Some Properties of Operator Valued Frames in Quaternionic Hilbert Spaces

Abstract

Quaternionic Hilbert spaces play an important role in applied physical sciences especially in quantum physics. In this paper, the operator valued frames on quaternionic Hilbert spaces are introduced and studied. In terms of a class of partial isometries in the quaternionic Hilbert spaces, a parametrization of Parseval operator valued frames is obtained. We extend to operator valued frames many of the properties of vector frames on quaternionic Hilbert spaces in the process. Moreover, we show that all the operator valued frames can be obtained from a single operator valued frame. Finally, several results for operator valued frames concerning duality, similarity of such frames on quaternionic Hilbert spaces are presented.

1. Introduction

Frames satisfy the well-known property of perfect reconstruction, in that any vector of the Hilbert space can be synthesized back from its inner products with the frame vectors. They were originally introduced for the study of nonharmonic Fourier analysis [1,2,3]. With the emergence of wavelet theory, they have been widely used in signal and image processing, time-frequency analysis and sampling theory [4,5]. By the commonly used definition of a countable frame in a separable Hilbert space a set ${\left\{f_i\right\}}_{i\in \mathbb{I}}\subset H$ is said to be a frame of the Hilbert space H if there exist two constants $0<\alpha \le \beta <+\infty$ such that the inequalities

$${\alpha \parallel f\parallel }_H\le \parallel \left\{{⟨f,f_i⟩}_H\right\}{\parallel }_{{\ell }^2\left(\mathbb{I}\right)}\le \beta {\parallel f\parallel }_H$$

hold for every $f\in H$. Here, ${\parallel ·\parallel }_H$ and ${⟨·,·⟩}_H$ denote the norm and inner product on H, respectively. Furthermore, ${\ell }^2\left(\mathbb{I}\right)$ is the Hilbert space of square-summable sequences on index set $\mathbb{I}$. For more information about theory and applications of frames, we refer to [6,7,8,9].

We observe that various generalizations of frames on quaternionic Hilbert spaces have been proposed recently. As well known, the extension of frame theory to quaternionic Hilbert spaces is an important issue. The classical frames have been extended to quaternionic Hilbert spaces in a variety of ways. One way is to use the unconditional convergence property of frame operators [10,11], which seems to be the core of Hilbert space frame theory. Another way is to generalize the frame inequality directly, such as [12,13,14]. The third way is to generalize the dual frame pairs in Hilbert spaces [15,16,17]. The operator valued frames introduced in this paper is a natural generalization of vector valued frames. The theoretical framework established in this article can be used to extend the vector valued results as well as the classical results of wavelet and dual frame theory to the context of quaternionic Hilbert spaces.

In [18], Khokulan, Thirulogasanthar and Srisatkunarajah first introduced frames for finite dimensional quaternionic Hilbert spaces. Sharma and Virender [15] studied some different types of dual frames of a given frame in a finite dimensional quaternionic Hilbert space. These works lead to a generalization of frames in separable quaternionic Hilbert spaces which were studied in [11]. Recently, Chen, Dang and Qian [19] had studied frames for Hardy spaces in the contexts of the quaternionic space and the Euclidean space in the Clifford algebra. Charfi and Ellouz [12] extend the concepts of atomic systems for operators and K-frames in separable complex Hilbert spaces to separable quaternionic Hilbert spaces. In [13], Ellouz devoted himself to the study of some properties of K-frames in quaternionic Hilbert spaces. After that, he derive a precise description to the concept of dual K-Bessel sequences of a given K-frame in quaternionic Hilbert spaces [20]. As needed for the construction of rank n continuous frames on a right quaternionic Hilbert space the so-called S-spectrum of a right quaternionic operator was studied in [21] by Khokulana, Thirulogasanthar and Muraleetharan. Moreover, Zhang and Li [17] introduced the notion of approximately dual frames in quaternionic Hilbert spaces and characterized approximately dual frames.

A main purpose of this paper is to provide a language for the study of operator valued frames on quaternionic Hilbert spaces. The operator valued frames as a natural generalization of vector valued frame theory were developed by Kaftal, Larson and Zhang in [22]. In the present paper, we define operator-valued frames on quaternionic Hilbert spaces. We obtain a parametrization of all the operator valued frames on a certain quaternionic Hilbert space. We study similarity of operator valued frames. The similarity obtained by multiplying an operator valued frame from the right generalizes the one usual in the vector case and inherits its main properties. We characterize the case when two operator valued frames are similar and parameterize the dual of operator valued frames.

2. Preliminaries

Let $\mathbb{Q}$ be the skew field of quaternions and let $a_0,a_1,a_2,a_3\in \mathbb{R}$. The elements of $\mathbb{Q}$ are of the form:

$$q=\sum _{k=0}^3{a}_k{q}_k,$$

where $q_0=1$ and $q_1,q_2,q_3$ are called imaginary units and they define arithmetic rules in $\mathbb{Q}$. By definition, the elements $q_0,q_1,q_2,q_3$ satisfy

$$q_1{q}_2=-q_2{q}_1=q_3,q_2{q}_3=-q_3{q}_2=q_1,q_3{q}_1=-q_1{q}_3=q_2,q_1^2=q_2^2=q_3^2=-q_0.$$

Natural operations of addition and multiplication in $\mathbb{Q}$ turn it into a skew field. The main involution in $\mathbb{Q}$, the quaternion conjugation, is defined by

$$\overline{q_0}=q_0,\overline{q_k}=-q_k,k=1,2,3,$$

and it extends onto $\mathbb{Q}$ by $\mathbb{R}$-linearity, i.e., for $q\in \mathbb{Q}$,

$$\overline{q}=a_0-\sum _{k=1}^3{a}_k{q}_k.$$

Note that

$$\overline{q}q=q\overline{q}=\sum _{k=0}^3{a}_k^2={\left|q\right|}_{\mathbb{Q}}^2.$$

Therefore, for $q\in \mathbb{Q}\setminus \left\{0\right\}$ the quaternion $q^{-1}=\frac{\overline{q}}{{\left|q\right|}_{\mathbb{Q}}^2}$ is an inverse to q. Whereas the above mentioned properties are analogous to the complex one-dimensional case, we have for the quaternion conjugation that for any $q,p\in \mathbb{Q},\overline{qp}=\overline{p}\overline{q}$. For more details about Quaternion analysis, we refer to [23,24,25,26,27].

Hereafter, $\mathbb{H}$ denotes a vector space over the skew filed of quaternions $\mathbb{Q}$, that is, $\mathbb{H}$ is the additive group and the multiplication of the vector $x\in \mathbb{H}$ on scalars $\alpha ,\beta \in \mathbb{Q}$ from the right satisfies the axioms of the associativity and distributivity. It is called a quaternionic inner product space if there exists a Hermitian quaternionic scalar product $⟨·\mid ·⟩:\mathbb{H}\times \mathbb{H}\mapsto \mathbb{Q}$ which satisfies the following properties:

• $⟨x\mid y⟩=\overline{⟨y\mid x⟩}$ for all $x,y\in \mathbb{H}$;
• $⟨x\mid x⟩>0$ for all $x\in \mathbb{H}$ unless $x=0$;
• $⟨x\mid yp+zq⟩>=⟨x\mid y⟩p+⟨x\mid z⟩q$ for all $x,y,z\in \mathbb{H}$.

If $\mathbb{H}$ is complete relative to the norm ${\parallel x\parallel }_{\mathbb{H}}=\sqrt{⟨x\mid x⟩},x\in \mathbb{H},$ then $\mathbb{H}$ is called a quaternion Hilbert space. It has been mentioned in [11,13,28,29] that quaternionic Hilbert spaces share many of the standard properties of complex Hilbert spaces such as Hilbert basis, Cauchy-Schwarz inequality and parallelogram identity.

Let $\mathbb{H}$ and $\mathbb{K}$ be two quaternionic Hilbert spaces. A right $\mathbb{Q}$-linear operator is a map $T:\mathbb{H}\mapsto \mathbb{K}$ such that

$$T(xp+yq)=\left(Tx\right)p+\left(Ty\right)q\mathrm{if}x,y\in \mathbb{H}\mathrm{and}p,q\in \mathbb{Q}.$$

The operator T is called bounded if there exists $M>0$ such that for any $x\in \mathbb{H}$, ${\parallel Tx\parallel }_{\mathbb{K}}\le M{\parallel x\parallel }_{\mathbb{H}}$. The adjoint operator $T^{*}$ of T is defined by $⟨x\mid Ty⟩=⟨T^{*}x\mid y⟩,\forall x,y\in \mathbb{H},$ and T is said to be self-adjoint if $T=T^{*}$.

3. Main Results

In this section, we investigate operator valued frames on quaternionic Hilbert spaces. Before stating the results, we first introduce our notation and then give the definition of operator valued frames on quaternionic Hilbert spaces. Throughout the paper, symbol $\mathbb{I}$ will denote generic countable index set. Let $\mathbb{H}$ and $\mathbb{K}$ be separable quaternionic Hilbert spaces and let $\mathbb{B}(\mathbb{H},\mathbb{K})$ be the space of all the bounded right $\mathbb{Q}$-linear operators from $\mathbb{H}$ to $\mathbb{K}$ (if $\mathbb{H}=\mathbb{K}$ we write $\mathbb{B}\left(\mathbb{H}\right)$). The symbols ran$(·)$ and $I_{\mathbb{H}}$ refer, respectively, to the range of an operator and the identity operator on $\mathbb{H}$.

Definition 1.

Let $\mathbb{H}$ and $\mathbb{K}$ be two quaternion Hilbert spaces. A collection ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ of right $\mathbb{Q}$-linear operators ${\mathsf{\Lambda }}_i\in \mathbb{B}(\mathbb{H},\mathbb{K})$ is called an operator valued frame on quaternion Hilbert space $\mathbb{H}$ with respect to $\mathbb{K}$ if the series

$$\sum _{i\in \mathbb{I}}{\mathsf{\Lambda }}_i^{*}{\mathsf{\Lambda }}_i$$

converges to a positive bounded invertible operator $S_{\mathsf{\Lambda }}$, where the convergence is in the strong operator topology.

Clearly, if ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ is an operator valued frame on $\mathbb{H}$, then there exist two positive constants $\alpha ,\beta >0$ such that $\alpha I_{\mathbb{H}}\le S_{\mathsf{\Lambda }}\le \beta I_{\mathbb{H}}$. We call $\alpha$ and $\beta$ the operator valued frame bounds. An operator valued frame ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ on $\mathbb{H}$ is said to be tight if $\alpha =\beta$ and Parseval if it is tight with $\alpha =\beta =1$.

Regarding the existence of operator valued frames on $\mathbb{H}$, we have the following example.

Example 1.

Let $\mathbb{K}$ be an infinite dimension quaternion Hilbert space and let ${\left\{V_i\right\}}_{i\in \mathbb{I}}$ be a collection partial isometries on $\mathbb{K}$ with mutually orthogonal range projections. Assume that ${\sum }_{i\in \mathbb{I}}{V}_i^{*}{V}_i=I_{\mathbb{K}}$ and $V_i^{*}{V}_i=Q$. Let $P\in \mathbb{B}\left(\mathbb{K}\right)$ be a nonzero projection and let ${\mathsf{\Lambda }}_i=V_i^{*}P\in \mathbb{B}(P\mathbb{K},Q\mathbb{K})$. Then

$$\sum _{i\in \mathbb{I}}{\mathsf{\Lambda }}_i^{*}{\mathsf{\Lambda }}_i=P\sum _{i\in \mathbb{I}}{V}_i^{*}{V}_{i}P{\mid }_{P\mathbb{K}}.$$

It follows that the collection ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ is a Parseval operator valued frame on $P\mathbb{K}$ with respect to $Q\mathbb{K}$.

Given two right quaternionic Hilbert spaces $\mathbb{L}$ and $\mathbb{K}$, define

$$V_i:\mathbb{K}\mapsto \mathbb{L}\otimes \mathbb{K},V_{i}k=e_i\otimes k,$$

where ${\left\{e_i\right\}}_{i\in \mathbb{I}}$ is the standard Hilbert basis of $\mathbb{L}$. It is easy to see that $V_i$ is a partial isometry. In particular, $V_j^{*}{V}_i=I_{\mathbb{K}}$ if $i=j$ and $V_j^{*}{V}_i=\mathbf{0}$ if $i\ne j$. It follows that

$$\sum _{i\in \mathbb{I}}{V}_i{V}_i^{*}=I_{\mathbb{L}}\otimes I_{\mathbb{H}}.$$

Proposition 1.

Let ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ be an operator valued frame on $\mathbb{H}$ with respect to $\mathbb{K}$. Then the series ${\sum }_{i\in \mathbb{I}}{V}_i{\mathsf{\Lambda }}_i$ converges in $\mathbb{B}(\mathbb{H},\mathbb{L}\otimes \mathbb{K})$, where the convergence is in the strong operator topology.

Proof. 

For every $x\in \mathbb{H}$, we have

$$⟨S_{\mathsf{\Lambda }}x\mid x⟩=\sum _{i\in \mathbb{I}}\parallel {\mathsf{\Lambda }}_i{x\parallel }^2=\sum _{i\in \mathbb{I}}{\parallel V_i{\mathsf{\Lambda }}_{i}x\parallel }^2=\parallel \sum _{i\in \mathbb{I}}{V}_i{\mathsf{\Lambda }}_{i}x{\parallel }^2.$$

□

Next, we provide some fundamental facts about operator valued frames in quaternion Hilbert spaces.

Proposition 2.

Given an operator valued frame ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ on $\mathbb{H}$ with respect to $\mathbb{K}$, let ${\mathsf{\Theta }}_{\mathsf{\Lambda }}={\sum }_{i\in \mathbb{I}}{V}_i{\mathsf{\Lambda }}_i$. Then

1. 

$S_{\mathsf{\Lambda }}={\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }};$

2. 

${\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-\frac{1}{2}}$ is an isometry;

3. 

${\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$ is the range projection of ${\mathsf{\Theta }}_{\mathsf{\Lambda }};$

4. 

${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ is Parseval if and only if ${\mathsf{\Theta }}_{\mathsf{\Lambda }}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$ is a projection;

5. 

${\mathsf{\Lambda }}_i=V_i^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }},i\in \mathbb{I}$.

Proof. 

The statements 1, 2 and 3 immediately follow from Proposition 1. For the proof of 4, it is easy to see that ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ is Parseval if and only if ${\mathsf{\Theta }}_{\mathsf{\Lambda }}$ is an isometry. Note that

$$V_j^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}=V_j^{*}(\sum _{i\in \mathbb{I}}{V}_i{\mathsf{\Lambda }}_i)=\sum _{i\in \mathbb{I}}{V}_j^{*}{V}_i{\mathsf{\Lambda }}_i={\mathsf{\Lambda }}_j,$$

these identities prove 5. □

To state our next result, we need the following notation. For the sake of brevity, let $P_{\mathsf{\Lambda }}={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$ and $\mathbb{M}=\mathbb{L}\otimes \mathbb{K}$. Given an operator valued frame ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ on $\mathbb{H}$ with respect to $\mathbb{K}$, define

$${\mathbb{A}}_{\mathsf{\Lambda }}=\{A\in \mathbb{B}(P_{\mathsf{\Lambda }}\mathbb{M},\mathbb{M})\mid A\mathrm{is}\mathrm{left}\mathrm{invertible}\}$$

and

$${\mathbb{A}}_{\mathsf{\Lambda }}^{\prime }=\{A^{\prime }\in \mathbb{B}\left(\mathbb{M}\right)\mid A^{\prime }=A^{\prime }{P}_{\mathsf{\Lambda }},A^{\prime }{}^{*}{A}^{\prime }{\mid }_{P_{\mathsf{\Lambda }}\mathbb{M}}\mathrm{is}\mathrm{invertible}\}.$$

Theorem 1.

Let ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ be an operator valued frame on $\mathbb{H}$ with respect to $\mathbb{K}$. Then ${\mathbb{A}}_{\mathsf{\Lambda }}={\mathbb{A}}_{\mathsf{\Lambda }}^{\prime }$.

Proof. 

Clearly, ${\mathbb{A}}_{\mathsf{\Lambda }}\supseteq {\mathbb{A}}_{\mathsf{\Lambda }}^{\prime }$. Now suppose that $A\in {\mathbb{A}}_{\mathsf{\Lambda }}$, then there exists an non-zero operator $X\in \mathbb{B}(P_{\mathsf{\Lambda }}\mathbb{M},\mathbb{M})$ such that $XA=I_{\mathbb{M}}$. It follows that for any $x\in \mathbb{H}$, $\parallel X\parallel \parallel Ax\parallel \ge \parallel XAx\parallel =\parallel x\parallel$. This implies that A is a bounded below operator and thus A is injective. Hence $A^{*}A{\mid }_{P_{\mathsf{\Lambda }}\mathbb{M}}$ is invertible. □

In what follows, we denote by ${\mathbb{F}}_{OPV}$ (${\mathbb{F}}_{P-OPV}$) the set of all the (Parseval) operator valued frames on $\mathbb{H}$ with respect to $\mathbb{K}$ and indexed by $\mathbb{I}$. Let ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{OPV}$. For all ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{OPV}$, define ${\mathsf{\Phi }}_{\mathsf{\Lambda }}:{\mathbb{F}}_{OPV}\mapsto {\mathbb{A}}_{\mathsf{\Lambda }}$,

$${\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right)={\mathsf{\Theta }}_{\mathsf{\Gamma }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}.$$

Theorem 2.

The following statements hold.

1. 

${\mathsf{\Phi }}_{\mathsf{\Lambda }}$ is invertible and for all $A\in {\mathbb{A}}_{\mathsf{\Lambda }}$, ${\mathsf{\Phi }}_{\mathsf{\Lambda }}^{-1}\left(A\right)={\left\{V_i^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}\right\}}_{i\in \mathbb{I}}$.

2. 

If ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{OPV}$ and $A={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right)$, then ${\mathsf{\Theta }}_{\mathsf{\Gamma }}=A{\mathsf{\Theta }}_{\mathsf{\Lambda }}$ and $V_A=A{\left(A^{*}A\right)}^{-\frac{1}{2}}$ is a partial isometry such that $P_{\mathsf{\Lambda }}=V_A^{*}{V}_A$ and $P_{\mathsf{\Gamma }}=V_A{V}_A^{*}$.

Proof. 

Let ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{OPV}$ and let $A={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right)={\mathsf{\Theta }}_{\mathsf{\Gamma }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$. Then

$$A{P}_{\mathsf{\Lambda }}={\mathsf{\Theta }}_{\mathsf{\Gamma }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}={\mathsf{\Theta }}_{\mathsf{\Gamma }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}=A.$$

Suppose that $\alpha$ is a lower operator valued frame bound for ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ and $\beta$ is an upper operator valued frame bound for ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$, then

$$\begin{array}{cc}{A}^{*}A\hfill & ={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Gamma }}^{*}{\mathsf{\Theta }}_{\mathsf{\Gamma }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}\hfill \\ \hfill & ={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{S}_{\mathsf{\Gamma }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}\hfill \\ \hfill & \ge \alpha {\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}\hfill \\ \hfill & \ge \frac{\alpha }{\beta }{\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}\hfill \\ \hfill & =\frac{\alpha }{\beta }{P}_{\mathsf{\Lambda }}.\hfill \end{array}$$

It follows that $A^{*}A$ is invertible in $\mathbb{B}\left(P_{\mathsf{\Lambda }}\mathbb{M}\right)$ and therefore $A\in {\mathbb{A}}_{\mathsf{\Lambda }}$. Now assume that ${\left\{{\Delta }_i\right\}}_{i\in \mathbb{I}}$ is another operator valued frame in ${\mathbb{F}}_{OPV}$ such that

$${\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right)={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\Delta }_i\right\}}_{i\in \mathbb{I}}\right),$$

then we have

$$\begin{array}{cc}{\mathsf{\Theta }}_{\mathsf{\Gamma }}\hfill & ={\mathsf{\Theta }}_{\mathsf{\Gamma }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}\hfill \\ \hfill & =A{\mathsf{\Theta }}_{\mathsf{\Lambda }}\hfill \\ \hfill & ={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right){\mathsf{\Theta }}_{\mathsf{\Lambda }}\hfill \\ \hfill & ={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\Delta }_i\right\}}_{i\in \mathbb{I}}\right){\mathsf{\Theta }}_{\mathsf{\Lambda }}\hfill \\ \hfill & ={\mathsf{\Theta }}_{\Delta }{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}\hfill \\ \hfill & ={\mathsf{\Theta }}_{\Delta }.\hfill \end{array}$$

Moreover, it is readily to observe that two operator valued frames ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{{\Delta }_i\right\}}_{i\in \mathbb{I}}$ are identical if and only if ${\mathsf{\Theta }}_{\mathsf{\Gamma }}={\mathsf{\Theta }}_{\Delta }$. Therefore, ${\mathsf{\Gamma }}_i={\Delta }_i$ for all $i\in \mathbb{I}$.

In following we prove that ${\mathsf{\Phi }}_{\mathsf{\Lambda }}$ is surjective. For any $A\in {\mathbb{A}}_{\mathsf{\Lambda }}$, define ${\mathsf{\Gamma }}_i=V_i^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}$ for all $i\in \mathbb{I}$. Then

$$\begin{array}{cc}\sum _{i\in \mathbb{I}}{\mathsf{\Gamma }}_i^{*}{\mathsf{\Gamma }}_i\hfill & =\sum _{i\in \mathbb{I}}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{A}^{*}{V}_i{V}_i^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}\hfill \\ \hfill & ={\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{A}^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}\hfill \\ \hfill & \le {\parallel A\parallel }^2{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}\hfill \\ \hfill & ={\parallel A\parallel }^2{S}_{\mathsf{\Lambda }}.\hfill \end{array}$$

Since $A^{*}A\ge {\parallel {\left(A^{*}A\right)}^{-1}\parallel }^{-1}$, we obtain

$$\sum _{i\in \mathbb{I}}{\mathsf{\Gamma }}_i^{*}{\mathsf{\Gamma }}_i\ge {\parallel {\left(A^{*}A\right)}^{-1}\parallel }^{-1}{S}_{\mathsf{\Lambda }},$$

which proves that ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{OPV}$. Moreover,

$${\mathsf{\Theta }}_{\mathsf{\Gamma }}=\sum _{i\in \mathbb{I}}{V}_i{\mathsf{\Gamma }}_i=\sum _{i\in \mathbb{I}}{V}_i{V}_i^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}.$$

By definition of ${\mathsf{\Phi }}_{\mathsf{\Lambda }}$, we know that ${\mathsf{\Theta }}_{\mathsf{\Gamma }}={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right){\mathsf{\Theta }}_{\mathsf{\Lambda }}.$ Therefore

$$A{\mathsf{\Theta }}_{\mathsf{\Lambda }}={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right){\mathsf{\Theta }}_{\mathsf{\Lambda }}$$

and thus

$$\begin{array}{c}A=A{P}_{\mathsf{\Lambda }}=A{\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right){\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right).\hfill \end{array}$$

Hence ${\mathsf{\Phi }}_{\mathsf{\Lambda }}$ is surjective and ${\mathsf{\Phi }}_{\mathsf{\Lambda }}^{-1}\left(A\right)={\left\{V_i^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}\right\}}_{i\in \mathbb{I}}$ for any $A\in {\mathbb{A}}_{\mathsf{\Lambda }}$.

Recall that ${\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-\frac{1}{2}}$ is an isometry, therefore ${\mathsf{\Theta }}_{\mathsf{\Lambda }}$ is injective. By definition, $P_{\mathsf{\Gamma }}$ is the range projection of ${\mathsf{\Theta }}_{\mathsf{\Gamma }}=A{\mathsf{\Theta }}_{\mathsf{\Lambda }}$ and hence of A. Due to

$$V_A^{*}{V}_A={\left(A^{*}A\right)}^{-\frac{1}{2}}\left(A^{*}A\right){\left(A^{*}A\right)}^{-\frac{1}{2}}=P_{\mathsf{\Lambda }},$$

we obtain $V_A$ is a partial isometry. Since ${\left(A^{*}A\right)}^{-\frac{1}{2}}$ is an invertible operator, we have $\mathrm{ran}\left(V_A\right)=\mathrm{ran}\left(A\right)$. It follows that $V_A{V}_A^{*}=P_{\mathsf{\Gamma }}$, as desired. □

As another application of Theorem 2, we have the following result in the framework of quaternionic Hilbert spaces.

Corollary 1.

Let ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}},{\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}},{\left\{{\Delta }_i\right\}}_{i\in \mathbb{I}}$ be operator valued frames in ${\mathbb{F}}_{OPV}$. Then

$${\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\Delta }_i\right\}}_{i\in \mathbb{I}}\right)={\mathsf{\Phi }}_{\mathsf{\Gamma }}\left({\left\{{\Delta }_i\right\}}_{i\in \mathbb{I}}\right){\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right).$$

Proof. 

The desired result follows from equality

$${\mathsf{\Theta }}_{\Delta }{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}={\mathsf{\Theta }}_{\Delta }{S}_{\mathsf{\Gamma }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Gamma }}^{*}{\mathsf{\Theta }}_{\mathsf{\Gamma }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}.$$

□

We now give a parametrization result of Parseval operator valued frames in the quaternionic Hilbert spaces.

Theorem 3.

Suppose that ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ is a Parseval operator valued frame in ${\mathbb{F}}_{P-OPV}$, then

$${\mathbb{F}}_{P-OPV}=\{{\left\{V_i^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}\right\}}_{i\in \mathbb{I}}\mid A\in \mathbb{B}\left(\mathbb{M}\right),A^{*}A=P_{\mathsf{\Lambda }}\}.$$

In particular, if $A\in \mathbb{B}\left(\mathbb{M}\right)$, $A^{*}A=P_{\mathsf{\Lambda }}$ and ${\{{\mathsf{\Gamma }}_i=V_i^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{P-OPV}$, then $A={\mathsf{\Theta }}_{\mathsf{\Gamma }}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$.

Proof. 

Let ${\mathsf{\Gamma }}_i=V_i^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}$ for all $i\in \mathbb{I}$. First note that ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{P-OPV}$ if and only if ${\mathsf{\Theta }}_{\mathsf{\Gamma }}{\mathsf{\Theta }}_{\mathsf{\Gamma }}^{*}$ is a projection. Since

$${\mathsf{\Theta }}_{\mathsf{\Gamma }}{\mathsf{\Theta }}_{\mathsf{\Gamma }}^{*}=A{\mathsf{\Theta }}_{\mathsf{\Lambda }}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{A}^{*}=A{P}_{\mathsf{\Lambda }}{A}^{*}=A{A}^{*},$$

we see that ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is a Parseval operator valued frame if and only if A is a partial isometry. In particular,

$$A={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right)={\mathsf{\Theta }}_{\mathsf{\Gamma }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}={\mathsf{\Theta }}_{\mathsf{\Gamma }}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}.$$

□

Next, we introduce the dual and canonical dual of operator valued frames on quaternion Hilbert spaces.

Definition 2.

Let ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ be operator valued frames in ${\mathbb{F}}_{OPV}$. Then ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is called a dual of ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ if ${\mathsf{\Theta }}_{\mathsf{\Gamma }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}=I_{\mathbb{H}}$. The operator valued frame ${\left\{{\mathsf{\Lambda }}_i{S}_{\mathsf{\Lambda }}^{-1}\right\}}_{i\in \mathbb{I}}$ is called the canonical dual of ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$.

Proposition 3.

Let ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ be operator valued frames in ${\mathbb{F}}_{OPV}$ and $A={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right)$. Then the following statements are equivalent.

1. 

${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is a dual of ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$.

2. 

${\sum }_{i\in \mathbb{I}}{\mathsf{\Gamma }}_i^{*}{\mathsf{\Lambda }}_i$.

3. 

$P_{\mathsf{\Lambda }}A{P}_{\mathsf{\Lambda }}={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-2}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$.

Proof. 

Clearly, we have 1 ⇔ 2.

1 ⇒ 3. By Theorem 2, there exists a unique operator $A\in \mathbb{B}\left(\mathbb{M}\right)$ such that ${\mathsf{\Gamma }}_i=V_i^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}$ for all $i\in \mathbb{I}$, where $A=A{P}_{\mathsf{\Lambda }}$ and $A^{*}A{\mid }_{P_{\mathsf{\Lambda }}\mathbb{M}}$ is invertible. In particular, we have ${\mathsf{\Theta }}_{\mathsf{\Gamma }}=A{\mathsf{\Theta }}_{\mathsf{\Lambda }}$. If ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is a dual of ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$, then ${\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}=I_{\mathbb{H}}$. Therefore,

$$P_{\mathsf{\Lambda }}A{P}_{\mathsf{\Lambda }}={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}A{\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-2}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}.$$

3 ⇒ 1. This is obvious. □

The following Proposition 4 gives a parametrization the canonical dual of operator valued frames in quaternion Hilbert spaces.

Proposition 4.

Let ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ be operator valued frames in ${\mathbb{F}}_{OPV}$. Then ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is the canonical dual of ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ if and only if $A=P_{\mathsf{\Lambda }}A{P}_{\mathsf{\Lambda }}$.

Proof. 

Since ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is the canonical dual of ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ if and only if ${\mathsf{\Theta }}_{\mathsf{\Gamma }}={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-2}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}$, by Theorem 2 and Proposition 3, we have ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is the canonical dual of ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ if and only if $A=P_{\mathsf{\Lambda }}A{P}_{\mathsf{\Lambda }}$. □

Driven by the idea of canonical dual, we introduce the notion of similar operator valued frames in quaternion Hilbert spaces.

Definition 3.

Let ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ be operator valued frames in ${\mathbb{F}}_{OPV}$. We say that ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is similar to ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ if there is an invertible operator $T\in \mathbb{B}\left(\mathbb{H}\right)$ such that ${\{{\mathsf{\Gamma }}_i={\mathsf{\Lambda }}_{i}T\}}_{i\in \mathbb{I}}$.

Before state our next result, we will provide some fundamental observations about similarity of operator valued frames in quaternion Hilbert spaces..

Theorem 4.

Suppose that ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ is an operator valued frame in ${\mathbb{F}}_{OPV}$ with bounds α and β. Let $T\in \mathbb{B}\left(\mathbb{H}\right)$ be an invertible operator and ${\mathsf{\Gamma }}_i={\mathsf{\Lambda }}_{i}T$ for all $i\in \mathbb{I}$. Then

1. 

${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is an operator valued frame in ${\mathbb{F}}_{OPV}$ with bound $\frac{\alpha }{\parallel T^{-1}{\parallel }^2}$ and ${\beta \parallel T\parallel }^2$.

2. 

If T is unitary, then ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is an operator valued frame in ${\mathbb{F}}_{OPV}$ with bound α and β.

3. 

If ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{P-OPV}$, then ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{P-OPV}$ if and only if T is unitary.

4. 

${\mathsf{\Theta }}_{\mathsf{\Gamma }}={\mathsf{\Theta }}_{\mathsf{\Lambda }}T$ and $S_{\mathsf{\Gamma }}=T^{*}{S}_{\mathsf{\Lambda }}T$.

5. 

${\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right)={\mathsf{\Theta }}_{\mathsf{\Lambda }}T{S}_{\mathsf{\Gamma }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$.

Proof. 

1. Assume that ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ is an operator valued frames in ${\mathbb{F}}_{OPV}$ with bounds $\alpha$ and $\beta$. Then

$$S_{\mathsf{\Gamma }}=\sum _{i\in \mathbb{I}}{\mathsf{\Gamma }}_i^{*}{\mathsf{\Gamma }}_i=\sum _{i\in \mathbb{I}}{T}^{*}{\mathsf{\Lambda }}_i^{*}{\mathsf{\Lambda }}_{i}T=T^{*}{S}_{\mathsf{\Lambda }}T,$$

which implies that $\frac{\alpha }{\parallel T^{-1}{\parallel }^2}\le S_{\mathsf{\Gamma }}\le \beta {\parallel T\parallel }^2$.

2. This is immediately follows from 1.

3. If ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\in {\mathbb{F}}_{P-OPV}$, then $S_{\mathsf{\Gamma }}=T^{*}T=I_{\mathbb{H}}.$ It follows that ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ is Parseval if and only if T is unitary.

4. Indeed,

$${\mathsf{\Theta }}_{\mathsf{\Gamma }}=\sum _{i\in \mathbb{I}}{V}_i{\mathsf{\Gamma }}_i=\sum _{i\in \mathbb{I}}{V}_i{\mathsf{\Lambda }}_{i}T={\mathsf{\Theta }}_{\mathsf{\Lambda }}T.$$

5. Clearly. □

Finally, we give some characterizations of similar operator valued frames in quaternion Hilbert spaces.

Proposition 5.

Assume that ${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ are operator valued frames in ${\mathbb{F}}_{OPV}$ and $A={\mathsf{\Phi }}_{\mathsf{\Lambda }}\left({\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}\right)$. Then the following statements are equivalent.

1. 

${\left\{{\mathsf{\Lambda }}_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{{\mathsf{\Gamma }}_i\right\}}_{i\in \mathbb{I}}$ are similar.

2. 

${\mathsf{\Theta }}_{\mathsf{\Gamma }}={\mathsf{\Theta }}_{\mathsf{\Lambda }}T$ for some invertible operator $T\in \mathbb{B}\left(\mathbb{H}\right)$.

3. 

$A=P_{\mathsf{\Lambda }}A{P}_{\mathsf{\Lambda }}$ is invertible in $\mathbb{B}\left(P_{\mathsf{\Lambda }}\mathbb{M}\right)$.

4. 

$P_{\mathsf{\Lambda }}=P_{\mathsf{\Gamma }}$.

Proof. 

1 ⇔ 2. This is given by Theorem 4.

2 ⇒ 3. Let $B={\mathsf{\Theta }}_{\mathsf{\Lambda }}T{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$. Then

$$\begin{array}{cc}{B}^{*}B\hfill & ={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{T}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}T{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}\hfill \\ \hfill & \ge {\alpha \parallel T\parallel }^2{\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-2}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}\hfill \\ \hfill & \ge \frac{\alpha }{\beta }{\parallel T\parallel }^2{P}_{\mathsf{\Lambda }},\hfill \end{array}$$

and $B{P}_{\mathsf{\Lambda }}={\mathsf{\Theta }}_{\mathsf{\Lambda }}T{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{P}_{\mathsf{\Lambda }}=B$. Therefore $B\in {\mathbb{A}}_{\mathsf{\Lambda }}$. By the injectivity of ${\mathsf{\Phi }}_{\mathsf{\Lambda }}$, we obtain $A={\mathsf{\Theta }}_{\mathsf{\Lambda }}T{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$. Since $P_{\mathsf{\Lambda }}A={\mathsf{\Theta }}_{\mathsf{\Lambda }}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}T{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}=A$, we arrived at $A=P_{\mathsf{\Lambda }}A{P}_{\mathsf{\Lambda }}$. Now write $C={\mathsf{\Theta }}_{\mathsf{\Lambda }}{T}^{-1}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}$. Then

$$AC={\mathsf{\Theta }}_{\mathsf{\Lambda }}T{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}{T}^{-1}{S}_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}=P_{\mathsf{\Lambda }}=CA.$$

Hence $P_{\mathsf{\Lambda }}A{P}_{\mathsf{\Lambda }}$ is invertible in $\mathbb{B}\left(P_{\mathsf{\Lambda }}\mathbb{M}\right)$.

3 ⇒ 4. Suppose that A is invertible in $\mathbb{B}\left(P_{\mathsf{\Lambda }}\mathbb{M}\right)$, then $P_{\mathsf{\Lambda }}$ is the range projection of A and hence $P_{\mathsf{\Lambda }}=P_{\mathsf{\Gamma }}$.

4 ⇒ 2. Let $T=S_{\mathsf{\Lambda }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Lambda }}^{*}{\mathsf{\Theta }}_{\mathsf{\Gamma }}$ and $S=S_{\mathsf{\Gamma }}^{-1}{\mathsf{\Theta }}_{\mathsf{\Gamma }}^{*}{\mathsf{\Theta }}_{\mathsf{\Lambda }}$. Then it is readily to verify that $TS=ST=I_{\mathbb{H}}$ and ${\mathsf{\Theta }}_{\mathsf{\Gamma }}={\mathsf{\Theta }}_{\mathsf{\Lambda }}T$. □

4. Discussion

The main purpose of this paper is to introduce some of the natural generalizations of operator valued frame theory to quaternionic Hilbert spaces and study some important properties of operator valued frames in this setting. The operator valued frames as a natural generalization of vector valued frame theory were developed by Kaftal, Larson and Zhang in [18]. The theoretical framework established in this article can be used to extend the vector valued results as well as the classical results of wavelet and dual frame theory to the context of quaternionic Hilbert spaces.

5. Conclusions

In this paper, we have defined compact and Parseval operator value frames on quaternionic Hilbert spaces and obtained the parameterization of Parseval operator value frames according to a class of partial isometry in quaternionic Hilbert spaces. As the main result, we have obtained a parameterization of all operator valued frames on quaternionic Hilbert spaces, which allows us to study the similarity between operator valued frames. The similarity of operator valued frames obtained by multiplying operators from right extends the general similarity in vector case, and inherits its main properties. Moreover, we have characterized the case that the two operator value frames are similar and parameterized the dual operator value frames.