# Woven Frames in Quaternionic Hilbert Spaces

Authors: S. K. Sharma; Nitin Sharma; Khole Timothy Poumai
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, we introduce and study woven frames in quaternionic Hilbert spaces. We also give some properties of woven frames and give some conditions on family of frames under which it is woven in quaternionic Hilbert spaces. Also, a characterization of weaving frames in terms of a surjective-bounded right linear operator is given.


---

## Body

## 1. Introduction and Preliminaries

In 1952, Duffin and Schaeffer [1] defined frames as follows:“ifℋ is a Hilbert space, then unn∈ℕ⊂ℋ is said to be a frame for ℋ if there exist finite constants cl and cu with 0<cl≤cu such that(1)clu2≤∑n∈ℕu,un2≤cuu2,for allu∈ℋ.”The positive constantscl and cu are called lower and upper frame bounds for the frame unn∈ℕ, respectively. Inequality (1) is called the frame inequality for the frame unn∈ℕ. A frame unn∈ℕ in ℋ is said to be(i)tight: if it is possible to choose cl=cu(ii)Parseval: if it is a tight frame with cl=cu=1Frames have many properties of bases, but lack a very important one, namely, uniqueness. In fact, a frame is an overcomplete sequence of vectors. Members of a frame need not be linearly independent, but still serve as building blocks and are able to recover every vector of the space completely. This property of frames turns out to be very useful particularly in image and signal processing and in communications in general, since this redundancy yields robustness, i.e., less sensitivity to truncation and transmission errors. Frames can be approached in different ways. Two most important ways are to consider frame theory as a branch of functional analysis, which belongs to the branch of pure mathematics, and to consider a class of frames, which is best suited in applications, which lies in the branch of applied mathematics.Dependencies among the coefficients of the overcomplete representations guarantee a better stability in presence of noise, quantization, and erasures, as well as greater freedom of design comparing to the bases. The redundant counterpart of a basis is called a frame. The role played by redundancy varies with specific applications. One important role is its robustness. That is, by spreading our information over a wider range of vectors, we are more capable to sustain losses and still have accurate reconstruction.Frame work is an important tool in the study of signal and image processing [2], filter bank theory [3], wireless communications [4], and sigma-delta quantization [5], refer [4, 6–12] for more literature, applications, and various generalizations of frames in Hilbert spaces.


### 1.1. Notations

Throughout this paper, we will denoteQ to be the noncommutative field of quaternions, ℕ be the set of natural numbers, and ℍRQ be a separable right quaternionic Hilbert space. By the term “right linear operator,” we mean a “right Q-linear operator,” and BℍRQ denotes the set of all bounded (right Q-linear) operators of ℍRQ and ℕm denotes the set of first m natural numbers.


### 1.2. Quaternionic Hilbert Spaces

The noncommutative field of quaternionsQ is a four-dimensional real algebra with unity. In Q, 0 denotes the null element and 1 denotes the identity with respect to multiplication. It also includes three so-called imaginary units, denoted by i,j,andk, i.e.,(2)Q=r0+r1i+r2j+r3k:r0,r1,r2,r3∈ℝ,where i2=j2=k2=−1,ij=−ji=k,jk=−kj=i, and ki=−ik=j. For each q=r0+r1i+r2j+r3k∈Q, define conjugate of q denoted by q¯ as q¯=r0−r1i−r2j−r3k∈Q. If q=r0+r1i+r2j+r3k is a quaternion, then r0 is called the real part of q and r1i+r2j+r3k is called the imaginary part of q. The modulus of a quaternion q=r0+r1i+r2j+r3k is defined as(3)q=q¯q1/2=qq¯1/2=r02+r12+r22+r32.Definition 1.
A right quaternionic pre-Hilbert space or right quaternionic inner product spaceVRQ is a right quaternionic vector space together with the binary mapping .|.:VRQ×VRQ⟶Q (called the Hermitian quaternionic inner product) which satisfies the following properties:(a)
v1|v2¯=v2|v1 for all v1,v2∈VRQ(b)
v|v>0ifv≠0(c)
v|v1+v2=v|v1+v|v2 for all v,v1,v2∈VRQ(d)
v|uq=v|uq for all v,u∈VRQ and q∈Q
A right quaternionic inner product spaceVRQ also has the following properties:(i)
vq|u=q¯v|u for all v,u∈VRQ and q∈Q(ii)
v1p+v2q∈VRQ, for all v1,v2∈VRQ and p,q∈Q
LetVRQ be right quaternionic inner product space with the Hermitian inner product .|.. Define the quaternionic norm .:VRQ⟶ℝ+ on VRQ by(4)u=u|u,u∈VRQ.Definition 2.
The right quaternionic pre-Hilbert space is called a right quaternionic Hilbert space if it is complete with respect to norm (4) and is denoted by ℍRQ.Theorem 1 (the Cauchy–Schwarz inequality, see [13]).
IfℍRQ is a right quaternionic Hilbert space, then(5)u|v2≤u|uv|v,for allu,v∈ℍRQ.
For more literature related to quaternionic Hilbert spaces, one may refer to [13, 14].


### 1.3. Frames in Quaternionic Hilbert Spaces

Khokulan et al. [15] introduced and studied frames for finite-dimensional quaternionic Hilbert spaces, which were further studied in [16]. Recently, Sharma and Goel [17] introduced and studied frames for separable quaternionic Hilbert spaces. Frames in separable right quaternionic Hilbert spaces ℍRQ are defined as follows.Definition 3.
(see [17]). Let ℍRQ be a right quaternionic Hilbert space and uii∈ℕ be a sequence in ℍRQ. Then, uii∈ℕ is said to be a frame for ℍRQ if there exist two finite real constants with 0<r1≤r2 such that(6)r1u2≤∑i∈ℕui|u2≤r2u2,fo allu∈ℍRQ.
The positive constantsr1 and r2 are called lower frame and upper frame bounds for the frame uii∈ℕ, respectively. Inequality (6) is called frame inequality for the frame uii∈ℕ. A sequence uii∈ℕ is called a Bessel sequence for the right quaternionic Hilbert space ℍRQ with bound r2 if uii∈ℕ satisfies the right-hand side of inequality (6). A sequence uii∈ℕ is a tight frame for right quaternionic Hilbert space ℍRQ if there exist positive r1 and r2 satisfying inequality (6) with r1=r2, Parseval frame if it is tight with r1=r2=1, and exact if it ceases to be a frame in case any one of its element is removed.
Ifuii∈ℕ is a frame for ℍRQ, then the right linear operator T:ℓ2Q⟶ℍRQ defined by(7)Tqii∈ℕ=∑i∈ℕuiqi,qi∈ℓ2Q,is called the (right) synthesis operator and the adjoint operator T∗ is called the (right) analysis operator is given by(8)T∗u=ui|ui∈ℕ,u∈ℍRQ.
Also, the (right) frame operatorS:ℍRQ⟶ℍRQ for the frame uii∈ℕ is a right linear operator given by(9)Su=TT∗u=Tui|ui∈ℕ=∑i∈ℕuiui|u,u∈ℍRQ.


### 1.4. Woven Frames in Hilbert Spaces

Woven frame for separable Hilbert spaces were introduced and studied by Bemrose et al. [18]. They gave the following definition:Definition 4.
(see [18]). A family of frames F=uiji∈ℕ:j∈ℕm is said to be woven frame for a Hilbert space ℋ if there are universal constants cl and cu such that, for every partition P=σjj∈ℕm of ℕ, the family FP=uiji∈σj,j∈ℕm is a frame for ℋ with lower and upper frame bounds cl and cu, respectively.
Distributed signal processing is the main motivation behind the concept of woven frames. Furthermore, woven frames have inherent applications in wireless sensor networks which are based upon distributed processing under family of frames. For more literature on woven frame, one may refer to [19–21].


### 1.5. Outline of the Paper

In this paper, we introduce and study woven frames in quaternionic Hilbert spaces. Example is given in support of its existence. We also discuss some properties of woven Bessel family and woven frames. Some conditions under which a family of frames is woven in quaternionic Hilbert spaces are given. Furthermore, a characterization of weaving frames in terms of a surjective-bounded right linear operator is given.


## 1.1. Notations

Throughout this paper, we will denoteQ to be the noncommutative field of quaternions, ℕ be the set of natural numbers, and ℍRQ be a separable right quaternionic Hilbert space. By the term “right linear operator,” we mean a “right Q-linear operator,” and BℍRQ denotes the set of all bounded (right Q-linear) operators of ℍRQ and ℕm denotes the set of first m natural numbers.


## 1.2. Quaternionic Hilbert Spaces

The noncommutative field of quaternionsQ is a four-dimensional real algebra with unity. In Q, 0 denotes the null element and 1 denotes the identity with respect to multiplication. It also includes three so-called imaginary units, denoted by i,j,andk, i.e.,(2)Q=r0+r1i+r2j+r3k:r0,r1,r2,r3∈ℝ,where i2=j2=k2=−1,ij=−ji=k,jk=−kj=i, and ki=−ik=j. For each q=r0+r1i+r2j+r3k∈Q, define conjugate of q denoted by q¯ as q¯=r0−r1i−r2j−r3k∈Q. If q=r0+r1i+r2j+r3k is a quaternion, then r0 is called the real part of q and r1i+r2j+r3k is called the imaginary part of q. The modulus of a quaternion q=r0+r1i+r2j+r3k is defined as(3)q=q¯q1/2=qq¯1/2=r02+r12+r22+r32.Definition 1.
A right quaternionic pre-Hilbert space or right quaternionic inner product spaceVRQ is a right quaternionic vector space together with the binary mapping .|.:VRQ×VRQ⟶Q (called the Hermitian quaternionic inner product) which satisfies the following properties:(a)
v1|v2¯=v2|v1 for all v1,v2∈VRQ(b)
v|v>0ifv≠0(c)
v|v1+v2=v|v1+v|v2 for all v,v1,v2∈VRQ(d)
v|uq=v|uq for all v,u∈VRQ and q∈Q
A right quaternionic inner product spaceVRQ also has the following properties:(i)
vq|u=q¯v|u for all v,u∈VRQ and q∈Q(ii)
v1p+v2q∈VRQ, for all v1,v2∈VRQ and p,q∈Q
LetVRQ be right quaternionic inner product space with the Hermitian inner product .|.. Define the quaternionic norm .:VRQ⟶ℝ+ on VRQ by(4)u=u|u,u∈VRQ.Definition 2.
The right quaternionic pre-Hilbert space is called a right quaternionic Hilbert space if it is complete with respect to norm (4) and is denoted by ℍRQ.Theorem 1 (the Cauchy–Schwarz inequality, see [13]).
IfℍRQ is a right quaternionic Hilbert space, then(5)u|v2≤u|uv|v,for allu,v∈ℍRQ.
For more literature related to quaternionic Hilbert spaces, one may refer to [13, 14].


## 1.3. Frames in Quaternionic Hilbert Spaces

Khokulan et al. [15] introduced and studied frames for finite-dimensional quaternionic Hilbert spaces, which were further studied in [16]. Recently, Sharma and Goel [17] introduced and studied frames for separable quaternionic Hilbert spaces. Frames in separable right quaternionic Hilbert spaces ℍRQ are defined as follows.Definition 3.
(see [17]). Let ℍRQ be a right quaternionic Hilbert space and uii∈ℕ be a sequence in ℍRQ. Then, uii∈ℕ is said to be a frame for ℍRQ if there exist two finite real constants with 0<r1≤r2 such that(6)r1u2≤∑i∈ℕui|u2≤r2u2,fo allu∈ℍRQ.
The positive constantsr1 and r2 are called lower frame and upper frame bounds for the frame uii∈ℕ, respectively. Inequality (6) is called frame inequality for the frame uii∈ℕ. A sequence uii∈ℕ is called a Bessel sequence for the right quaternionic Hilbert space ℍRQ with bound r2 if uii∈ℕ satisfies the right-hand side of inequality (6). A sequence uii∈ℕ is a tight frame for right quaternionic Hilbert space ℍRQ if there exist positive r1 and r2 satisfying inequality (6) with r1=r2, Parseval frame if it is tight with r1=r2=1, and exact if it ceases to be a frame in case any one of its element is removed.
Ifuii∈ℕ is a frame for ℍRQ, then the right linear operator T:ℓ2Q⟶ℍRQ defined by(7)Tqii∈ℕ=∑i∈ℕuiqi,qi∈ℓ2Q,is called the (right) synthesis operator and the adjoint operator T∗ is called the (right) analysis operator is given by(8)T∗u=ui|ui∈ℕ,u∈ℍRQ.
Also, the (right) frame operatorS:ℍRQ⟶ℍRQ for the frame uii∈ℕ is a right linear operator given by(9)Su=TT∗u=Tui|ui∈ℕ=∑i∈ℕuiui|u,u∈ℍRQ.


## 1.4. Woven Frames in Hilbert Spaces

Woven frame for separable Hilbert spaces were introduced and studied by Bemrose et al. [18]. They gave the following definition:Definition 4.
(see [18]). A family of frames F=uiji∈ℕ:j∈ℕm is said to be woven frame for a Hilbert space ℋ if there are universal constants cl and cu such that, for every partition P=σjj∈ℕm of ℕ, the family FP=uiji∈σj,j∈ℕm is a frame for ℋ with lower and upper frame bounds cl and cu, respectively.
Distributed signal processing is the main motivation behind the concept of woven frames. Furthermore, woven frames have inherent applications in wireless sensor networks which are based upon distributed processing under family of frames. For more literature on woven frame, one may refer to [19–21].


## 1.5. Outline of the Paper

In this paper, we introduce and study woven frames in quaternionic Hilbert spaces. Example is given in support of its existence. We also discuss some properties of woven Bessel family and woven frames. Some conditions under which a family of frames is woven in quaternionic Hilbert spaces are given. Furthermore, a characterization of weaving frames in terms of a surjective-bounded right linear operator is given.


## 2. Woven Frames in Quaternionic Hilbert Spaces

We begin this section with the following definition of woven frames in quaternionic Hilbert spaces.Definition 5.
Letℕm be the set of first m natural numbers, ℍRQ be a right quaternionic Hilbert space, and F=uiji∈ℕ:j∈ℕm be a family of frames for ℍRQ. Then, F is said to be woven if there are universal positive real numbers r1 and r2 so that, for every partition P=σjj∈ℕm of ℕ, the family FP=uiji∈σj,j∈ℕm is a frame for ℍRQ with lower and upper frame bounds r1 and r2, respectively. Each family FP is called a weaving. If every weaving is a Bessel sequence, then F is called a woven Bessel sequence for ℍRQ.
Next, we give the following lemma:Lemma 1.
Letvii∈ℕ be a frame for ℍRQ with frame bounds r1 and r2 and qii∈ℕ⊂Q be a sequence such that r3<infi∈ℕ|qi|<supi∈ℕ|qi|<r4. Define a family F=uiji∈ℕ=jviqii∈ℕ:j∈ℕm. Then, F is a woven frame for ℍRQ.Proof.
For any partitionP=σjj∈ℕm of ℕ, we have(10)r1r32u2≤∑i∈ℕviqi|u2≤∑j∈ℕm∑i∈σjjviqi|u2≤∑j∈ℕm∑i∈σjuij|u2≤m2∑j∈ℕm∑i∈σjviqi|u2≤m2r2r42u2,u∈ℍRQ.
Thus,F is a woven frame for ℍRQ with lower and upper frame bounds r1r32 and m2r2r42, respectively.
Next, we generalize Lemma1 in the form of the following theorem.Theorem 2.
LetF=uiji∈ℕ:j∈ℕm be a woven frame for ℍRQ with universal frame bounds r1 and r2 and Q=qiji∈ℕ:j∈ℕm be a family of sequences in Q such that 0<r3≤infqij2≤supqij2≤r4, j∈ℕm. Then, FQ=uijqiji∈ℕ:j∈ℕm is a woven frame for ℍRQ.Proof.
For any partitionP=σjj∈ℕm of ℕ, we have(11)r1r3u2≤r3∑j∈ℕm∑i∈σjuij|u2≤∑j∈ℕm∑i∈σj|uijqij|u|2≤r4∑j∈ℕm∑i∈σjuij|u2≤r2r4u2,u∈ℍRQ.
Thus,FQ is a woven frame for ℍRQ with universal lower and upper frame bounds r1r3 and r2r4, respectively.
Next, we prove that any finite family of Bessel sequences in a quaternionc Hilbert space is a family of woven Bessel family.Theorem 3.
LetF=uiji∈ℕ:j∈ℕm be a family of Bessel sequences for ℍRQ with Bessel bounds rj, j∈ℕm. Then, F is woven Bessel sequence with universal Bessel bound ∑j∈ℕmrj.Proof.
For any partitionP=σjj∈ℕm of ℕ, we have(12)∑j∈ℕm∑i∈σjuij|u2≤∑j∈ℕm∑i∈ℕuij|u2≤∑j∈ℕmrju2,u∈ℍRQ.
Thus,F is woven Bessel sequence with universal Bessel bound ∑j∈ℕmrj.
LetP=σjj∈ℕm be any partition of ℕ, and define the space(13)⊕j∈ℕmℓ2Qσj=qiji∈σj,j∈ℕm⊂Q:∑j∈ℕm∑i∈σjqij2<∞.
Then,⊕j∈ℕmℓ2Qσj is a right quaternionic Hilbert space with the quaternionic inner product:(14)piji∈σj,j∈ℕm|qiji∈σj,j∈ℕm=∑j∈ℕm∑i∈σjpijqij¯.
LetF=Uj=uiji∈ℕ:j∈ℕm be a woven frame for ℍRQ, for any partition P=σjj∈ℕm of ℕ, then FP=uiji∈σj,j∈ℕm is a frame for ℍRQ and the operator TFP:⊕j∈ℕmℓ2Qσj⟶ℍRQ defined by(15)TFPqiji∈σj,j∈ℕm=∑j∈ℕm∑i∈σjuijqij∈ℍRQ,is the synthesis operator of frame FP. The adjoint operator TFP∗:ℍRQ⟶⊕j∈ℕmℓ2Qσj of TFP is given by(16)TFP∗u=uij|ui∈σj,j∈ℕm,u∈ℍRQ.
In view of the above discussion, if we considerDσj to be a diagonal matrix such that(17)dii=1,ifi∈σj,0,otherwise,then one can observe that(18)TFPqiji∈σj,j∈ℕm=∑j∈ℕmTUjDσjqiji∈σj,j∈ℕm,TFP∗u=∑j∈ℕmDσjTUj∗u,where TUj is the synthesis operator of the frame Uj.
The frame operatorSFP:ℍRQ⟶ℍRQ of the frame FP is given by(19)SFPu=TFPTFP∗u=TFPuij|ui∈σj,j∈ℕm=∑j∈ℕm∑i∈σjuijuij|u,u∈ℍRQ.
One may easily verify that, for any partitionP, the frame operator SFP is positive, self-adjoint, and invertible.
In the next result, we give a characterization of weaving tight frame.Theorem 4.
LetF=Uj=uiji∈ℕ:j∈ℕ2 be a woven frame for ℍRQ, where TUj is the synthesis operator of the frame Uj, j∈ℕ2. For any partition P=σ1,σ2=ℕ∼σ1, the weaving ui1i∈σ1∪ui2i∈σ2 is a tight frame for ℍRQ with the frame bound r if and only if ∑j∈ℕ2TUjDσjTUj∗=rℐℍRQ, where TUj is the synthesis operator of the frame Uj and Dσj, a diagonal matrix such that dii=1,ifi∈σj,0,otherwise.Proof.
For any partitionP=σjj∈ℕ2 of ℕ, the synthesis operator of the tight frame U=ui1i∈σ1∪ui2i∈σ2 is ∑j∈ℕ2TUjDj. Therefore, the frame operator of SU is(20)rℐℍRQ=SU=∑j∈ℕ2TUjDj∑j∈ℕ2TUjDj∗=∑j∈ℕ2TUjDσjTUj∗.
In the following, we construct a woven frame with the help of a given woven frame.Theorem 5.
LetF=Uj=uiji∈ℕ:j∈ℕ2 be a woven frame for ℍRQ with universal frame bounds r1 and r2 and SUj be the frame operators for the frames Uj, j∈ℕ2. If, for j∈ℕ2, SUj−1SU1−SU2<r1/r2, then G=SUj−1Uj=SUj−1uiji∈ℕ:j∈ℕ2 is also a woven frame for ℍRQ.Proof.
LetSU2−1SU1−SU2<r1/r2. For any partition P=σjj∈ℕ2 of ℕ and u∈ℍRQ, we have(21)∑j∈ℕ2∑i∈σjSUj−1uij|u21/2=∑j∈ℕ2∑i∈σjuij|SUj−1u21/2=∑i∈σ1ui1|SU1−1u2+∑i∈σ2ui2|SU1−1+SU2−1−SU1−1u21/2≥∑i∈σ1ui1|SU1−1u2+∑i∈σ2ui2|SU1−1u21/2−∑i∈σ2ui2|SU2−1−SU1−1u21/2≥r1SU1−1u−∑i∈ℕui2|SU2−1−SU1−1u21/2≥r1SU1−1u−r2SU2−1−SU1−1u≥r1SU1−1−r2SU2−1−SU1−1u.
Also,(22)∑j∈ℕ2∑i∈σjSUj−1uij|u2≤∑j∈ℕ2∑i∈ℕSUj−1uij|u2≤r2∑j∈ℕ2SUj2u2.
Next, we give a characterization for the existence of the woven frame in terms of bounded surjective right linear operator.Theorem 6.
LetF=uiji∈ℕ:j∈ℕm be a family of woven frames for ℍRQ with universal frame bounds r1 and r2 and T be a bounded right linear operator on ℍRQ. Then, TF=Tuiji∈ℕ:j∈ℕm is a family of woven frames for ℍRQ if and only if T is surjective.Proof.
AsTF=Tuiji∈ℕ:j∈ℕm is a family of woven frames for ℍRQ, therefore, for the partition P=σjj∈ℕm,(23)∑j∈ℕm∑i∈σjTuijTuij|u=TSPT∗u,u∈ℍRQ,where SP be the frame operator for the frame uiji∈σj,j∈ℕm. Therefore, TSPT∗ is the frame operator of Tuiji∈σj,j∈ℕm, so TSPT∗ is invertible and T is surjective.
Conversely, letT is surjective. Then, ranT=ℍRQ is closed. Therefore, by Theorem 2.19 in [13], kerT⊕ranT∗=ℍRQ=kerT∗⊕ranT. Now, we shall show that TT∗ is bijective.
IfTT∗u=0,u∈ℍRQ, then T∗u∈kerT∩ranT∗=0. Therefore, T∗u=0. Now, let u∈kerT∗=ranT⊥=ℍRQ⊥=0. This give u=0. Thus, TT∗ is injective.
AsTT∗ is a positive invertible operator on ℍRQ, therefore,(24)0≤TT∗−1≤TT∗−1ℐℍRQ.
So, we have(25)TT∗−TT∗−1−1ℐℍRQ≥0.
This gives(26)T∗u2≥TT∗−1−1u2,u∈ℍRQ.
Thus, for any partitionP=σjj∈ℕm of ℕ, we have(27)r1TT∗−1−1u2≤r1T∗u2≤∑j∈ℕm∑i∈σjuij|T∗u2=∑j∈ℕm∑i∈σjTuij|u2≤r2T2u2,u∈ℍRQ.
Hence,TF=Tuiji∈ℕ:j∈ℕm is a family of woven frames for ℍRQ with universal lower and upper frame bounds r1TT∗−1−1 and r2T2, respectively.Corollary 1.
LetF=uiji∈ℕ:j∈ℕm be a family of woven frames for ℍRQ. Then, one of the frame in the family F is considered to be Parseval by considering the family SλF, where Sλ is the frame operator of the frame uiλi∈ℕ, for some λ∈ℕm.
Next, we give a condition under which a subfamily from a woven frame is removed so that remaining family is a woven frame.Theorem 7.
LetF=uiji∈ℕ:j∈ℕm be a family of woven frames for ℍRQ with universal lower and upper frame bounds r1 and r2, respectively. If J⊂ℕ and for some λ∈ℕm, there exists positive real constant r3<r1 such that(28)∑i∈Juiλ|u2≤r3u2,u∈ℍRQ.
Then,Fℕ,J=uiji∈ℕ,J:j∈ℕm is a family of woven frames for ℍRQ.Proof.
For any partitionP=ηjj∈ℕm of ℕ,J, we have(29)∑j∈ℕm∑i∈ηjuij|u2≤∑j∈ℕm∑i∈ηj∪Juij|u2≤r2u2,u∈ℍRQ.
Also,(30)∑j∈ℕm∑i∈ηjuij|u2=∑i∈ηλ∪Juiλ|u2−∑i∈Juiλ|u2+∑j∈ℕmj≠λ∑i∈ηjuij|u2≥r1−r3u2,u∈ℍRQ.
Hence,Fℕ,J is a family of woven frames for ℍRQ.
In the next result, we give a positive right linear operator with the help of family of frames, whose existence ensures the weaving of frame family.Theorem 8.
LetF=uiji∈ℕ:j∈ℕm be a family of frames for ℍRQ with frame bounds r1j and r2j, j∈ℕm. For any J⊂ℕ and fix λ∈ℕm, let(31)Tju=∑i∈Juijuij|u−∑i∈Juiλuiλ|u,j∈ℕm,j≠λ.
IfTj is a positive right linear operator, then F is a woven family.Proof.
LetP=ηjj∈ℕm be a partition of ℕ. Then, for any u∈ℍRQ, we have(32)r1λu2≤∑i∈ℕuiλ|u2=∑i∈η1uiλ|u2+⋯+∑i∈ηjuiλ|u2+⋯+∑i∈ηmuiλ|u2=∑i∈η1uiλ|u2+⋯+u|∑i∈ηjuiλuiλ|u+⋯+∑i∈ηmuiλ|u2≤∑i∈η1uiλ|u2+⋯+u|∑i∈ηjuijuij|u−Tju+⋯+∑i∈ηmuiλ|u2≤∑i∈η1uiλ|u2+⋯+u|∑i∈ηjuijuij|u+⋯+∑i∈ηmuiλ|u2=∑i∈η1uiλ|u2+⋯+∑i∈ηjuij|u2+⋯+∑i∈ηmuiλ|u2≤∑i∈η1uij|u2+⋯+∑i∈ηjuij|u2+⋯+∑i∈ηmuij|u2≤∑j∈ℕmr1ju2.
Hence,F is a family of woven frames for ℍRQ with universal lower and upper frame bounds r1λ and ∑j∈ℕmr1j, respectively.



---

*Source: 6694892-2021-02-12.xml*