Abstract
In the present paper, we discuss some properties of K-frames in quaternionic Hilbert spaces such as the invertibility of the frame operator as well as the interchangeability of two Bessel sequences. Further, we propose several approaches to construct K-frames and we show that a T-frame can be constructed from a K-frame by the perturbation of a bounded linear operator T. Finally, we study the stability of K-frames under some perturbations.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The theory of frames, dates backs to 1952, was introduced in the pioneeristic paper of Duffin and Schaeffer [11] in the context of nonharmonic Fourier series and popularized later by Daubechies et al. [10] in 1986. A frame is a countable family of elements in a separable Hilbert space which allows stable and not necessarily unique decompositions of arbitrary elements in an expansion of frame elements. It can be viewed as a generalization of Riesz basis intensely studied in literature [4, 5, 12, 13, 18]. The redundancy and flexibility offered by frames have spurred their applications in a variety of areas such as wavelet and frequency analysis theories, filter bank theory, signal and image processing.
In the last two decades, the theory of frames has been a focus of study for several authors such as Christensen [8], Jeribi [18] and Young [22]. Further, many generalizations of this notion have been introduced and developed in numerous paper as [2, 3, 6, 14, 15, 21].
Mainly in [15], Găvruţa introduced the concept of K-frames to study the atomic systems with respect to a bounded linear operator K in separable Hilbert spaces. This generalization allows to write every element of the range of K, which need not be closed, as a superposition of elements which do not necessarily belong to the range of K. However, since a K-frame may not be in general a frame, unless if \(K=I\), many properties of ordinary frames such as the surjectivity of the corresponding synthesis operator, the isomorphism of the frame operator, the interchangeability of the alternate dual reconstruction pair; may not hold for K-frames as shown in [21]. Several methods to construct K-frames and the stability of perturbations for the K-frames have been discussed in [17, 19].
Recently, Charfi and Ellouz [7] extend the concepts of atomic systems for operators and K-frames in separable complex Hilbert spaces to separable quaternionic Hilbert spaces. They have studied the existence of atomic systems for operators and they gave a formal definition of K-frames. Moreover, they have characterized them. These concepts lead to a generalization of frames which were recently studied in [20] and allow us to reconstruct elements from the range of a linear and bounded operator in a quaternionic Hilbert space.
In the present paper, we investigate some properties of K-frames in quaternionic Hilbert spaces. More precisely, we will focus on the study of the isomorphism of the frame operator as well as the interchangeability of the two Bessel sequences in the alternate dual reconstruction for K-frames. Further, we give several approaches for constructing new K-frames from given ones. Finally, we discuss the stability of perturbations for the K-frames.
Throughout this paper, we will adopt such notations. \(V_{R}(\mathfrak {Q})\) is a separable right quaternionic Hilbert space; I is the identity operator for \(V_{R}(\mathfrak {Q})\); \(\mathcal{L}(V_{R}(\mathfrak {Q}),V_{1,R}(\mathfrak {Q}))\) is a collection of all bounded linear operators from \(V_{R}(\mathfrak {Q})\) to \(V_{1,R}(\mathfrak {Q})\) , where \(V_{R}(\mathfrak {Q})\), \(V_{1,R}(\mathfrak {Q})\) are two quaternionic Hilbert spaces, and if \(V_{R}(\mathfrak {Q})=V_{1,R}(\mathfrak {Q})\), \(\mathcal{L}(V_{R}(\mathfrak {Q}),V_{1,R}(\mathfrak {Q}))\) is denoted by \(\mathcal{L}(V_{R}(\mathfrak {Q}))\). Let \(K \in \mathcal{L}(V_{R}(\mathfrak {Q}))\). The range of K is denoted by R(K) and the pseudo-inverse of K is denoted by \(K^{\dagger }\).
2 Quaternionic Hilbert Space
As the quaternions are non-commutative in nature therefore there are two different types of quaternionic Hilbert spaces, the left quaternionic Hilbert space and the right quaternionic Hilbert space depending on positions of quaternions. This fact can entail several problems. For example, when a Hilbert space H is one-sided (either left or right) the set of linear operators acting on it does not have a linear structure. In this section, we will study some basic notations about the algebra of quaternions, right quaternionic Hilbert spaces and operators acting on these spaces.
In the next, let \(\mathfrak {Q}\) denotes the skew field of quaternions, whose elements are in the form \(q = x_0 + x_1i + x_2j + x_3k\), where \(x_0,x_1,x_2\) and \(x_3\) are real numbers and i, j, k are called imaginary units and obey the following multiplication rules:
For more information about the properties of quaternions, we refer the readers to [1, 9, 16].
Let \(V_{R}(\mathfrak {Q})\) be a linear vector space over the skew field of quaternions under right scalar multiplication. It is called a quaternionic pre-Hilbert space if there exists a Hermitian quaternionic scalar product; that is, a map
satisfying, for every \(u,v,w \in V_{R}(\mathfrak {Q})\) and \(p,q \in \mathfrak {Q}\), the following properties:
- (i)
\(\langle u,v\rangle =\overline{\langle v,u\rangle }\),
- (ii)
\(\langle u,u\rangle > 0\) unless \(u = 0\),
- (iii)
\(\langle u,vp+ wq\rangle =\langle u,v\rangle p +\langle u,w\rangle q\).
Suppose that \(V_{R}(\mathfrak {Q})\) is equipped with such a Hermitian quaternionic scalar product. Then, we can define the quaternionic norm \(\Vert \, . \, \Vert :V_{R}(\mathfrak {Q})\rightarrow \mathbb {R}_{+}\) by setting
It has been mentioned, in [9, 16], that the quaternionic norm satisfies all properties of a norm, including Cauchy-Schwarz inequality and parallelogram identity.
The right quaternionic pre-Hilbert space \(V_{R}(\mathfrak {Q})\) is said to be a right quaternionic Hilbert space, if it is complete with respect to the norm (2.1).
It should be noted here that quaternionic Hilbert spaces share many of the standard properties of complex Hilbert spaces such as Hilbert basis. So, let us recall the following results from [9, 16].
Proposition 2.1
Let \(V_{R}(\mathfrak {Q})\) be a right quaternionic Hilbert space and N be a subset of \(V_{R}(\mathfrak {Q})\) such that, for \(z,z'\in N\), \(\langle z,z'\rangle =0\) if \(z \ne z'\) and \(\langle z,z'\rangle =1\). Then, the following assertions are equivalent:
- (i)
For every \(u,v\in V_{R}(\mathfrak {Q})\), the series \(\sum _{z\in N}\langle u,z\rangle \langle z,v\rangle \) converges absolutely and it holds:
$$\begin{aligned} \langle u,v\rangle = \displaystyle \sum _{z\in N}\langle u,z\rangle \langle z,v\rangle . \end{aligned}$$ - (ii)
\(\Vert u\Vert ^2 = \sum _{z\in N}\vert \langle z,u\rangle \vert ^2\) for every \(u\in V_{R}(\mathfrak {Q})\).
- (iii)
\(N^{\bot } :=\left\{ v \in V_{R}(\mathfrak {Q}) :\langle v,z\rangle =0 , ~~\forall z \in N\right\} =\{0\}\).
- (iv)
Span(N) is dense in \(V_{R}(\mathfrak {Q})\).
\(\diamondsuit \)
Remark 2.1
The subset N in Proposition 2.1 is called a Hilbert basis.\(\diamondsuit \)
Proposition 2.2
Every right quaternionic Hilbert space admits a Hilbert basis, and two Hilbert bases have the same cardinality. Furthermore, if N is a Hilbert basis of \(V_{R}(\mathfrak {Q})\), then every \(u\in V_{R}(\mathfrak {Q})\) can be uniquely decomposed as follows
where the series \(\sum _{z\in N} z\langle z,u\rangle \) converges absolutely in \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
Now, we shall define right \(\mathfrak {Q}\)-linear operators and recall some basic properties.
A mapping \( T : V_{R}(\mathfrak {Q}) \longrightarrow V_{R}(\mathfrak {Q})\) is said to be a right linear operator if for all \(u,v\in V_{R}(\mathfrak {Q})\) and \(p\in \mathfrak {Q}\),
Such an operator is called bounded if there exists \(K \ge 0\) such that for all \(u\in V_{R}(\mathfrak {Q})\),
As in the complex case, the norm of a bounded right linear operator T is defined by
The set of all bounded right linear operators on \(V_{R}(\mathfrak {Q})\) is denoted by \(\mathcal{L}(V_{R}(\mathfrak {Q}))\), which is a complete normed space with the norm defined by (2.2) (see [9, 16]). For every \(T \in \mathcal{L}(V_{R}(\mathfrak {Q}))\), there exists a unique operator \(T^* \in \mathcal{L}(V_{R}(\mathfrak {Q}))\), which is called the adjoint of T, such that, for all \(u,v \in V_{R}(\mathfrak {Q})\),
3 Properties of K-Frames
In this part, we explore some properties of K-frame. To this interest, let us introduce some basic facts about the concept of atomic systems and K-frames.
Throughout this paper, \(I\subseteq \mathbb {N}\) denotes a finite or countable index set.
Let’s begin with the definition of frame and Bessel sequence generalized by Sharma and Goel in [20] to separable right quaternionic Hilbert spaces \(V_{R}(\mathfrak {Q})\).
Definition 3.1
[20] A family \(\{f_n\}_{n\in I}\) is said to be a frame for \(V_{R}(\mathfrak {Q})\), if there exist two positive constants \(0< A \le B<\infty \) such that
The numbers A and B are called lower and upper frame bounds. If only the right inequality of Eq. (3.1) holds, \(\{f_n\}_{n\in I}\) is called a Bessel sequence.\(\diamondsuit \)
For a Bessel sequence \(\{f_n\}_{n\in I}\), we define its synthesis operator \(T: l^2(\mathfrak {Q}) \rightarrow V_{R}(\mathfrak {Q})\) by
where
equipped with the following quaternionic inner product
defines a right quaternionic Hilbert space.
The adjoint operator of T, \(T^* : V_{R}(\mathfrak {Q})\rightarrow l^2(\mathfrak {Q})\) defined by \(T^*f=\{\langle f_n,f\rangle \}_{n\in I}\) for \(f\in V_{R}(\mathfrak {Q})\), is called the analysis operator. By composing T with its adjoint \(T^*\), we obtain the frame operator
Now, we recall the concept of atomic systems and K-frames introduced in [7].
Definition 3.2
[7] A family \(\{f_n\}_{n\in I}\) of \(V_{R}(\mathfrak {Q})\) is called an atomic system for \(K\in \mathcal{L} (V_{R}(\mathfrak {Q}))\) if the following statements hold:
- (i)
the series \(\sum _{n\in I}f_nc_n\) converges for all \(c=\{c_n\}_{n\in I}\in l^2(\mathfrak {Q})\);
- (ii)
there exists \(C>0\) such that for every \(x\in V_{R}(\mathfrak {Q})\) there exists \(a_x=\{a_{n}\}_{n\in I}\in l^2(\mathfrak {Q})\) such that \(\Vert a_x\Vert _{l^2(\mathfrak {Q})}\le C\Vert x\Vert \) and \(Kx=\sum _{n\in I}f_na_n\).\(\diamondsuit \)
Proposition 3.1
[7] Let \(\{f_n\}_{n\in I}\subset V_{R}(\mathfrak {Q})\). Then, the following statements are equivalent:
- (i)
\(\{f_n\}_{n\in I}\) is an atomic system for K;
- (ii)
there exist \(A, B>0\) such that
$$\begin{aligned} A\Vert K^*x\Vert ^2\le \displaystyle \sum _{n\in I}\vert \langle f_n,x\rangle \vert ^2\le B\Vert x\Vert ^2,~~\forall x\in V_{R}(\mathfrak {Q}); \end{aligned}$$(3.2) - (iii)
\(\{f_n\}_{n\in I}\) is a Bessel sequence and there exists a Bessel sequence \(\{g_n\}_{n\in I}\) such that
$$\begin{aligned} Kx=\displaystyle \sum _{n\in I}f_n\langle g_n,x\rangle . \end{aligned}$$(3.3)
\(\diamondsuit \)
Definition 3.3
[7] A family \(\{f_n\}_{n\in I}\) is said to be a K-frame for \(V_{R}(\mathfrak {Q})\) with lower and upper K-frame bounds A and B, if \(\{f_n\}_{n\in I}\) satisfies Eq. (3.2).\(\diamondsuit \)
Remark 3.1
Due to Definition (3.3), a sequence satisfying any of the conditions in Proposition 3.1 is also called a K-frame for \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
In the next, we give an example of a K-frame.
Example 3.1
Let \(\{e_n\}_{n=1}^3\) be a Hilbert basis for a three dimensional right quaternionic Hilbert space \(V_{R}(\mathfrak {Q})\) and \(K\in \mathcal{L}(V_{R}(\mathfrak {Q}))\) be defined as
Let \(\{f_n\}_{n=1}^3=\{e_1,e_1,e_2\}\). Clearly, we have
Further, by simple calculations, we can see that the adjoint of K is given as
Then, we get
So, Eqs. (3.4) and (3.5) imply that \(\{f_n\}_{n=1}^3\) is a K-frame for \(V_{R}(\mathfrak {Q})\). However, \(\{f_n\}_{n=1}^3\) is not a frame for \(V_{R}(\mathfrak {Q})\) since it does not possess a lower frame bound.
Remark 3.2
Since K-frames are not frames in general, many properties of ordinary frames may not hold for K-frames, such as the corresponding synthesis operator for K-frames is not surjective, the frame operator for K-frames is not isomorphic and the alternate dual reconstruction pair for K-frames is not interchangeable in the quaternionic Hilbert space \(V_{R}(\mathfrak {Q})\). However, they are satisfied on a subspace R(K) of \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
Proposition 3.2
Let K be with a closed range and \(\{f_n\}_{n\in I}\) be a K-frame for \(V_{R}(\mathfrak {Q})\). Then, the frame operator for the K-frame \(\{f_n\}_{n\in I}\) is invertible on a subspace R(K) of \(V_{R}(\mathfrak {Q})\). \(\diamondsuit \)
To prove this result, we need the following lemma which is a slight modification of [8, Lemma 2.5.1]. The proof of this Lemma is similar to the one in complex case.
Lemma 3.1
Let \(V_{R}(\mathfrak {Q})\) and \(V_{1,R}(\mathfrak {Q})\) be two right quaternionic Hilbert spaces and suppose that \(U:V_{R}(\mathfrak {Q})\rightarrow V_{1,R}(\mathfrak {Q})\) is a bounded operator with closed range R(U). Then, there exists a bounded operator \(U^{\dagger }:V_{1,R}(\mathfrak {Q})\rightarrow V_{R}(\mathfrak {Q})\) for which
\(\diamondsuit \)
Proof of Proposition 3.2
As \(\{f_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\), then there exist \(A, B>0\) such that
Further, since \(K\in \mathcal{L}(V_{R}(\mathfrak {Q}))\) and R(K) is closed, then in view of Lemma 3.1, there exists a pseudo-inverse \(K^{\dagger }\) of K such that
Therefore, we obtain
Hence, for \(f\in R(K)\) we have
which implies that
So, using Eq. (3.7) we get
Combining Eqs. (3.6) and (3.8), we obtain
Hence, \(S:R(K)\rightarrow S(R(K))\) is a homeomorphism; that is if \(f\in S(R(K))\) then \(S^{-1}f\in R(K)\) and consequently, Eq. (3.9) yields
Therefore, we get
\(\square \)
Proposition 3.3
Suppose that K is with closed range and \(\{f_n\}_{n\in I}\) and \(\{g_n\}_{n\in I}\) are as in Eq. (3.3). Then there exists a sequence \(\{h_n\}_{n\in I}=\{(K_{|R(K)}^{\dagger })^*g_n\}_{n\in I}\) derived by \(\{g_n\}_{n\in I}\) such that
Moreover, \(\{h_n\}_{n\in I}\) and \(\{f_n\}_{n\in I}\) are interchangeable for any \(f\in R(K)\).\(\diamondsuit \)
Proof
As \(K\in \mathcal{L}(V_{R}(\mathfrak {Q}))\) and R(K) is closed, then it follows from Lemma 3.1 that there exists a pseudo-inverse \(K^{\dagger }\) of K such that \(f = KK^{\dagger }f\), \(\forall f\in R(K)\). So, Eq. (3.3) yields
where \(\{f_n\}_{n\in I}\) and \(\{g_n\}_{n\in I}\) are Bessel sequences in \(V_{R}(\mathfrak {Q})\) satisfying Eq. (3.3).
Now let \(h_n =( K_{|R(K)}^{\dagger })^*g_n\). Clearly, \(\{h_n\}_{n \in I}\subset R(K)\). In fact, it suffices to see that
and so we may obtain
Further, \(\{h_n\}_{n\in I}\) is a Bessel sequence in R(K). Indeed, let \(f\in R(K)\subset V_{R}(\mathfrak {Q})\) we have
As \(\{g_n\}_{n\in I}\) is a Bessel sequence, then there exists \(B>0\) such that
To complete our proof, it suffices to show that \(\{h_n\}_{n\in I}\) and \(\{f_n\}_{n\in I}\) are interchangeable on R(K). To this interest, let \(f,g\in R(K)\subset V_{R}(\mathfrak {Q})\). We have
that is, \(f = \sum \nolimits _{n\in I}h_n\langle f_n,f\rangle ,\)\(\forall f \in R(K)\).
\(\square \)
4 Construction of K-Frames
In this part, we are concerning with the construction of new K-frames. We begin first by obtaining a K-frame from ordinary frame.
Proposition 4.1
Suppose that \(\{f_n\}_{n\in I}\) is an ordinary frame for \(V_{R}(\mathfrak {Q})\), then \(\{Kf_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
Proof
By Theorem 3.1, it suffices to show that \(\{Kf_n\}_{n\in I}\) is an atomic system for K.
Since \(\{f_n\}_{n\in I}\) is an ordinary frame for \(V_{R}(\mathfrak {Q})\), then for any \(f \in V_{R}(\mathfrak {Q})\) we have
where S is the frame operator for \(\{f_n\}_{n\in I}\). Thus, Eq. (4.1) implies the following equality
Now, it remains to show that \(\{Kf_n\}_{n\in I}\) is a Bessel sequence and \(\Vert \{\langle S^{-1}f_n,f\rangle \}\Vert _{l^2(\mathfrak {Q})} \le C \Vert f\Vert \), where C is a positive constant. In fact, since \(\{f_n\}_{n\in I}\) is an ordinary frame for \(V_{R}(\mathfrak {Q})\), we let A, B be its lower and upper frame bounds, then it follows from [20] that \(\{S^{-1}f_n\}_{n\in I}\) is a frame for \(V_{R}(\mathfrak {Q})\) with bounds \(B^{-1},A^{-1}>0\). Therefore, for any \(f \in V_{R}(\mathfrak {Q})\) we obtain
On the other hand, we have
Consequently, \(\{Kf_n\}_{n\in I}\) is an atomic system for K.
\(\square \)
Corollary 4.1
Assume that \(\{e_n\}_{n\in I}\) is a Hilbert basis for \(V_{R}(\mathfrak {Q})\), then \(\{Ke_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
Proposition 4.2
If \(T \in \mathcal{L}(V_{R}(\mathfrak {Q}))\) and \(\{f_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\), then \(\{Tf_n\}_{n\in I}\) is a TK-frame for \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
Remark 4.1
It should be noted that the result of Proposition 4.2 does not hold for ordinary frames in general. More precisely, if we assume that \(T \in \mathcal{L}(V_{R}(\mathfrak {Q}))\) and \(\{f_n\}_{n\in I}\) is an ordinary frame for \(V_{R}(\mathfrak {Q})\), then \(\{Tf_n\}_{n\in I}\) is not an ordinary frame for \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
Proof of Proposition 4.2
Let \(T \in \mathcal{L}(V_{R}(\mathfrak {Q}))\) and \(\{f_n\}_{n\in I}\) be a K-frame for \(V_{R}(\mathfrak {Q})\) with frame bounds A and B, i.e.,
By observing that
Eq. (4.2) yields
Hence, \(\{Tf_n\}_{n\in I}\) is a TK-frame for \(V_{R}(\mathfrak {Q})\).
\(\square \)
Corollary 4.2
If \(\{f_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\), then \(\{K^Nf_n\}_{n\in I}\) is a \(K^N\)-frame for \(V_{R}(\mathfrak {Q})\), where \(N \ge 1\) is a fixed integer. \(\diamondsuit \)
Now, we construct new K-frames from old ones by taking sums.
Theorem 4.1
Let \(\{f_n\}_{n\in I}\) be a K-frame for \(V_{R}(\mathfrak {Q})\), \(T \in \mathcal{L}(V_{R}(\mathfrak {Q}))\) and \(TKK^*\) be a positive operator. Then \(\{f_n + Tf_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
Proof
Suppose that \(\{f_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\) with frame bounds A and B. Then, for \(f \in V_{R}(\mathfrak {Q})\), we obtain
Now, for \(f\in V_{R}(\mathfrak {Q})\) we have
Using Eqs. (4.3) and (4.4), we get
On the other hand, we have
Since \(TKK^*\) is a positive operator, then
So, we obtain
Combining Eqs. (4.5) and (4.6), we get that \(\{f_n + Tf_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\).
\(\square \)
Theorem 4.2
Suppose that \(\{f_n\}_{n\in I}\) and \(\{g_n\}_{n\in I}\) are K-frames for \(V_{R}(\mathfrak {Q})\). Let \(U_1\) and \(U_2\) be the synthesis operators of \(\{f_n\}_{n\in I}\) and \(\{g_n\}_{n\in I}\), respectively. If \(U_1U_2^*=0\) and T or L is surjective operator in \(\{K\}'\), then \(\{Tf_n+Lg_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\). \(\diamondsuit \)
Proof
Let \(f\in V_{R}(\mathfrak {Q})\). We have
As \(U_1U_2^*=0\), then for \(f,g\in V_{R}(\mathfrak {Q})\) we get
Therefore, we obtain
Since \(\{f_n\}_{n\in I}\) and \(\{g_n\}_{n\in I}\) are K-frames for \(V_{R}(\mathfrak {Q})\), then there exist \(A_i,B_i>0,i=1,2,\) such that
and
So, we get
Without loss of generality, assume that T is surjective. Then, there exists \(C>0\) such that
Since \(TK=KT\), thus \(T^*K^*=K^*T^*\). So,
Hence, Eqs. (4.7) and (4.8) entail that \(\{Tf_n+Lg_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\).
\(\square \)
Let \(L=0\). We get the following corollary.
Corollary 4.3
Suppose that \(\{f_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\). If T is surjective and \(TK=KT\), then \(\{Tf_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
Let \(T=L=I\). The following result holds.
Corollary 4.4
Suppose that \(\{f_n\}_{n\in I}\) and \(\{g_n\}_{n\in I}\) are K-frames for \(V_{R}(\mathfrak {Q})\). Let \(U_1\) and \(U_2\) be the synthesis operators of \(\{f_n\}_{n\in I}\) and \(\{g_n\}_{n\in I}\), respectively. If \(U_1U_2^*=0\), then \(\{f_n+g_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\).\(\diamondsuit \)
We close this part by showing that a T-frame can be constructed from a K-frame by the perturbation of a bounded right linear operator T.
Theorem 4.3
Let K be with a closed range and let \(\{f_n\}_{n\in I}\) be a K-frame for \(V_{R}(\mathfrak {Q})\). Let \(T\in \mathcal{L}(V_{R}(\mathfrak {Q}),V_{1,R}(\mathfrak {Q}))\). If \(R(T^*)\subset R(K)\), then \(\{Tf_n\}_{n\in I}\) is a T-frame for \(V_{1,R}(\mathfrak {Q})\).\(\diamondsuit \)
The following result extends [19, Lemma 2.2] to quaternionic Hilbert spaces. We omit the proof since it is similar to the complex case.
Lemma 4.1
Let \(V_{R}(\mathfrak {Q})\), \(V_{1,R}(\mathfrak {Q})\) be two right quaternionic Hilbert spaces, and suppose that \(T \in \mathcal{L} (V_{R}(\mathfrak {Q}),V_{1,R}(\mathfrak {Q}))\) has a closed range, then
\(\diamondsuit \)
Proof of Theorem 4.3
Assume that \(\{f_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\) with frame bounds A and B. Hence, we have
Now, let \(g\in V_{1,R}(\mathfrak {Q}) \). As \(T\in \mathcal{L}(V_{R}(\mathfrak {Q}),V_{1,R}(\mathfrak {Q}))\), then \(T^*g\in V_{R}(\mathfrak {Q})\). So, we obtain
Since \(K \in \mathcal{L}(V_{R}(\mathfrak {Q}))\) has a closed range and \(R(T^*) \subset R(K)\), by Lemma 4.1, we have
This implies that
Consequently, \(\{Tf_n\}_{n\in I}\) is a T-frame for \(V_{1,R}(\mathfrak {Q})\).
\(\square \)
5 Stability of Perturbation of K-Frames
In this section, we prove a stability result for K-frames. To this interest, we recall the following lemma due to Sharma and Goel [20].
Lemma 5.1
[20] A sequence of vectors \(\{f_n\}_{n\in I}\subset V_{R}(\mathfrak {Q})\) is a Bessel sequence for \(V_{R}(\mathfrak {Q})\) with bound B if and only if the right linear operator \(T : l^2(\mathfrak {Q})\rightarrow V_{R}(\mathfrak {Q})\) given by
is a well defined and bounded operator with \(\Vert T\Vert \le \sqrt{B}\).\(\diamondsuit \)
Now, we are ready to state our result.
Theorem 5.1
Suppose that K is with closed range, \(\{f_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\) with frame bounds A and B and \(\alpha ,\beta ,\gamma \in [0,\infty )\), such that \(\max \{\alpha +\gamma \sqrt{A^{-1}}\Vert K^{\dagger }\Vert ,\beta \}<1\). If \(\{g_n\}_{n\in I}\subset V_{R}(\mathfrak {Q})\) and satisfy
for all finite quaternions \(q_n\in l^2(\mathfrak {Q})\), \(n\in J\subseteq I\) with \(\vert J\vert <+\infty \), then \(\{g_n\}_{n\in I}\) is a \(P_{Q(R(K))}K\)-frame for \(V_{R}(\mathfrak {Q})\), with frame bounds
where \(P_{Q(R(K))}\) is the orthogonal projection operator from \(V_R(\mathfrak {Q})\) to Q(R(K), \(Q=UT^*,T,U\) are the synthesis operators for \(\{f_n\}_{n\in I}\) and \(\{g_n\}_{n\in I}\), respectively.\(\diamondsuit \)
Proof
Let \(J\subseteq I\) with \(|J|<+\infty \). Then, we have
Furthermore, we have
Hence, using Eqs. (5.2) and (5.3), we obtain
for all finite quaternions \(q_n\in l^2(\mathfrak {Q})\), \(n\in J\). So, we can define a bounded operator
Clearly, U is well defined and bounded with \(\Vert U\Vert \le \frac{\alpha +1}{1-\beta }\sqrt{B}+\frac{\gamma }{1-\beta }\). Therefore, it follows from Lemma 5.1 that \(\{g_n\}_{n\in I}\) is a Bessel sequence for \(V_{R}(\mathfrak {Q})\).
Further, as \(\{f_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\), then we consider its synthesis operator
Setting \(Q=UT^*:R(K)\rightarrow R(Q)\). We prove that Q is invertible. To this interest, let begin with the closure of R(Q). Let \(\{y_n\}_{n\in I}\subset R(Q)\) such that \(y_n\rightarrow y,\)\(y\in V_{R}(\mathfrak {Q})\), then there exists \(x_n\in R(K)\) such that
Although, it follows from Eq. (5.1) that
So, if we consider \(q_f=T^*f\in l^2(\mathfrak {Q})\), \(f\in R(K)\), Eq. (5.5) implies that
On the other hand, by Eq. (3.9) we have
Combining Eqs. (5.6) and (5.7), we obtain
Using the triangle inequality, Eq. (5.8) yields
Hence, it follows from Eqs. (3.9) and (5.9) that for any \(f\in R(K)\) we have
Consequently, Eqs. (5.4) and (5.10) imply that
where \(C=\frac{(1-(\alpha +\gamma \sqrt{A^{-1}}\Vert K^{\dagger }\Vert ))A\Vert K^{\dagger }\Vert ^{-2}}{1+\beta }\). Since \(\{y_n\}_{n\in I}\) is a cauchy sequence, Eq. (5.11) entails that \(\{x_n\}_{n\in I}\) is a cauchy sequence. Hence, there exists \(x\in R(K)\) such that \(x_n\rightarrow x\). As Q is bounded, we have \(y_n=Q(x_n)\rightarrow Q(x)\). By the uniqueness of the limit, we obtain \(y=Q(x)\in R(Q)\) which implies that R(Q) is closed. On the other hand, it follows from Eq. (5.1) that Q is injective on R(K). As a consequence, we conclude that \(Q:R(K)\rightarrow R(Q)\) is invertible. Taking into accounts Eqs. (5.9) and (5.10), we get
which implies that
Now, let \(f\in V_{R}(\mathfrak {Q})\). Since
then
Using Eqs. (5.12), (5.13) and (5.14), we get
Consequently, we obtain
\(\square \)
Corollary 5.1
Suppose that K is with closed range and \(\{f_n\}_{n\in I}\) is a K-frame for \(V_{R}(\mathfrak {Q})\) with frame bounds A and B and assume further that there exists \(0<R<A\). If \(\{g_n\}_{n\in I}\subset V_{R}(\mathfrak {Q})\) and satisfies
for all quaternions \(q_n\in l^2(\mathfrak {Q})\),then \(\{g_n\}_{n\in I}\) is a \(P_{Q(R(K))}K\)-frame for \(V_{R}(\mathfrak {Q})\) with frame bounds
where \(P_{Q(R(K))}\) is the orthogonal projection operator from \(V_{R}(\mathfrak {Q})\) to Q(R(K), \(Q=UT^*,T,U\) are the synthesis operators for \(\{f_n\}_{n\in I}\) and \(\{g_n\}_{n\in I}\), respectively.\(\diamondsuit \)
Proof
It suffices to take \(\alpha =\beta =0\) and \(\gamma =\sqrt{R}\) in Theorem 5.1.\(\square \)
References
Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, New York (1995)
Cahill, J., Casazza, P.G., Li, S.: Non-orthogonal fusion frames and the sparsity of fusion frame operators. J. Fourier Anal. Appl. 18, 287–308 (2012)
Casazza, P.G., Kutyniok, G.: Frames of Subspaces. Wavelets, Frames and Operator Theory. Contemporary Mathematics, vol. 345, pp. 87–113. American Mathematical Society, Providence (2004)
Charfi, S., Jeribi, A., Walha, I.: Riesz basis property of families of nonharmonic exponentials and application to a problem of a radiation of a vibrating structure in a light fluid. Numer. Funct. Anal. Optim. 32(4), 370–382 (2011)
Charfi, S., Ellouz, H.: Riesz basis of eigenvectors for analytic families of operators and application to a non-symmetrical Gribov operator. Mediterr. J. Math. 15, 1–16 (2018)
Charfi, S., Ellouz, H.: Frame of exponentials related to analytic families operators and application to a non-self adjoint problem of radiation of a vibrating structure in a light fluid. Complex Anal. Oper. Theory 13, 839–858 (2019)
Charfi, S., Ellouz, H.: Frames for operators in quaternionic Hilbert spaces (Submitted)
Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis, 2nd edn. Birkhäuser, Basel (2016)
Colombo, F., Gantner, J., Kimsey, David P.: Spectral Theory on the \(S\)-Spectrum for Quaternionic Operators. Operator Theory: Advances and Applications, 270, p. ix+356. Birkhäuser, Cham (2018)
Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 24, 1271–1283 (1986)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Ellouz, H., Feki, I., Jeribi, A.: On a Riesz basis of exponentials related to the eigenvalues of an analytic operator and application to a non-selfadjoint problem deduced from a perturbation method for sound radiation. J. Math. Phys. 54, 112101 (2013)
Ellouz, H., Feki, I., Jeribi, A.: On a Riesz basis of exponentials related to a family of analytic operators and application. J. Pseudo Differ. Oper. Appl. (2018). https://doi.org/10.1007/s11868-018-0262-z
Ellouz, H., Feki, I., Jeribi, A.: Non-orthogonal fusion frames of an analytic operator and application to a one-dimensional wave control system. Mediterr. J. Math. 16, 52 (2019)
Găvruţa, L.: Frames for operators. Appl. Comput. Harmon. Anal. 32, 139–144 (2012)
Ghiloni, R., Moretti, V., Perotti, A.: Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 25, 1350006 (2013)
Guo, X.: Canonical dual K-Bessel sequences and dual K-Bessel generators for unitary systems of Hilbert spaces. J. Math. Anal. Appl. 444, 598–609 (2016)
Jeribi, A.: Denseness, Bases and Frames in Banach Spaces and Applications. De Gruyter, Berlin (2018)
Jia, M., Zhu, Y.-C.: Some results about the operator perturbation of a K-frame. Results Math. 73(4), 138 (2018)
Sharma, S.K., Goel, S.: Frames in quaternionic Hilbert spaces. J. Math. Phys. Anal. Geom. 15(3), 395–411 (2019)
Xiao, X., Zhu, Y., Găvruţa, L.: Some properties of K-frames in Hilbert spaces. Results Math. 63, 1243–1255 (2013)
Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, London (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Daniel Aron Alpay.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the topical collection “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.
Rights and permissions
About this article
Cite this article
Ellouz, H. Some Properties of K-Frames in Quaternionic Hilbert Spaces. Complex Anal. Oper. Theory 14, 8 (2020). https://doi.org/10.1007/s11785-019-00964-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-019-00964-5