The concept of b-frame which is a generalization of the frame in Hilbert spaces generated by the bilinear mapping is considered. b-frame operator is defined; analogues of some well-known results of frame theory are obtained in Hilbert spaces. Conditions for the existence of b-frame in Hilbert spaces are obtained; the relationship between the definite bounded surjective operator and b-frame is also studied. The concept of b-orthonormal b-basis is introduced.

1. Introduction

Frames for a Hilbert space were formally defined by Duffin and Schaeffer to study some problems in nonharmonic Fourier series [1]. Frames are applied in various branches of natural sciences, such as in signal processing, in the image processing, and in data compression. More details about the application of frames can be found, for example, in [2–5]. Numerous works have been dedicated to frames (see, e.g., [6–8]). More details about frame theory can be found in the monographs [9, 10] and also in review articles [11, 12]. One of the important facts of the theory of frames is the decomposition of each vector with respect to the frame in Hilbert space. It is a remarkable fact that in this case a sequence of coefficients has a minimal property. In frame theory, an important tool for obtaining frames is the stability of frames. Stability of frames in Hilbert spaces is studied in [13, 14] and so forth. Extension of the concept of frame to Banach spaces belongs to Gröchenig [15]. The concepts of a Banach frame and an atomic decomposition are introduced in [15]. Frames in Banach spaces have been also studied in [16–18].

The concept of g-frame as a generalization of the frame in Hilbert spaces has been considered by Sun in [19]. Analogues of most properties of frames were obtained for the g-frame in [20, 21]. Another generalization of the frame in Hilbert spaces is a concept of a b-frame which is introduced in the works [22, 23] by the bilinear mapping. The concept of t-frame is introduced and studied in [22], where t is tensor mapping. Let us note that the approximate concepts associated with the linear mapping and related results have been introduced in [24, 25]. An atomic decomposition of Lebesgue spaces in the trigonometric systems with degenerate coefficients has been studied in [24, 25].

This paper is devoted to the study of the characteristic properties of b-frame in Hilbert spaces. Analogues of the known results of the frame are established. Namely, a criterion of b-frameness for the given sequence is established. The concept of b-orthonormal b-basis is introduced. In this paper, b-frameness of the sequence, generated by the matrix operator, is studied in Hilbert spaces.

2. Some Notations and Auxiliary Facts

We will use the standard notation. Let X and Z be Hilbert spaces with scalar products (·,·)X and (·,·)Z, respectively. As usual, by L(X,Z) we denote the space of all bounded linear operators acting from X to Z. The kernel and the range of the operator T∈L(X,Z) are denoted by N(T) and R(T), respectively. If R(T)=Z and N(T)=0, then by Banach theorem ∃T-1∈L(Z,X). In the case when R(T) is closed, there exists T+∈L(Z,X) such that TT+z=z, ∀z∈R(T). Indeed, T+ is a continuation of the operator T1-1:R(T)→(N(T))⊥ by zero to all Z, where T1=T|(N(T))⊥. The adjoint of the operator T is denoted by T∗.

By l2(X), denote the linear space of sequences of vectors x→=xkk∈N from X, with coordinatewise linear operations, such that ∑k=1∞xkX2<+∞. l2(X) is a Hilbert space with the scalar product(1)x→′,x→′′l2X=∑k=1∞xk′,xk′′X,x→′=xk′k∈N,x→′′=xk′′k∈N.Obviously, ∀x→=xkk∈N∈l2(X) has a decomposition(2)x→=∑k=1∞δikxki∈N, where δik is the Kronecker symbol.

Let Y be a Banach space with a norm ·Y. Consider the bilinear mapping b:X×Y→Z, satisfying the condition(3)∃M>0:bx,yZ≤MxXyY,∀x∈X,y∈Y.For each pair z∈Z and y∈Y, let us consider the functional fz,y in X, defined by the formula fz,y(x)=(b(x,y),z)Z. Let us show that fz,y∈X∗. Linearity of fz,y is obvious. Further, ∀x∈X; taking into account (3), we have(4)fz,yx=bx,y,zZ≤bx,yZzZ≤MxXyYzZ.Hence, fz,y≤MyYzZ. According to the Riesz theorem, there exists the unique element x∗∈X such that fz,y(x)=(x,x∗)X. Further, this element will be called the b-scalar product of z and y and will be denoted by z,y. It is clear that z,y is linear and continuous for z. It directly follows from z,yX=fz,y that(5)z,yX≤MyYzZ.

We will also need some concepts and statements from [6, 22, 23].

Definition 1.

Sequence ykk∈N⊂Y is said to be b-orthonormal in Z, if (6)bx,yk,yj=δkjx,∀x∈X,k,j∈N.

By Lb(ykk∈N), denote the set of all finite sums of the form ∑k=1nb(xk,yk), xk∈X.

Definition 2.

Sequence ykk∈N⊂Y is said to be b-complete in Z, if the closure Lb(ykk∈N)¯ coincides with Z.

Let us provide the following criterion for b-completeness of the sequence.

Theorem 3.

Sequence ykk∈N⊂Y is b-complete in Z if and only if ∀z∈Z the equality z,yk=0, ∀k∈N, implies z=0.

Proof.

Let the sequence ykk∈N⊂Y be b-complete in Z. Let us show that, ∀z∈Z from the equality z,yk=0, ∀k∈N it follows that z=0. Assume the contrary that is let there exists an element z∈Z such that z,yk=0, ∀k∈N, but z≠0. Then, ∀x∈X and ∀k∈N, we have(7)0=x,z,ykX=bx,yk,zZ. Thus, z is orthogonal to Lb(ykk∈N), and hence, by its continuity, z will be orthogonal to Lb(ykk∈N)¯. Since Lb(ykk∈N)¯ coincides with Z, then, in particular, (z,z)Z=0, and therefore z=0, which contradicts the assumption.

Conversely, let the sequence ykk∈N⊂Y be such that, from the equation z,yk=0, ∀k∈N it follows that z=0. Let us prove its b-completeness in Z. If Lb(ykk∈N)¯ does not coincide with Z, then there exists a nonzero element z0∈Z such that (z,z0)Z=0, ∀z∈Lb(ykk∈N)¯. Then, ∀x∈X, we have(8)0=bx,yk,z0Z=x,z0,ykX,∀k∈N, and, consequently, z0,yk=0, ∀k∈N. Then, z0=0, and this gives a contradiction. Therefore, ykk∈N is b-complete in Z.

The theorem is proved.

Definition 4.

The sequence ykk∈N⊂Y is said to be b-basis in Z, if ∀z∈Z uniquely represented in the form (9)z=∑k=1∞bxk,yk,xk∈X.

In the case of b-basis, ykk∈N is an orthonormal; then, it is called a b-orthonormal b-basis in Z.

It is clear that if the system ykk∈N forms a b-orthonormal b-basis for Z, then ∀z∈Z has a unique representation(10)z=∑k=1∞bz,yk,yk. Indeed, taking into account the continuity of b-scalar product z,y with respect to z, from (9) we obtain(11)z,yj=limn→∞∑k=1nbxk,yk,yj=xj,∀j∈N.

The following theorem is an analogue of the criterion for an orthonormal basis.

Theorem 5.

Let ykk∈N⊂Y be a b-orthonormal sequence. Then, the following conditions are equivalent:

ykk∈N is b-orthonormal b-basis in Z.

ykk∈N is b-complete in Z.

∀z∈Z, the b-Parseval identity zZ2=∑k=1∞z,ykX2 holds.

Proof.

Let condition (1) hold. Then, it is clear that Lb(ykk∈N)¯ coincides with Z; that is, condition (2) is fulfilled. The equivalence of conditions (1) and (3) follows from the relation(12)z-∑k=1nbz,yk,ykZ2=zZ2-∑k=1nz,ykZ2. It remains to show that condition (3) follows from condition (2). For every x1,x2,…,xn∈X, we have(13)z-∑k=1nbxk,ykZ2=z-∑k=1nbxk,yk,z-∑k=1nbxk,ykZ=zZ2-∑k=1nz,yk,xkX-∑k=1nxk,z,ykX+∑k=1nxkX2=zZ2+∑k=1nz,yk-xkX2-∑k=1nz,ykX2≥zZ2-∑k=1nz,ykX2.Hence, it is easy to see that if the system ykk∈N is b-complete in Z, then condition (3) is true.

The theorem is proved.

In theory of frames, often use the following theorem, which describes some properties of the adjoint operator.

Theorem 6.

Let T∈L(X,Z). Then,

T∗∈L(Z,X) and T∗=T;

R(T) is closed if and only if R(T∗) is closed;

T is surjective if and only if ∃A>0:T∗zZ≥czZ, ∀z∈Z.

3. Main Results

Let X and Z be a Hilbert space with scalar products (·,·)X and (·,·)Z, respectively, and let Y be a Banach space with the norm ·Y. Suppose that b:X×Y→Z is bilinear mapping and satisfies condition (3). Consider a sequence of vector ykk∈N from Y.

Definition 7.

Sequence ykk∈N is called a b-frame in Z, if there exist constants A,B>0 such that (14)AzZ2≤∑k=1∞z,ykX2≤BzZ2,∀z∈Z.

Constants A and B are called the bounds of b-frame. When the right-hand side of (14) is fulfilled, then the sequence ykk∈N is called b-Besselian in Z with a bound B.

Let us provide some examples to b-frames.

Example 8.

Let X, Z be Hilbert spaces and Y=L(X,Z). Assume b(x,Λ)=Λ∗(x), where x∈X and Λ∈Y. Let us recall that (see. [16]) the sequence Λkk∈N⊂L(X,Z) is called g-frame in Z, if there exist constants A,B>0 such that(15)AzZ2≤∑k=1∞ΛkzX2≤BzZ2,∀z∈Z. Then, b-frame Λkk∈N⊂Y in Z is g-frame in Z.

Example 9.

Let X, Y be Hilbert spaces and Z=X⊗Y is their Hilbert tensor product. Assume that b(x,y)=x⊗y, where x⊗y is a tensor product of elements x∈X and y∈Y. Then, b-frame ykk∈N⊂Y in Z forms a t-frame for Z [19].

Let us provide a characteristic property of b-Besselian sequence in Z.

Theorem 10.

Sequence ykk∈N⊂Y is b-Besselian in Z with a bound B if and only if the bounded operator T:l2(X)→Z is defined:(16)Txkk∈N=∑k=1∞bxk,ykand T≤B. Moreover, an adjoint operator T∗:Z→l2(X) is determined by T∗(z)=z,ykk∈N.

Proof.

Let ykk∈N be b-Besselian in Z with a bound B. The series ∑k=1∞b(xk,yk) converges for every x→=xkk∈N∈l2(X). Indeed, for m>n, we have(17)zm-znZ=∑k=n+1mbxk,ykZ=supz=1∑k=n+1mbxk,yk,zZ=supz=1∑k=n+1mxk,z,ykX≤supz=1∑k=n+1mxkXz,ykX≤supz=1∑k=n+1mz,ykX21/2∑k=n+1mxkX21/2≤B∑k=n+1mxkX21/2.

Since the right-hand side tends to zero, as n→∞, then the sequence znn∈N satisfies the Cauchy criterion and also converges. So the operator T(xkk∈N)=∑k=1∞b(xk,yk) is defined. It is easy to show that(18)Txkk∈NZ≤Bxkk∈Nl2X. Hence, we get T≤B.

Conversely, suppose that the bounded operator T:l2(X)→Z, T(xkk∈N)=∑k=1∞b(xk,yk) is defined, and T≤B. Let us prove that the system ykk∈N is b-Besselian in Z with a bound B. For arbitrary x→=xkk∈N∈l2(X) and z∈Z, consider (T(xkk∈N),z)Z. We have(19)Txkk∈N,zZ=∑k=1∞bxk,yk,zZ=∑k=1∞xk,z,ykX=xkk∈N,z,ykk∈NX.Hence, we directly obtain an expression for the adjoint operator(20)T∗z=z,ykk∈N,z∈Z. We have(21)∑k=1∞z,ykX2=T∗zl2X2≤T∗2zZ2=T2zZ2≤BzZ2.

The theorem is proved.

Let the sequence ykk∈N⊂Y form a b-frame for Z with the bounds A and B. Then, a bounded operator S:Z→Z is defined by the following formula:(22)Sz=∑k=1∞bz,yk,yk,∀z∈Z. Operator S is called b-frame operator for ykk∈N. Many of the properties of the ordinary frames are valid in this case.

Theorem 11.

Let the sequence ykk∈N⊂Y form a b-frame for Z with bounds A, B and with b-frame operator S. Then S is a positive, self-adjoint, bounded invertible operator and B-1≤S-1≤A-1.

Proof.

For every z∈Z, we have(23)Sz,zZ=∑k=1∞bz,yk,yk,zZ=∑k=1∞z,ykX2; that is, S is a positive operator. Let the operator T:l2(X)→Z be defined by the expression T(xkk∈N)=∑k=1∞b(xk,yk). Then, by Theorem 10, it directly follows that T∗(z)=z,ykk∈N, and therefore S=TT∗. Hence, we obtain that S is a self-adjoint operator. Then, for every z∈Z, the following inequality holds:(24)AzZ2≤∑k=1∞z,ykX2=Sz,zZ≤SzZzZ, and consequently AzZ≤SzZ. Thus, as surjective and injective operator, by Banach theorem, S is a bounded invertible operator. Since A≤S≤B holds, then from the relation(25)AS-1zZ≤SS-1zZ=zZ,zZ=SS-1zZ≤BS-1zZ it follows that B-1≤S-1≤A-1.

Theorem is proved.

Theorem 12.

Sequence ykk∈N⊂Y forms a b-frame for Z if and only if the bounded surjective operator,(26)T:l2X⟶Z,Txkk∈N=∑k=1∞bxk,yk, is defined.

Proof.

Let the sequence ykk∈N form a b-frame for Z and let S be its b-frame operator. Then, it is a b-Besselian in Z and therefore, by Theorem 10, the bounded operator T:l2(X)→Z, T(xkk∈N)=∑k=1∞b(xk,yk), is defined. It remains to show that R(T)=Z. Since TT∗=S and R(S)=Z, then R(T)=Z.

Conversely, suppose that a bounded surjective operator T:l2(X)→Z, T(xkk∈N)=∑k=1∞b(xk,yk) is defined. Let us show that ykk∈N forms a b-frame for Z. According to Theorem 10, the sequence ykk∈N is b-Besselian in Z. Let us take an arbitrary z∈Z. So z=TT+z; then supposing T+z=tkk∈N, we have(27)zZ4=z,zZ2=Ttkk∈N,zZ2=∑k=1∞btk,yk,zZ2=∑k=1∞tk,z,ykX2≤∑k=1∞tkXz,ykX2≤∑k=1∞tkX2∑k=1∞z,ykX2=T+zZ2∑k=1∞z,ykX2≤T+2zZ2∑k=1∞z,ykX2.Thus,(28)1T+2zZ2≤∑k=1∞z,ykX2.

The theorem is proved.

Let us provide the criterion of b-frameness with bounds A and B.

Theorem 13.

Sequence ykk∈N⊂Y forms a b-frame for Z with the bounds A and B if and only if the following conditions are fulfilled:

ykk∈N is b-complete in Z.

The bounded operator,(29)T:l2X⟶Z,Txkk∈N=∑k=1∞bxk,yk,

is defined, such that (30)A∑k=1∞xkX2≤Txkk∈NZ2≤B∑k=1∞xkX2,∀xkk∈N∈NT⊥.Proof.

Let ykk∈N form a b-frame for Z with bounds A and B. Let us prove the b-completeness of the sequence ykk∈N in Z. Assume the contrary; that is, the sequence ykk∈N is not b-complete in Z. Then, by Theorem 3, there exists a nonzero element z0∈Z such that z0,yk=0, ∀k∈N. According to (14), we obtain that z0=0, and hence the assumption is incorrect. Then, from the closedness of R(T), closedness of R(T∗) follows. Therefore, R(T∗)=(N(T))⊥. Let us take an arbitrary element xkk∈N∈(N(T))⊥. Let z∈Z, such that xkk∈N=T∗z=z,ykk∈N. We have(31)∑k=1∞z,ykX22=TT∗z,zZ2≤TT∗zZ2zZ2≤1ATT∗zZ2∑k=1∞z,ykX2.Hence,(32)A∑k=1∞z,ykX2≤TT∗zZ2. On the other hand,(33)TT∗zZ2≤T2T∗zl2X2≤B∑k=1∞z,ykX2.

Conversely, suppose that conditions (1) and (2) hold. Let us show that T is a surjective operator. Indeed, let z∈R(T)¯ and zn∈R(T) such that zn→z as n→∞. Then, there exists x→n∈(N(T))⊥ such that zn=Tx→n. From (2), it is easy to see that the sequence x→n is fundamental and therefore converges to x→∈(N(T))⊥. Considering continuity of an operator T, we obtain zn=Tx→n→Tx→=z; that is, R(T) is closed. Future, from b-completeness of ykk∈N in Z and from Lb(ykk∈N)⊂R(T), it follows that R(T)=Z. As(34)∀x→∈l2X;T+Tx→∈NT⊥, then, from (2), we have(35)AT+Tx→l2X2≤TT+Tx→Z2=Tx→Z2. Consequently, T+≤1/A. We have(36)RT∗+=RT+∗=NT+⊥=RT=Z, and thus ∀z∈Z we obtain z=(T∗)+T∗z and(37)zZ2=T∗+T∗zZ2≤T∗+2T∗zl2X2=T+∗2T∗zl2X2=T+2T∗zl2X2≤1AT∗zl2X2=1A∑k=1∞z,ykX2.

The theorem is proved.

The following property of frames is a generalization of the similar properties with respect to the Riesz basis.

Theorem 14.

Let the sequence ekk∈N⊂Y form a b-orthonormal b-basis for Z. Then, the system ykk∈N⊂Y forms a b-frame for Z if and only if there exists a bounded surjective operator U:Z→Z such that U(b(x,ek))=b(x,yk), ∀x∈X and ∀k∈N.

Proof.

Let ykk∈N form a b-frame for Z. By Theorem 12, the bounded surjective operator T:l2(X)→Z is defined:(38)Txkk∈N=∑k=1∞bxk,yk. Consider the operator Φ:Z→l2(X) by the formula Φ(z)=z,ekk∈N. It is clear that Φ(b(x,ej))=δjkxk∈N. Let us define the operator U:Z→Z as follows: U=TΦ. It is obvious that U is a bounded surjective operator and(39)Ubx,ek=TΦbx,ek=Tδjkxk∈N=bx,yk,∀x∈X,∀k∈N is fulfilled.

Conversely, suppose that there is a bounded surjective operator U:Z→Z, such that U(b(x,ek))=b(x,yk), ∀x∈X, ∀k∈N. Let us show that ykk∈N is a b-frame in Z. Let z∈Z be an arbitrary element. We have(40)z,yk=U∗z,ek.Indeed, ∀x∈X we have(41)x,z,ykX=bx,yk,zZ=Ubx,ek,zZ=bx,ek,U∗zZ=x,U∗z,ekX,and therefore holds (40). Then,(42)∑k=1∞z,ykX2=∑k=1∞U∗z,ekX2=U∗zl2X2.By virtue of the surjectivity of U, there exists A>0 such that U∗zl2(X)≥AzZ. On the other hand, U∗zl2(X)≤UzZ. Thus, from (42) it follows that ykk∈N forms a b-frame for Z.

Theorem is proved.

Let the sequence ykk∈N⊂Y form a b-frame for Z and consider the operator-matrix U=(ulk), i,k∈N, where ulk∈L(X,X). Assume that the system φll∈N⊂Y satisfies the following condition:(43)z,φl=∑k=1∞ulkz,yk.

The following theorem is true, which of an independent interest.

Theorem 15.

Let the sequence ykk∈N⊂Y form a b-frame for Z with bounds A, B and a b-frame operator S; φll∈N⊂Y forms a b-frame for Z with bounds A1 and B1. If ulk(x)=S-1(b(x,yk)),φl, i,k∈N, then U=(ulk) is a bounded operator in l2(X).

Proof.

Take ∀x→=xkk∈N∈l2(X).

We have(44)Ux→l2X2=∑l=1∞∑k=1∞ulkxkX2=∑l=1∞∑k=1∞S-1bxk,yk,φlX2=∑l=1∞∑k=1∞S-1bxk,yk,φlX2≤B1∑k=1∞S-1bxk,ykZ2=B1supz=1∑k=1∞S-1bxk,yk,zZ2=B1supz=1∑k=1∞xk,S-1z,ykX2≤B1supz=1∑k=1∞xkXS-1z,ykX2≤B1supz=1∑k=1∞xkX2∑k=1∞S-1z,ykX2≤B1S-12x→l2X2.Thus, the operator U is bounded and U≤B1S-1.

The theorem is proved.

The following theorem in a certain sense is the inverse of Theorem 15.

Theorem 16.

Let the sequence ykk∈N⊂Y form a b-frame for Z with bounds A and B, let U=(ulk) be a bounded operator in l2(X), and the sequence φll∈N⊂Y is such that the relation (43) holds. Then, φll∈N forms a b-frame for Z if and only if (45)∃c>0:∑l=1∞z,φlX2≥c∑k=1∞z,ykX2,∀z∈Z.

Proof.

Let φll∈N be a b-frame in Z with the bounds A1 and B1. Then, ∀z∈Z, we have(46)∑l=1∞z,φlX2≥A1zZ2≥A1B∑k=1∞z,ykX2.

Conversely, suppose that inequality (45) holds. Let us show that φll∈N is b-frame in Z. It is clear that(47)Uz,ykk∈N=z,φll∈N,∀z∈Z. Consequently,(48)∑l=1∞z,φlX2=Uz,ykk∈Nl2X2≤U2∑k=1∞z,ykX2≤U2BzZ2.Then, according to (45) and (14), we obtain(49)∑l=1∞z,φlX2≥c∑k=1∞z,ykX2≥cAzZ2; that is, φll∈N forms a b-frame for Z.

The theorem is proved.

The following theorem establishes a relationship between the frames and systems of systems ykk∈N and φll∈N.

Theorem 17.

Let the sequence ykk∈N⊂Y form a b-frame for Z with bounds A and B, the operators ulk∈L(X,X), i,k∈N, are such that the inequality ulk(x)X≥alkxX is valid, alk≥0, ∀x∈X, and the following conditions are fulfilled:(50)b=supk∑j=1∞∑l=1∞ulj∗ulk<+∞;a=infk∑l=1∞alk2-∑j≠k∑l=1∞ulj∗ulk>0. Then, φll∈N is a b-frame in Z with bounds aA and bB.

Proof.

Take ∀z∈Z. Firstly, let us prove that φll∈N is b-Besselian in Z with a bound bB. We have(51)∑l=1∞z,φlX2=∑l=1∞∑k=1∞ulkz,ykX2=∑l=1∞∑k=1∞ulkz,yk,∑j=1∞uljz,yjZ=∑l=1∞∑k=1∞∑j=1∞ulj∗ulkz,yk,z,yjZ=∑k=1∞∑j=1∞∑l=1∞ulj∗ulkz,yk,z,yjZ≤∑k=1∞∑j=1∞∑l=1∞ulj∗ulkz,ykXz,yjX≤∑k=1∞∑j=1∞∑l=1∞ulj∗ulkz,ykX21/2∑k=1∞∑j=1∞∑l=1∞ulj∗ulkz,yjX21/2=∑k=1∞∑j=1∞∑l=1∞ulj∗ulkz,ykX2≤b∑k=1∞z,ykX2≤bBzZ2.Now, let us establish for φll∈N the left-hand side of inequality (14). We obtain(52)∑l=1∞z,φlX2=∑l=1∞∑k=1∞ulkz,ykX2=∑l=1∞∑k=1∞ulkz,yk,∑j=1∞uljz,yjZ=∑l=1∞∑k=1∞ulkz,ykX2+∑l=1∞∑k=1∞∑j≠k∞ulj∗ulkz,yk,z,yjZ=I1+I2.Consider I1 and I2, separately. We have(53)I1≥∑l=1∞∑k=1∞alk2z,ykX2,I2=∑k=1∞∑j≠k∞∑l=1∞ulj∗ulkz,yk,z,yjZ≤∑k=1∞∑j≠k∞∑l=1∞ulj∗ulkz,ykXz,yjX≤∑k=1∞∑j≠k∞∑l=1∞ulj∗ulkz,ykX21/2∑k=1∞∑j≠k∞∑l=1∞ulj∗ulkz,yjX21/2=∑k=1∞∑j≠k∞∑l=1∞ulj∗ulkz,ykX2.Consequently,(54)∑l=1∞z,φlX2≥∑l=1∞∑k=1∞alk2z,ykX2-∑k=1∞∑j≠k∞∑l=1∞ulj∗ulkz,ykX2≥a∑k=1∞z,ykX2≥aAzZ2.

The theorem is proved.

In particular, in the case of g-frame, we obtain the following result.

Theorem 18.

Let the sequence Λkk∈N⊂L(X,Z) form a g-frame for Z with bounds A and B, the operators ulk∈L(X,X), i,k∈N, are such that ulk(x)X≥alkxX, alk≥0, ∀x∈X, and the following condition is fulfilled:(55)b=supk∑j=1∞∑l=1∞ulj∗ulk<+∞;a=infk∑l=1∞alk2-∑j≠k∑l=1∞ulj∗ulk>0. Then, the bounded operators Φl=∑k=1∞ulkΛk are defined in X and Φll∈N are g-frames in Z with bounds aA and bB.

Competing Interests

The authors declare that they have no competing interests.

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