Soft Frames in Soft Hilbert Spaces

: In this paper, we use soft linear operators to introduce the notion of discrete frames on soft Hilbert spaces, which extends the classical notion of frames on Hilbert spaces to the context of algebraic structures on soft sets. Among other results, we show that the frame operator associated to a soft discrete frame is bounded, self-adjoint, invertible and with a bounded inverse. Furthermore, we prove that every element in a soft Hilbert space satisﬁes the frame decomposition theorem. This theoretical framework is potentially applicable in signal processing because the frame coefﬁcients serve to model the data packets to be transmitted in communication networks.


Introduction
Discrete frame theory on Hilbert spaces is relatively recent. This theory was introduced by Duffin and Schaeffer in 1952 [1] as a tool to study problems related to non-harmonic Fourier series. Almost 30 years later, it was developed by Daubechies, Grossman and Meyer [2]; they used frames to find expansions in series of functions on L 2 (R) similar to the expansion in series that is done using orthonormal bases. As a result, the main advantage is that a good discrete frame can behave almost like an orthonormal basis; with an additional, they do not require the uniqueness of the coefficients when writing a Hilbert space vector as a linear combination of the frame elements (frame decomposition theorem). For this reason, in the literature, they are often called over-complete bases. Consequently, frames have turned out to be a powerful tool in signal processing, image processing, data understanding, sampling theory, and wavelet analysis [3] (see also [4][5][6]). Frame theory has been extended to the context of Krein's spaces, which are a generalization of Hilbert spaces that have a wide variety of applications in physics (see [7,8]).
Soft set theory was introduced by Molodtsov [9] in 1999 as a new mathematical tool for dealing with uncertainties while modeling problems in engineering, physics, computer science, economics, social science, and medical sciences. This theory began to receive special attention in 2002 when Maji et al. [10] applied the soft sets to decision-making problems, using rough mathematics, and later in 2003, with the definitions of various operations of soft sets [11]. Since then, research works in soft sets theory and its applications in various fields have been progressing rapidly because this theory is free from the many difficulties that have troubled the usual theoretical approaches. This is how research related to soft sets has been carried out in several directions among which we can mention the following: information systems, decision making, nonlinear neutral differential equations and algebraic structures, fuzzy sets, and rough sets, as we can see in the papers [12][13][14][15][16]. In particular, regarding mathematical analysis and its applications, the concepts and results of soft real sets, soft real numbers, soft complex sets, soft complex numbers, soft linear spaces, soft metric spaces, soft normed spaces, soft inner product, soft Hilbert spaces, soft linear operator, soft linear functional, soft Banach algebra, soft topology, etc., have originated (see [11,[17][18][19][20][21][22][23][24][25][26]). In the book [27], various applications of soft sets are discussed in problems related to data filling in incomplete information systems, and as is well known, frames are also a suitable tool to deal with these same types of problems, for which it is interesting to address frame theory together with soft sets theory to propose a new mathematical tool (soft frames) that is more efficient in the absence of partial information. This theoretical framework motivates the study of certain topics based on operator theory in spaces with a complete inner product (Hilbert spaces), which is potentially applicable in signal processing, where the frame coefficients serve to model the data packets to be transmitted; hence, the elimination of an element of the frame is equivalent to a packet that is not delivered in the communication, but this could be recovered using the frame decomposition theorem.
This manuscript was designed as follows. Section 2 corresponds to the preliminaries, where we cover all the theory of soft sets necessary to study soft frames, in addition to introducing the notions of soft inner product, soft operators, soft Hilbert spaces, etc. In Section 3, we develop the definition of soft frame on soft Hilbert spaces, and state the most important result: the soft frame decomposition theorem.

Preliminaries
Throughout this paper, X denotes a non-empty set (possibly without algebraic structure), P (X) the power set of X, and A a non-empty set of parameters. Definition 1 ([9]). A soft set on X is a pair (F, A) where F is a mapping given by F : A → P (X).
In this way, we can see a soft set as the following: where G A F is the graph of F with respect to A.
Note that a soft set is determined by knowing F(λ) for all λ ∈ A. Therefore, it is common to find in the literature that F : A → P (X) is called a soft set on X, but it should not be a cause for confusion. 2, 3, 4, 5, 6, 7, 8, 9} and A = {Z 4 = λ 1 , Z 6 = λ 2 , Z 8 = λ 3 }. If F : A → P (X) describes the generating elements of the cyclic group. Then, it is easy to see that 3, 5, 7}. Hence, (F, A) is a soft set on X seen as follows:

Definition 2 ([23]).
A soft set (F, A) on X is said to be a null soft set if F(λ) = ∅ for all λ ∈ A, and in this case we write (F, A) := Φ.
If F(λ) = ∅ for some λ ∈ A, then the soft set (F, A) is said to be a non-null soft set on X.

Definition 3 ([23]).
A soft set (F, A) on X is said to be an absolute soft set if F(λ) = X for all λ ∈ A; in this case, we write (F, A) :=X. This convention of absolute soft set is adopted throughout the present work.

Definition 4 ([23]).
A soft element on X is a function : A → X. Now if (λ) ∈ F(λ) for all λ ∈ A, the soft element is said to belong to the soft set (F, A) on X, which we denote by ∈ F. Proposition 1 ([23]). Let (F, A) be a soft set on X and be a soft element on X, which belongs to (F, A). Given λ ∈ A, we have We denote the collection of all the soft elements of a soft set (F, A) by SE((F, A)); this is SE((F, A)) := { : ∈ F} = { : (λ) ∈ F(λ), ∀λ ∈ A}.
Definition 5 ([23]). Let K = R or K = C, and let A be a non-empty set of parameters. Consider the set then, a mapping F : A → B(K) ⊂ P (K) is called a soft K-set. This is denoted by (F, A). Furthermore, if for all λ ∈ A it is satisfied that F(λ) is a singleton, then by identifying (F, A) with its corresponding soft element, we call this soft element a soft K-number.
We denote the set of all the soft R-numbers or soft real numbers by R(A). In addition, we use the symbols α, β, etc., to denote soft K-numbers such that they behave as constants, that is α(λ) = α for all λ ∈ A. Similarly, we symbolize the set of all soft C-numbers or soft complex numbers by C(A).

Definition 7 ([23]
). For two soft real numbers α : A → R and β : A → R we define the following: Note that if we assign to X a structure of vector space, it is interesting to think of the possible structure of F(λ) as a subset of X for all λ ∈ A. This motivates the following important definition. Definition 8 ([18,24]). Let X be a K-vector space (typically K = R or K = C), A be a nonempty set of parameters and (F, A) be a soft set on X. The soft set (F, A) is said to be a soft K-vector space on X if F(λ) is a vector subspace of X for all λ ∈ A.
The importance of the above definition is that it allows us to relate the usual linear algebra to soft set theory. Furthermore, it gives us the tools to define important concepts in the classical functional analysis, such as norm, inner product, Banach space, and Hilbert space, among others, but from this context. Definition 9 ([18,24]). Let (F, A) be soft K-vector space. A soft element of (F, A) is said to be a soft vector of (F, A). Similarly, a soft element : A → K is said to be a soft scalar, where K is the scalar field. Definition 10 ( [18,24]). Let (F, A) be a soft K-vector space on X. A soft vector x of (F, A) is said to be a null soft vector if x(λ) = θ, ∀λ ∈ A, where θ is the zero element of X. This is denoted by Θ.
Definition 11 ([18,24]). Let (F, A) be a soft K-vector space on X and x, y be two soft vectors of (F, A) and k be a soft scalar. Then, the addition x + y of x with y, and scalar multiplication k · x of k and x are defined by (x + y)(λ) = x(λ) + y(λ), (k · x)(λ) = k(λ) · x(λ). Evidently, x + y, k · x are soft vectors of (F, A). Theorem 1 ([18,24]). Let X be a K-vector space, A be a nonempty set of parameters and (F, A) be a soft K-vector space on X. Then, we have the following: 1.
(−1) · x = −x, for all x∈ F. Definition 12 ([19]). LetX be an absolute soft K-vector space, then a mapping · : SE(X) → R(A) is said to be a soft norm onX if · satisfies the following conditions: The soft K-vector spaceX with a soft norm · onX is said to be a soft K-normed space and is denoted by (X, · , A) or (X, · ).
The two lemmas above allow us to prove the following important theorem about soft normed spaces.

Theorem 2 ([19] Decomposition theorem).
If (X, · , A) is a soft normed space, then for ech λ ∈ A, the mapping · λ : X → R + defined by ξ λ := x (λ) where x is such that x(λ) = ξ is a norm on X for all λ ∈ A. Definition 13 ([24]). IfX is an absolute soft vector space, then a binary operation ·, · : SE(X) × SE(X) → C(A) is said to be a soft inner product onX if it satisfies the following conditions: (I1) x, x ≥ 0 for all x ∈X and x, x = 0 if and only if x = Θ, (I2) x, y = y, x for all x, y ∈X, (I3) α · x, y = α · x, y for all x, y ∈X and for all soft scalar α, (I4) x + y, z = x, z + y, z for all x, y, z ∈X.
The soft vector spaceX with a soft inner product ·, · onX is said to be a soft inner product space and is denoted by (X, ·, · , A) or (X, ·, · ).

Definition 14 ([17]).
A soft inner product space is said to be complete if it is complete with respect to the soft metric defined by the soft inner product. A complete soft inner product space is said to be a soft Hilbert space.
Theorem 12 ([24]). LetȞ be a soft Hilbert space, T : SE(Ȟ) → SE(Ȟ) be a continuous soft linear operator and T * be the adjoint operator of T. Then, the following properties hold: 1. T * is unique; 2.
T * is a soft linear operator; 3.
T * is a continuous soft operator with T * ≤ T ; 4.
TT * = T 2 9. ( Theorem 14 ([17]). Let {α n } n∈N be an orthonormal sequence on a soft Hilbert spaceȞ having a finite set of parameters A. Then, the infinite series where c 1 , c 2 , · · · , c n , · · · are soft scalars, is convergent if and only if the series ∑ ∞ i=1 |c i | 2 is convergent. Theorem 15 ([17]). Let {α n } n∈N be an orthonormal sequence on a soft Hilbert spaceȞ having a finite set of parameters A. Then, for any x ∈Ȟ, Definition 22 ([17]). Let B be a non-null collection of orthonormal soft elements ofȞ. Then, B is said to be complete orthonormal if there exists a non-orthonormal set D such that D is a proper subset of B. If the set B contains only a countable number of soft elements then we call it a complete orthonormal sequence.
In the following theorem, we consider S the collection of all soft vectors x ofȞ such that x(λ) = θ for all λ ∈ A, together with the null soft vector Θ. In symbols, {α n } n∈N is complete, 2.
If for all x ∈Ȟ and for all i ∈ N we have x ⊥ α i then x = Θ, 3. For Corollary 1. Let G be a soft Banach algebra. If x ∈ G and e − x < 1, then there exists x −1 and x −1 = e + ∑ ∞ j=1 (e − x) j .

Soft Frames in Soft Hilbert Spaces
Definition 23 ([25]). Let (Ȟ, ·, · Ȟ , A) be a soft Hilbert space. We say that a self-adjoint soft linear operator T is positive if T(x), x Ȟ ≥ 0 for all x ∈ SE(Ȟ). In notation, we write T ≥ 0Id. Also, T ≥ cId means that T(x), x Ȟ ≥ cIdx, x Ȟ for all x ∈ SE(Ȟ) and all soft scalar c.   (Ȟ, d), where d is the soft metric onȞ induced by ·, · . We must show that {h n } n∈N is convergent in SE(Ȟ). Indeed, we assert that for all λ ∈ A, {h n (λ)} n∈N is a Cauchy sequence, because given >0-note that {h n } n∈N is a Cauchy sequence-there exists N ∈ N such that d(h n , h m ) < for all m, n ≥ N, that is, d(h n , h m )(λ) = d H (h n (λ), h m (λ)) < (λ) for all λ ∈ A and all m, n ≥ N, which proves the assertion. Thus, since (Ȟ, d) is complete, we have that {h n (λ)} n∈N is convergent for all λ ∈ A, for example, h n (λ) → h λ as n → ∞. Now, given > 0, there exists N λ ∈ N such that d H (h n (λ), h λ ) < (λ) for all n ≥ N λ .
Example 2. Let X = 2 (N). It is well known that X is a Hilbert space with respect to the inner product x, (N). Now, let x, y be soft elements of the absolute soft vector spaceX. Then, , y(λ) 2 for all λ ∈ A, is a soft inner product onX. Therefore, (X, ·, · , A), with A being a finite set of parameters, is a soft Hilbert space.
Next, we introduce the notion of soft discrete frame in soft Hilbert spaces. We study the pre-frame operator and the frame operator of a soft discrete frame. In addition, we establish the most important result, called the decomposition theorem of soft frames. Definition 24. Let {x n } n∈N be a sequence of soft vectors of a soft Hilbert spaceȞ having a finite set of parameters. We say that {x n } n∈N ⊆ SE(Ȟ) is a soft frame onȞ if there exist soft real numbers b ≥ a > 0 such that the following holds: The soft real numbers a and b are called bounds of the soft frame. These are not unique, as the optimal bounds are the largest possible value of a and the smallest possible value of b that satisfy (1). In the case that a = b, the soft frame is called tight.

Remark 1. The relationship between the previous definition and the discrete frames in usual
Hilbert spaces is that every discrete frame on a Hilbert space induces a soft discrete frame on a soft Hilbert space, with respect to any finite set of parameters. The proof of this claim is as follows: Let (H, ·, · H ) be any Hilbert space and let { f n } n∈N ⊆ H be a discrete frame on H with bounds b ≥ a > 0. Then, if A is a finite set of parameters, we know that, by virtue of Theorem 17, (Ȟ, ·, · , A) is a soft Hilbert space, where h 1 , h 2 (λ) = h 1 (λ), h 2 (λ) H for all λ ∈ A and all h 1 , h 2 ∈Ȟ, whereby, if for each n ∈ N we define x n : A → H by x n (λ) = f n for all λ ∈ A, we can affirm that {x n } n∈N ⊆ SE(Ȟ) is a soft frame onȞ with bounds b ≥ a > 0 defined by b, a : A → R, b(λ) = b, ∀λ ∈ A and a(λ) = a, ∀λ ∈ A. Indeed, for each h ∈ SE(Ȟ) and each λ ∈ A, we have the following: where we have used that { f n } n∈N ⊆ H is a frame on H. On the other hand, we have the following: Thus, we have proved the following: Therefore, {x n } n∈N ⊆ SE(Ȟ) is a soft frame onȞ.
Definition 25. Given a soft frame {x n } n∈N on a soft Hilbert spaceȞ having a finite set of parameters, we define the pre-frame operator by the following: Proposition 5. The pre-frame operator associated to {x n } n∈N is well defined and bounded.

Proposition 7.
The associated frame operator to the soft frame {x n } n∈N ⊆ SE(Ȟ) with bounds b ≥ a > 0 is bounded and self-adjoint.
Proof. Note the following: Hence, S is bounded. Additionally, the following holds: Thus, S is self-adjoint. x, x n Ȟ x n , for all x ∈ SE(Ȟ).

Remark 2.
Note that for all x ∈ SE(Ȟ) we have the following: Hence, the frame condition can be written in the form a x 2 ≤ S(x), x Ȟ ≤ b x 2 , for all x ∈ SE(Ȟ).

Lemma 6.
If {x n } n∈N ⊆ SE(Ȟ) is a soft frame on a soft Hilbert spaceȞ with bounds b ≥ a > 0, then {x n } n∈N is complete onȞ.
Proof. Let x ∈ SE(Ȟ) be such that x, x n Ȟ = 0 for all n ∈ N. Then, by the frame condition, we have and so x = Θ.
Proposition 9. Given a sequence {x n } n∈N ⊆ SE(Ȟ) in the soft Hilbert spaceȞ, the following statements are equivalent: S(x) = ∑ n∈N x, x n Ȟ x n is a positive and bounded soft linear operator of SE(Ȟ) to SE(Ȟ), which satisfies aId ≤ S ≤ bId.
Proof. (i)⇒(ii) Suppose that {x n } n∈N ⊆ SE(Ȟ) is a soft frame with bounds b ≥ a > 0, then S = TT * is bounded by Proposition 7 and since it follows by Remark 2 that S(x), x Ȟ = ∑ n∈N | x, x n Ȟ | 2 . Thus, and hence, aId ≤ S ≤ bId.
(ii)⇒(i) Suppose that (ii) is true, that is, for all x ∈ SE(Ȟ). Then {x n } n∈N ⊆ SE(Ȟ) is a soft frame with bounds b ≥ a > 0. Proof. Indeed, let x, y, z ∈ SE(Ȟ) and α be soft scalars. Then, the following hold: The following result is a version of the Cauchy-Schwarz inequality for positive soft linear operators. We will see its usefulness later. Proof. By Lemma 7, it is satisfied that for all > 0, [x, y] = (T + Id)x, y Ȟ defines a soft inner product onȞ. Thus, by the Cauchy-Schwarz inequality, we have the following: for all > 0 and all x, y ∈ SE(Ȟ); this is, Thus, making → 0, we obtain the following | T(x), y | 2 ≤ T(x), x T(y), y , for all x, y ∈ SE(Ȟ).
Proof. (i) Since aId ≤ S ≤ bId then and also, In other words, Then, Therefore, by Corollary 1, it follows that S is invertible. In addition, observe that by Remark 2 and Cauchy-Schwarz inequality, we have the following: Hence, S −1 ≤ a −1 . Thus, the following holds: Therefore, S −1 ≤ a −1 Id.
On the other hand, by Proposition 10 and frame condition, the following is satisfied: Id(x), x Ȟ and hence S −1 ≥ b −1 Id. In summary, we have proved that b −1 Id ≤ S −1 ≤ a −1 Id.
(ii) Since S −1 is positive and self-adjoint, we have the following: Note that Theorem 19 tells us that there is no restriction to write every element of a soft Hilbert space as a linear combination of the soft frame and the associated inverse soft frame operator, which differs from Theorem 16-(3).

Conclusions
In this article, we have introduced the concept of discrete frames in soft Hilbert spaces. In addition, we have studied the most important properties of frame theory, such as proving that the associated pre-frame operator to a soft frame is bounded. In addition, we have introduced a soft inner product that allows us to demonstrate a version of the Cauchy-Schwarz inequality for positive soft linear operators, a concept that we have also coined here. We have investigated some characterizations for soft frames and finished with the decomposition theorem for soft frames, which allows us to write or decompose every element of a soft Hilbert space, using the inverse of the associated frame operator. The theoretical framework established in this article can be used to extend the results given in [28] as well as the classical results of wavelet and dual frame theory to the context of soft Hilbert spaces.