, ν), [·, ·]) as described in (2.8). The associated Hilbert space has the inner product (3.6) [ It follows immediately from (3.2) that T is well defined and bounded. In analogy to the Hilbert space case [1,2,6], we refer to T as the analysis operator, to Tk as the the analysis transform of k ∈ K, and to k i (x) := [η i x , k] as the i-th analysis transform of k. Straightforward calculations show that its adjoint operator, the so-called pre-frame operator, is given by the weak integral With the bounded operators T * i : The most important result in frame theory is the frame decomposition theorem. For continuous frames, it states that each vector in the Krein (Hilbert) space can be represented by a weak integral obtained from the continuous frame and the inverse of the frame operator. An equivalent way of stating the frame decomposition theorem is that this weak integral yields the identity operator. x , η 2 x , . . . , η n x } x∈M be a continuous frame of rank n ∈ N for K with respect to (M, B, ν). Then Proof. It has been observed in (3.5) that S has a bounded inverse. Clearly, by (3.9), we have S * = S and thus S −1 * = S −1 . From (3.3) and Proposition 3.3.ii), it follows that for all k ∈ K. Now the definition of the weak integral gives the result.
By Theorem 3.5, each continuous frame {η 1 x , η 2 x , . . . , η n x } x∈M of rank n for (K, [·, ·]) with respect to (M, B, ν) gives rise to three other frames with slightly different frame operators. In the following, we will relate these frame operators to S presented in (3.3). First, consider the frame {Jη 1 x , Jη 2 x , . . . , Jη n x } x∈M for (K, [·, ·]). Denoting the corresponding frame operator by S 0 , we get from (3.3) Comparing so that S 2 = J S. The next two propositions show that continuous frames can be characterized by properties of the analysis and pre-frame operator. By Theorem 3.5, the proofs are literally the same as in the Hilbert space case, see e.g. [29]. In practice, it is not necessary to verify the frame condition (3.2) for all k ∈ K, it suffices to show (3.2) on a dense subspace. This will be proven in the next proposition. x , η 2 x , . . . , η n x } x∈M is a continuous frame of rank n ∈ N for K with respect to (M, B, ν).
We close this section by showing that any continuous frame of rank n can be reduced to a continuous frame of rank 1 with respect to a suitable the measure space.  ((i, x)) := η i x . Since the functions M ∋ x → η i x ∈ K, i = 1, . . . , n, are weakly measurable, it follows that σ is weakly measurable. Note that |ν n | = µ n ⊗ |ν|. Now, by Tonelli's theorem, we have for all k ∈ K. From this, we conclude that {σ(y)} y∈M n is a continuous frame of rank 1 for K with respect to (M n , B n , ν n ) admitting the same frame bounds as {η 1 x , η 2 x , . . . , η n x } x∈M . 3.3. The role of fundamental symmetry in the formula of frame operator. Given a continuous frame {η 1 x , η 2 x , . . . , η n x } x∈M of rank n for the Krein space (K, [·, ·]), let T and T * denote the analysis operator and pre-frame operator defined in (3.7) and (3.8), respectively. In this section, we will discuss the importance of the fundamental symmetry J ⊕ L 2 in the formula (3.9) of the frame operator: S = T * J ⊕ L 2 T. For comparison, consider the bounded linear operator S : K → K given by S := T * T. As the frame operator S, it is obviously self-adjoint in the Krein space (K, [·, ·]). Furthermore, by (3.7) and (3.8), (3.15) [ for all h, k ∈ K. Hence S coincides with the weak integral The operators S and S have different properties, for instance, S is always invertible, but S may not be and therefore cannot be used in the frame decomposition theorem. We illustrate this in the following example.
3.4. Reproducing kernels. Let {η 1 x , η 2 x , ..., η n x } x∈M be a continuous frame of rank n for the Krein space (K, [·, ·]) with frame bounds a ≤ b. Then, by Theorem (3.5), we may view {η 1 x , η 2 x , ..., η n x } x∈M as a continuous frame in the Hilbert space (K, [·, ·] J ) with the same frame bounds. Considering the analysis operator T defined in (3.7) as a Hilbert space operator , it follows from (3.14) that T is bounded, injective, and Rang T is closed. In particular, T(K) := Rang T with the inner product [·, ·] ⊕ J ⊕ L 2 is a Hilbert space by itself. As shown in [1] and [3], is actually a reproducing kernel Hilbert space. To see this, note that where • denotes matrix multiplication. Moreover, since As a consequence, the evaluation map T(K) ∋ Φ → φ i (x) ∈ C is continuous for all x ∈ M and i = 1, . . . , n. Moreover, Equations (3.7), (3.16) and (3.17) show that this evaluation map can be given by the following inner product: Since we are working in the Krein space setting, it would be desirable to have a Krein space analogue reproducing kernel Hilbert space. Clearly, (3.18) can be written ] ⊕ , and then (3.16) becomes (3.20) Φ However, although T(K) is always a closed subspace of n i=1 L 2 (M, ν) , [·, ·] ⊕ , the restriction of [·, ·] ⊕ to T(K) does not necessarily yield a Krein space. For instance, in Example 3, we have [Th, Tk] = [h, Sk] = 0 for all h, k ∈ K so that T(K) ∩ T(K) ⊥ = T(K). Therefore, by Proposition (2.1), we may view T(K), [·, ·] ⊕ as a reproducing kernel Krein space only if any element in n i=1 L 2 (M, ν) admits at least one J ⊕ L 2 -orthogonal projection onto T(K), or equivalently, if T(K) is ortho-complemented. In this case, (3.19) and (3.20) hold with the Krein space kernel function .
3.5. Similar continuous frames. It is not difficult to see that the proof of the equivalences in Theorem 3.5 hinges on the fact that J happens to be a unitary operator with respect to [·, ·] and [·, ·] J . Theorem 3.5 can be generalized by an application of the inverse mapping theorem as follows.
The proof of the proposition is straightforward by observing that |[Aη i x , k]| = |[η i x , A * k]|, A * k J 1 ≤ A k J 2 and k J 2 ≤ A −1 A * k J 1 for all k ∈ K 2 . Proposition 3.11 motivates the following definition. Equations (3.4) and (3.5) ensure that S belongs to B(K) and has a bounded inverse. Therefore, the family {S −1 η 1 x , S −1 η 2 x , . . . , S −1 η n x } x∈M in the Frame Decomposition Theorem 3.6 yields a continuous frame of rank n that is similar to {η 1 x , η 2 x , . . . , η n x } x∈M . The next proposition provides a criterion for a pair of frames to be similar. It is an analogue of Deguang and Larson's result for Hilbert spaces [20]. Proof. The proof of the "if" part is the same as in [22,Theorem 3.3] for K G = K 1 and 1 for all h ∈ K 2 , i = 1, . . . , n, and the bijectivity of A ∈ B(K 1 , K 2 ).
To prove the opposite direction, assume that Rang . Similarly to (3.14), we obtain from (3.2), (3.6) and (3.7) that where J i denotes a chosen fundamental symmetry on K i and 0 < a i ≤ b i are frame bounds for the frame in K i , i = 1, 2. From (3.21), we conclude that T 1 and T 2 are bounded, injective, and have closed range V in the Hilbert space . It follows that the operators T 1 ∈ B(K 1 , V) and T 2 ∈ B(K 2 , V) possess bounded inverses and the same holds for T * 1 ∈ B(V, K 1 ) and T * 2 ∈ B(V, K 2 ). Given k 1 ∈ K 1 , set k 2 := T −1 2 T 1 k 1 .
is a bounded bijective operator. This shows that the two continuous frames are similar.
3.6. Dual and Parseval continuous frames. This section is devoted to the description of dual and Parseval continuous frames and their relations to the the fundamental symmetry J, the frame operator S and coherent states. A similar analysis for discrete J-frames can be found in [28]. We start with the definition of dual continuous frames.
Note that we used the measure ν instead of |ν| to emphasize the Krein space setting. Obviously, this can be compensated by multiplying θ i x by the value of the Radon-Nikodym derivative j L 2 at the point x ∈ M. The following proposition is analogous to [21,Proposition 3.8] in the discrete case. However, the Radon-Nikodym derivative j L 2 will appear in the formulas since the definition of the frame operator forced us to include the fundamental symmetry J ⊕ L 2 , see Section 3.3. Proposition 3.15. Let (K, [·, ·]) be a Krein space with fundamental symmetry J and let {η 1 x , η 2 x , . . . , η n x } x∈M be a continuous frame of rank n ∈ N for (K, [·, ·]) with respect to (M, B, ν). With S denoting the corresponding frame operator given in (3.3) and j L 2 denoting the Radon-Nikodym derivative of ν with respect to |ν|, there are the following dual frames: i) The continuous frame  In the light of Definition 3.12, Proposition 3.15 states that any continuous frame is similar to a dual frame, and Proposition 3.16 shows that any continuous frame is similar to a Parseval frame. Moreover, combining Proposition 3.10 and Example 4, we can say that any continuous frame with respect to a positive measure space determines a set of coherent states.
The conclusion in Example 4 that the frame operator S of any Parseval frame satisfies S = J remains valid for continuous frames of rank n. Then (3.3) and Proposition 3.3.ii) yield From this, we conclude that, for any Parseval continuous frame {η 1 x , η 2 x , . . . , η n x } x∈M , a dual frame is given by 4. Continuous frames in Krein spaces arising from a non-regular W-metric Each Krein space (K, [·, ·]) with fundamental symmetry J can be given by considering the Hilbert (K, [·, ·] J ) and setting [h, k] := [k, Jh] J for all k, h ∈ K. More generally, if W is a bounded self-adjoint operator on a Hilbert space (H, ·, · ) with polar decomposition W = J |W| and such that 0 spec(W), then (4.1) [ f, g] := f, Wg , f, g ∈ H, defines a non-degenerate inner product on H. Since, by assumption, 0 spec(|W|), there exists ǫ > 0 such that ǫ1 ≤ |W| ≤ W 1. It follows that the norms · J = √ ·, |W|(·) and · = √ ·, · are equivalent. Therefore, (H, [·, ·]) yields a Krein space with fundamental symmetry J and fundamental decomposition H = P + H ⊕ P − H, where P ± = 1 2 (1 ± J), see for instance [7]. Clearly, Krein or Hilbert spaces with the topology given by equivalent norms admit the same collection of (continuous) frames. Hence, by Proposition 3.11 and Theorem 3.5, any continuous frame for (H, ·, · ) of rank n ∈ N yields one for (H, [·, ·]) and vice versa.
As in [21], the aim of our last section is to show how to transfer continuous frames for a Hilbert space H to a Krein space with a W-metric like the one defined in (4.1), but with a possibly unbounded operator W and allowing 0 ∈ spec(W). To begin, let W denote a self-adjoint operator with domain dom(W) ⊂ H, polar decomposition W = J |W|, and integral representation W = λ dE(λ), where E stands for the corresponding projectionvalued measure on the Borel Σ-algebra B(R). We assume that ker(W) = E({0}) = {0}, which implies that J is a unitary self-adjoint operator. Analogous to (4.1), we define are said to be regular. The next proposition summarizes some properties of H W . Its proof can be found in [21].  ), and if 0 ∈ spec(W), then the completion process will always require to add abstract elements to dom( √ |W|).