Biframes and some of their properties

Recently, frame multipliers, pair frames, and controlled frames have been investigated to improve the numerical efficiency of iterative algorithms for inverting the frame operator and other applications of frames. In this paper, the concept of biframe is introduced for a Hilbert space. A biframe is a pair of sequences in a Hilbert space that applies to an inequality similar to a frame inequality. Also, it can be regarded as a generalization of controlled frames and a special kind of pair frames. The basic properties of biframes are investigated based on the biframe operator. Then, biframes are classified based on the type of their constituent sequences. In particular, biframes for which one of the constituent sequences is an orthonormal basis {ek}k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{e_{k}\}_{k=1}^{\infty}$\end{document} are studied. Then, a new class of Riesz bases denoted by [{ek}]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\{e_{k}\}]$\end{document} is introduced and is called b-Riesz bases. An interesting result is also proved, showing that the set of all b-Riesz bases is a proper subset of the set of all Riesz bases. More precisely, b-Riesz bases induce an equivalence relation on [{ek}]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\{e_{k}\}]$\end{document}.


Introduction
Many years after the advent of frame theory, its importance and applications in various scientific fields have become clear to everyone.The aim of this theory, developed by Duffin and Schaeffer in 1952 [16], was to solve some problems related to the nonharmonic Fo8urier series.In fact, the frame theory developed Gabor's studies in signal analysis in a pure form, which was extensively studied by the fundamental paper of Daubechies, Grossmann, and Meyer [13] in 1986.Many researchers in various fields of pure and applied mathematics, engineering, medicine, etc. have studied frames.For more information on frame theory and its applications, we refer the readers to [11,14,18,19,28].
To further apply this theory, many generalizations of frames have been proposed.For example, generalized frames (g-frames), introduced by Sun [26,27], generalize not only the original concept of frame, but also other generalizations of frames, including bounded quasi-projectors [21,22], frames of subspaces [4,10], pseudo-frames [24], oblique frames [12,17], and outer frames [1].After that, Askarizadeh and Dehghan showed that every g-frame is an special frame [5].Also, frames and their generalization in Hilbert C * -modules and locally compact abelian groups have been studied in [2,3,6].Controlled frames [8] provide another generalization of frames, actually developing theories related to this concept which were previously introduced in [9] 2and used only as a tool for spherical wavelets.
pair frames are a concept for a pair of sequences in a Hilbert space, introduced in [20], and one of their important results is to obtain a new reconstruction formula for members of the Hilbert space.the concept proposed in this paper aims at a different study of a pair of sequences in a Hilbert space, that is called a biframe.To define a frame, only one sequence is used; but, to define a biframe two sequences are needed.In fact, the concept of biframe is proposed as a generalization of controlled frames and a special case of pair frames.A biframe is a pair ({f k } ∞ k=1 , {g k } ∞ k=1 ) of sequences in the Hilbert space H, if there exist positive constants A and B such that An operator associated with a biframe is presented, which is a special case of multiplier operators [7].In the simplest case, multiplier operators are well-defined when the sequences are Bessel sequences, which gives us the Bessel multipliers.Also, the invertibility of this operator is obtained when the sequences are Riesz bases.But, it is interesting that the biframe operator has these properties for sequences that are not even Bessel sequences.Also, the biframe operator associated with a biframe has properties very close to those of the frame operator.Therefore, the reconstruction of the elements of the Hilbert space, which is one of the important achievements of frame theory, is well-done.Due to the important role played by orthonormal bases in a Hilbert space, we are looking for sequences that form a biframe together with an orthonormal basis.This research leads us to sets that are actually bases.The new set of all these bases has a place between the set of all orthonormal bases and the set of all Riesz bases, but studying this topic in pair frames does not have such an achievement for us.
This paper is organized as follows.Section 2 contains some preliminary results and notations that are used throughout the paper.In Section 3, we will introduce the new concept of biframe for a Hilbert space, created by8 a pair of sequences, and we present several examples of biframes.In Section 4, by defining the biframe operator associated with a biframe, we discuss the properties of biframes from the perspective of operator theory.Moreover, we investigate those operators which preserve the biframe property.In Section 5, we classify the biframes whose constituent sequences of them are Bessel sequences, frames, and Riesz bases.Finally, in Section 6, we classify the biframes for which one of the constituent sequences is an orthonormal basis.This classification leads us to new sets, which are partitions of the set of all sequences with this property (namely, the property of forming a biframe with an orthonormal basis).We obtain new bases which are closely related to orthonormal bases and Riesz bases.

Notation and preliminaries
Throughout this paper, H denotes a separable Hilbert space.The notation B(H, K) denotes the set of all bounded linear operators from H to the Hilbert space K.If H = K, this set is denoted by B(H) and I denote the identity operator on H. Also, GL(H) is defined as the set of all invertible, bounded linear operators on H, and GL + (H) denotes the subset of GL(H) that consists of positive operators.Finally, B + b.b. (H) is defined as the set o2f all self-adjoint and positive operators on H which are bounded below.
A sequence {f k } ∞ k=1 is called a frame for H, if there exist positive constants A and B such that The constants A and B are known as lower and upper frame bounds, respectively.If k=1 is called a Bessel sequence if only the right inequality holds.an associated operator to frame F = {f k } ∞ k=1 is frame operator, defined by The frame operator S F is a self-adjoint operator that belongs to GL + (H).Two Bessel sequences {f k } ∞ k=1 and {g k } ∞ k=1 are dual frames for H if one of the following statements holds.
, where {e k } ∞ k=1 is an orthonormal basis for H and the operator U belongs to GL(H).
We refer the readers to [11] for more details on frames and bases.
For U ∈ GL(H), a sequence {f k } ∞ k=1 in H is a U-controlled frame [8], if there exist positive constants A and B such that [25], if there exist positive constants A and B such that , such that W and T are bounded operators on H such that T W * = I, p, q, r, t ∈ R, and r + t = 1, p + q = 1.

Biframes
In this section, we define the main concept of this paper, namely, the notion of biframe.Next, we present some examples to illustrate the concept of biframes and their constituent sequences more.
The numbers A and B are called lower and upper biframe bounds, respectively.The biframe After defining th8e concept of biframes, its relation to the previously defined notions must be examined.The following remark shows that biframes generalize ordinary frames and controlled frames.Remark 3.2.According to Definition 3.1, the following statements are true for a sequence (ii) If (F, UF ) is a biframe for some U ∈ GL(H), then F is a U-controlled frame for H.
(iii) If (T F, UF ) is a biframe for some T and U in GL(H), then F is a (T,U)controlled frame for H.
A biframe is a pair of sequences.Therefore, the relationship between the two sequences that form a biframe is very important.To investigate the relationship between these two sequences, we try to answer the following questions.
(Q 1 ) What kind of Bessel sequences make biframes?And how about frames, Riesz bases and orthonormal bases?
Are there sequences that are not Bessel sequences, frames, Riesz bases or orthonormal bases, but form a biframe?(Q 3 ) Are the types of sequences that form a biframe interdependent?For example, if one of the sequences is a Bessel sequence, does it necessarily follow that the other one is also a Bessel sequence?What about Riesz bases and orthonormal bases?
In the following sections, we will answer these questions.But before that, we present some examples of biframes.
(i) A biframe may be constructed using two non-Bessel sequences.
It is clear that (ii) Two Bessel sequences may not form a biframe.
It is easy to see that Now, if we set f = e 1 , then the summation in the definition of biframe is equal to 0: This shows that the summation does not have a non-zero lower bound.
(iii) Two frames may not form a biframe.Consider the sequences This implies that the summation does not have a non-zero lower bound.Therefore, the pair (iv) Two Riesz bases may not form a biframe.
Because these operators are unitary, the following sequences are Riesz bases for L 2 (0, 1).
For f ∈ L 2 (0, 1), we have Therefore, the summation does not have a non-zero lower bound, so the pair ({f k } k∈Z , {g k } k∈Z ) isnot a biframe for L 2 (0, 1).
(v) Two orthonormal bases may not from a biframe.
Let σ be a permutation of N other than the identity.Then, the pair ) is not a biframe.In fact, an easy calculation shows that the summation in the definition of biframe is always equal to 0 for such sequences.If we set f = e j for some j ∈ N such that σ(j) = j, then A pair frame is a concept related to pair (F, G) of sequences in H. Although, a biframe is introduced as a pair of sequences with the same symbol, but these two notions are not actually the same.In fact, biframes are special types of pair frames, as we show in Corollary 4.3.In Section 6, we focus on the differences between these concepts, where we specifically consider one of their constituent sequences as the orthonormal basis.
The next example shows that a pair frame is not necessarily a biframe.
Example 3.4.Consider the Hilbert space R 2 , and the sequences k=1 defined as follows.
First, we show that (F, G) is a pair frame by considering the properties of the pair frame operator S. For (x, y) ∈ R 2 , the definition of S is given by the following equalities.The matrix associated with the operator S is It is clear that this matrix is invertible (det(S) = −17 = 0).So, the operator S is well-defined and invertible and therefore, (F, G) is a pair frame.But, this pair is not a biframe.For (x, ), (x, y) = x 2 − y 2 + 8xy.

The biframe operator
Since finding the bounds of a biframe is not always easy in practice, to better understand and be able to work with biframes, we need to introduce an operator similar to the frame operator, one that has as many good and useful properties as the frame operator.In this section, we introduce the biframe operator associated with a biframe, and we examine its properties.Also, we characterize biframes based on the properties of this operator.
) be a biframe for H.The biframe operator S F,G is defined by In what follows, we present some properties of the biframe operator.
) be a biframe for H with bounds A and B.Then, the following statements are true.
(i) The operator S F,G is well-defined, bounded, positive, and invertible.
n=1 is a sequence of linear and bounded operators on H. Consider i, j ∈ N, i > j.Then, So, its associated sequence of partial sums is a Cauchy sequence.Thus, the sequence { S n f, f } ∞ n=1 is Cauchy, and therefore an operator S ∈ B(H) exists (see [23], for example) such that {S n } ∞ n=1 converges to S weakly: By our definition of the biframe operator S F,G and the uniqueness of limit, we conclude that S F,G = S, and therefore S F,G is a well-defined and bounded operator.By our definition of S F,G , for every Which implies that S F,G is a positive operator.
To prove that S F,G is an invertible operator, we need to show that S F,G and S * F,G are injective and have closed ranges [20].For f, g ∈ H, . By the definition of biframe, S F,G and S G,F are injective.To prove that S F,G has a closed range, let {h n } ∞ n=1 ⊂ R(S F,G ) be a sequence that converges to h ∈ H.Then, there exists a sequence for every n; this means that the sequence {S F,G (t n )} ∞ n=1 converges to h.The sequence {h n } is convergent, so it is a Cauchy sequence.Let ǫ > 0 be given.Then, there exists N > 0 such that Consider m, n ≥ N, n > m.Now, by (4.2) and the Cauchy-Schwarz inequality, Hence t n − t m ≤ ǫ A , which implies that the sequence {t n } ∞ n=1 is a Cauchy sequence in H, and so it converges to some t ∈ H. Since S F,G is bounded, Now by the uniqueness of limit we obtain S F,G (t) = h.So, h ∈ R(S F,G ) and therefore, R(S F,G ) is closed.Thus, R(S * F,G ) is closed.(ii) Let (F, G) be a biframe with bounds A and B. Then for every f ∈ H, So, (G, F ) is a biframe with bounds A and B. The converse of this statement can be proved similarly.Proof.Let (F, G) be a biframe.By Theorem 4.2, S F,G is well-defined and invertible.So, by the definition of pair frames, (F, G) is a pair frame.
Reconstruction of the elements of a Hilbert space from the frame coefficients is one of the most important achievements of frame theory.The next theorem deals with this reconstruction by using biframes.
) is a biframe for H with biframe operator S F,G .Then, for every f ∈ H the following reconstruction formula holds.
Moreover, in a complex Hilbert space, we can rewrite the first equation of (4.3) in the following form.
Proof.We observed in the proof of Theorem 4.
For the final assertion note that in a complex Hilbert space, every positive operator is self-adjoint.(The next example shows that this is not true in real Hilbert spaces.)So, we have the self-adjointness property for the positive operator S F,G in complex Hilbert spaces and therefore, the result follows clearly by using S F,G = S G,F in (4.3).
The biframe decomposition stated in (4.3) is the reconstruction formula of biframe theory.The sequence { f, S −1 F,G f k } ∞ k=1 is called the sequence of biframe coefficients.Example 4.5.Consider the space R 2 as a real Hilbert space.
Clearly, the adjoint of this matrix is A = 1 2 3 8 , and this shows that A is not selfadjoint.
As mentioned at the beginning of this section, proving that two sequences form a biframe by using the definition is by no means an easy task.Therefore, we try to characterize biframes based on the properties of the biframe operator.The next theorem presents these conditions.Theorem 4.6.Let {f k } ∞ k=1 and {g k } ∞ k=1 be sequences in a complex Hilbert space H.Then, ) is a biframe for H if and only if S F,G is a positive and bounded below operator.
) be a biframe for H.As we observed in Theorem 4.2, S F,G is a positive and invertible operator.Therefore, it is bounded below, too.Conversely, suppose that S F,G is a positive and bounded below operator on H. Since S F,G is a positive operator, it is self-adjoint.For f ∈ H, To obtain the lower bound, by the positivity of S F,G [23], we obtain the following inequality.
On the other hand, S F,G is bounded below with lower bound α: Using (4.5) and (4.6) we find that ) is a biframe for H.To answer the questions we posed in the previous section about the structural relationship between the two sequences that form a biframe, we establish the next theorem which determines the dependency of two Riesz bases that form a biframe.First, we show by some examples that this dependency is not necessarily true for Bessel sequences and frames. Also, ) is a biframe with bounds 1 and 2, although {f k } ∞ k=1 is a frame and {g k } ∞ k=1 is not.
(ii) A Parseval biframe constructed by a Bessel sequence and a non-Bessel sequence Consider the following sequences.For f ∈ H, we obtain the following equalities. Hence, k=1 be a Riesz basis.Then, there exists an operator V ∈ GL(H) such that f k= V e k , for all k ∈ N, where {e k } ∞ k=1 is an orthonormal basis for H. Define the operator Clearly, U ∈ GL(H) and for k ∈ N, we obtain Hence g k = Ue k , for all k ∈ N.This means that {g k } ∞ k=1 is a Riesz basis for H. Similarly, we can prove that {f k } ∞ k=1 is a Riesz basis if {g k } ∞ k=1 is a Riesz basis.Remark 4.9.As mentioned before, a positive operator in a complex Hilbert space is self-adjoint.To use the self-adjointness property of the positive biframe operator S F,G , from now on we assume that the Hilbert space H is complex.
To investigate the operators that preserve the biframe property, since we deal with a pair of sequences in the definition of a biframe, we can study the action of two different operators on the sequences in a biframe.The next theorem presents some operators that preserve the biframe property of a given biframe.
) is a biframe for H with biframe operator S F,G .Then, the following statements are true.
) is a biframe for H, for some operators U and V in B(H) if and only if there exist operators G , when p, q, r, t ∈ R such that p + q = 1 and r + t = 1.
) is a Parseval biframe for H, for some operators U and V in B(H) if and only if there exist operators T, W ∈ B(H) such that T W * = I and U = W S −p F,G , V = T S −q F,G , when p, q ∈ R such that p + q = 1.
Conversely, let Q ∈ B + b.b. (H), and let T and W be operators in B(H) such that T W * = I and Then, by Proposition 2.1, Q = V S F,G U * , which means that for every f ∈ H, Therefore, Q is of the form of a biframe operator.By the assumption ) is a biframe for H. (ii) This part is a direct consequence of the previous part.
Having in mind the final part of Theorem 4.10, we obtain some simple results for Parseval biframes.
Now, the desired result follows from the final assertion of Theorem 4.10.Also, the pair ({e k } ∞ k=1 , {e k } ∞ k=1 ) is a Parseval biframe and again, by the final assertion of Theorem 4.10, the statement is true.

Charactrization of biframes
In this section, we characterize the biframes whose constituent sequences are Bessel sequences, frames, and Riesz bases.Also, we obtain some results concerning Parseval biframes. (5.1) For the Parseval biframe ({e k } ∞ k=1 , {e k } ∞ k=1 ), Theorem 4.10 gives us operators T, W ∈ B(H) such that T W * = I, and by (4.8), whenever r, t ∈ R and r + t = 1.Now, by (5.1), we can write and Conversely, suppose that there exist an orthonormal basis {e k } ∞ k=1 and operators

These relations and Theorem
) is a biframe for H. Corollary 5.2.Two Bessel sequences {f k } ∞ k=1 and {g k } ∞ k=1 form a Parseval biframe for H if and only if there exist operators T, W ∈ B(H) such that T W * = I, f k = W e k , and g k = T e k , for k ∈ N, whenever {e k } ∞ k=1 is an orthonormal basis for H. Furthermore, each of the above assertions implies that {f k } ∞ k=1 and {g k } ∞ k=1 are dual frames .
Here we recall the concept of generalazied dual frames for H (g-dual frames).In 2013, Dehghan and Hasankhani [15] introduced the concept of g-dual frames as follows.A frame {g k } ∞ k=1 is a g-dual frame of frame {f k } ∞ k=1 for H, if there exists an invertible operator A ∈ B(H) such that for all f ∈ H, the equality f = ∞ k=1 Af, g k f k is valid.The reconstruction formula (4.4), for the members of a Hilbert space by a given biframe ) for H, and the concept of g-dual frames for H give us this idea that, characterize the biframes which are formed by two frames, by using the characterization of g-dual frames.
k=1 be a frame for H with frame operator S F .Then the following statements are equivalent.
k=1 is a frame for H and the pair ) is a biframe for H with biframe operator S F,G .By the reconstruction formula (4.4) and according to this point that S F,G = S G,F , for f ∈ H, we have Therefore, {f k } ∞ k=1 and {g k } ∞ k=1 are g-dual frames.By using the characterization of g-dual frames presented in [15], every g k has the following form.
where the sequence {h k } ∞ k=1 is a Bessel sequence in H.For the proof of (ii) ⇒ (i), assume that there exists an operator k=1 be a Riesz basis for H with frame operator S F .Then the following statements are equivalent.
In particular, each of conditions (i) and (ii) implies that {g k } ∞ k=1 is a Riesz basis for H.
Proof.For the proof of (i) k=1 is a Riesz basis for H, there is an invertible operator k=1 is an orthonormal basis for H.By Theorem 4.8, the sequence {g k } ∞ k=1 is a Riesz basis for H and for every k ∈ N, The proof of (ii) ⇒ (i) is easily obtained.

B-Riesz bases
Since orthonormal bases are among the most important sequences in H, we study those biframes for which one of the constituent sequences is an orthonormal basis.We will see that these types of biframes have interesting properties that distinguish them from the pair frames having a similar property.In fact, by collecting all the sequences that form a biframe together with a given orthonormal basis in a set, new bases can be obtained that find a special place between the set of all orthonormal bases and the set of all Riesz bases.But, such results cannot be established for pair frames.We begin our study of this subject with a definition.Definition 6.1.We consider an orthonormal basis {e k } ∞ k=1 for H and define the set [{e k }] as the following.
The next proposition shows us the way the elements of the set [{e k }] can be represented.Proposition 6.2.Let E = {e k } ∞ k=1 be an orthonormal basis for H.The sequence This means that S ∈ B + b.b. (H), and by Theorem 4.6, ({e k=1 be an orthonormal basis for H, and consider the sequence and also S F f = V 2 f.These relations allow us to conclude that U 2 = V 2 , and so U = V .Now, for k ∈ N, one obtains Hence, U(e k − δ k ) = 0, and so e k = δ k .
Here, a new class of sequences is introduced and we study elements of it.
) is a biframe for H. Now, we can use Proposition 6.2 and Proposition 6.3, and give some equivalent conditions for a sequence to be a b-Riesz basis.Theorem 6.5.Let {f k } ∞ k=1 be a sequence in H. then the following statements are equivalent.
(iii) There exist an orthonormal basis {e k } ∞ k=1 and an operator for precisely an orthonormal basis {e k } ∞ k=1 .If we denote the set of all orthonormal bases of H by O, the set of all Riesz bases by R, and the set of all b-Riesz bases by E, then by Proposition 6.2, these sets are ordered as O ⊂ E ⊂ R. The following example illustrates that they are proper subsets.Example 6.6.(i) We consider the Riesz basis {f k } 2 k=1 = {(−1, 2), (1, 0)} for R 2 .every orthonormal basis for R 2 is in the following form.For a ∈ [0, 1], If there exists a ∈ [0, 1] such that {f k } 2 k=1 ∈ [{e a k }], then there are positive numbers A and B such that for every + (x, y), (− √ 1 − a 2 , a) (1, 0), (x, y) ≤ B(x 2 + y 2 ), and then Now, set y = 1.We get a quadratic equation as follows.
This equation has positive ∆ = 2a √ 1 − a 2 + 9 > 0. Hence, there are none zero points (x, y) in R 2 such that So there is no orthonormal basis for R 2 such that form a biframe with Now we can see that ) is a biframe for R 2 with bounds 1 and 4.But it is clear that {f k } 2 k=1 is not an orthonormal basis for R In order to show another difference between biframes and pair frames, in what follows we study a similar problem, and we observe that the results obtained for biframes do not apply to pair frames.Definition 6.7.Consider the set E ′ of sequences in H as follows.
) is a pair frame, for some orthonormal basis {e k } ∞ k=1 .

For an orthonormal basis {e
) is a pair frame for H . Proposition 6.8.The set E ′ is the set of all Riesz bases.
) is a pair frame for H, that is, the pair frame operator S is a welldefined and invertible operator.For j ∈ N, k=1 is a Riesz basis for H. Conversely, suppose that the sequence {f k } ∞ k=1 is a Riesz basis for H.Then, there exists an invertible operator ) is a pair frame for H with the pair frame operator V .
The next example shows that unlike the subsets [{e k }] of E, the subsets [{e k }] ′ of E ′ are not distinct.Hence, the collection of them cannot be a partition for the set E ′ = R.
It is easy to check that the sequence ′ , because for every k ∈ N, In what follows, we examine the biframe property according to whether the constituent sequences are b-Riesz bases or are not.In this regard, the following questions arise.
(Q 4 ) Is it necessarily true that any two b-Riesz bases form a biframe?(Q 5 ) Is there a biframe for which none of the constituent sequences are b-Riesz bases?(Q 6 ) Is there a biframe for which just one of constituent sequences is a b-Riesz basis?
The next example shows that the answer to question (Q 4 ) is negative.In Proposition 6.11, we propose a necessary condition for such sequences to form a biframe.Also, we have a positive answer to question (Q 5 ) and (Q 6 ) that are illustrated in Example 6.12 and Example 6.13.Example 6.10.We consider the orthonormal {e k } 2 k=1 = {(1, 0), (0, 1)} for R 2 and two sequences {f k } 2 k=1 and {g k } 2 k=1 as the following.
In the next theorem, we characterize those biframes for which one of the constituent sequences is a b-Riesz basis for H. Theorem 6.14.Let {f k } ∞ k=1 be a b-Riesz basis for H and {g k } ∞ k=1 is a sequence in H.Then, the following statements are equivalent.
) is a biframe for H.

Theorem
are biframes, Theorem 4.6 allows us to conclude that the operators S F,G and S U F,V G are in B + b.b. (H), and also are in GL + (H).By (4.7) and Proposition 2.1, there exist operators T, W ∈ B(H) such that T W * = I and U

Theorem 5 . 1 .
Two Bessel sequences F = {f k } ∞ k=1 and G = {g k } ∞ k=1 form a biframe for H, if and only if there exist operators Q ∈ B + b.b. (H) and T, W ∈ B(H) such that T W * = I, f k = Q r W e k , and g k = Q t T e k , for all k ∈ N, whenever r, t ∈ R and r + t = 1 and E = {e k } ∞k=1 is an orthonormal basis for H.Proof.Let (F, G) = ({f k } ∞ k=1 , {g k } ∞ k=1 )be a biframe with biframe operator S F,G , formed by Bessel sequences {f k } ∞ k=1 and {g k } ∞ k=1 .Since {f k } ∞ k=1 and {g k } ∞ k=1 are Bessel sequences, there exist operators U, V ∈ B(H) and the orthonormal basis E = {e k } ∞ k=1 for H, such that and [{δ k }], by Proposition 6.2, there exist operators U, V ∈ B + b.b. (H) such that f k = Ue k and f k = V δ k , for all k ∈ N. On the other hand, the sequenceF = {f k } ∞k=1 is a frame for H with biframe operator S F , too.For f ∈ H,

(
ii) There exist U, Q ∈ B + b.b. (H) such that f k = Ue k and g k = QU −1 e k , for all k ∈ N, where {e k } ∞ k=1 is an orthonormal basis for H. Proof.For the proof of (i) ⇒ (ii), let ({f k } ∞ k=1 , {g k } ∞ k=1 ) be a biframe with biframe operator S F,G ∈ B + b.b. (H).{f k } k=1 is a b-Riesz basis and by Theorem 6.5, there exist an orthonormal basis {e k } ∞ k=1 for H, and an operator U ∈ B + b.b. (H) such that f k = Ue k , for all k ∈ H.For f ∈ H, we obtainS F,G f = ∞ i=1 f, f i g i = ∞ i=1 f, Ue i g i = ∞ i=1Uf, e i g i , and so for k ∈ N,S F,G U −1 e k = ∞ i=1 UU −1 e k , e i g i = ∞ i=1 e k , e i g i = g k .This implies the desired result.Now, suppose that (ii) holds.For f ∈ H,∞ k=1 f, f k g k = ∞ k=1 f, Ue k QU −1 e k = QU −1 ∞ k=1 Uf, e k e k = QU −1 Uf = Qf.This equality implies that SF,G = Q ∈ B + b.b. (H), and ({f k } ∞ k=1 , {g k } ∞ k=1) is a biframe by Theorem 4.6.Corollary 6.15.Let {f k } ∞ k=1 be a b-Riesz basis for H and {g k } ∞ k=1 be a sequence in H.Then, the following statements are equivalent.(i)({f k } ∞ k=1 , {g k } ∞ k=1) is a Parseval biframe for H.

(
ii) There exists U ∈ B + b.b. (H) such that f k = Ue k and g k = U −1 e k , for all k ∈ N, where {e k } ∞ k=1 is an orthonormal basis for H.The end corollary illusterates that corresponding to every b-Riesz basis exists b-Riesz bases for H. Corollary 6.16.Let {f k } ∞ k=1 be a b-Riesz basis for H. Then the canonical dual of {f k } ∞ k=1 , and also, every biorthogonal frame to {f k } ∞ k=1 , are b-Riesz bases for H. Proof.Since {f k } ∞ k=1 is a Riesz basis, it only has one dual frame, the canonical dual frame.Now, according to the proof of Lemma 4.11, every dual of {f k } ∞ k=1 , and also every biorthogonal frame to {f k } ∞ k=1 , form a Parseval biframe with {f k } ∞ k=1 .Hence, by Corollary they are b-Riesz bases for H.
2, so E = O.The Proposition 6.3, shows that the subsets[{e k }] of E are distinct whenever {e k } ∞ k=1 is an orthonormal basis for H. Also, it is clear that E = {e k } , hence the collection Γ = [{e k }] | {e k } isan orthonormal basis for H is a partition of E, and therefore induces an equivalence relation ∼ on the set E, i.e., this result leads us to equivalence classes on E. This is why we represented these subsets by the notation [{e k }].In fact, we have the following equivalence relation ∼ between sequences {f k } ∞ k=1 and {g k 12 and 4, and({e k } 2 k=1 , {g k } 2 k=1) is a biframe for R 2 with bounds 1 4 and 3.For (x, y) ∈ R 2 , Proposition 6.11.Let {e k } ∞ k=1 be an orthonormal basis for H, and suppose that {f k