Some properties of weaving $K$-frames in $n$-Hilbert space

1.   Introduction

Frame theory is based on the development of wavelet theory. Frames have gradually become an important tool in signal processing due to the need to solve increasingly complex real problems. A variety of new generalized frames have also emerged, and many researchers have studied $K$-frames in Hilbert spaces ^[1,2]. For instance, Bemrose et al. ^[3] introduced weaving frames in Hilbert spaces. There are also many results on weaving $K$-frames in Hilbert spaces ^[4,5].

S. Gähler ^[6], Diminnie et al. ^[7], H. Gunawan ^[8], and A. Misiak ^[9] introduced the concept of linear 2-normed spaces, 2-inner product spaces, n-normed spaces, and $n$-inner product spaces for $n\geq 2$.

For the last 30 years, research on 2-Hilbert space, and $n$-Hilbert space has been an important topic in the field of functional analysis. However, because wavelet theory and frame theory were developed relatively recently, and the classical results of frame theory are established in Hilbert spaces, few works have used frame theory for studying frames in $n$-Hilbert spaces. Wavelet theory researchers need to further explore this area.

Recently, A. Akbar et al. ^[10] studied frames in a 2-inner product space. P. Ghosh et al. ^[11] presented the notion of frames in $n$-Hilbert spaces.

Now, various generalized frames, such as $G$-frames, $K$-frames, and weaving (or woven) frames in Hilbert spaces are a hot topic in frame theory. For instance, Li et al. ^[13] discussed weaving $g$-frames in Hilbert spaces.

The motivation of this article is to study weaving $K$-frames in $n$-Hilbert spaces, we still introduce and characterize the concept of weaving $K$-frames in $n$-Hilbert spaces and present several new methods for their construction. We then present some stability and perturbation results for weaving $K$-frames in $n$-Hilbert spaces.

The systematic study of the theory of various generalized frames in $n$-Hilbert spaces, and in particular, the definition of various generalized frames in $n$-Hilbert spaces, and the study of the characterization, perturbation, stability, and constructive properties of these generalized frames, will enrich and expand the theory of frames.

Throughout this paper, suppose that $H$ denotes a separable Hilbert space with the inner product $\langle\cdot, \cdot\rangle$, and $B(H)$ denotes the space of all bounded linear operators on $H$. We also denote $R(T)$ as a range set of $T$, where $T\in B(H)$. Let $\mathbb{N}$ be an index set of natural numbers, and $\ell^2(\mathbb{N})$ denotes the space of square, summable scalar-valued sequences with the index set $\mathbb{N}$. For a given number, $m\in \mathbb{N}$, let $[m] = \{1, 2, \ldots, m\}$ and $[m]^{c} = \{m+1, m+2, \ldots\}$. As usual, we denote the set of all bounded linear operators from $H$ to another Hilbert space $K$ by $B(H, K)$, and if $H = K$, then $B(H, K)$ is abbreviated to $B(H)$.

2.   Preliminaries

Lemma 2.1. ^[14] Let $T_{1}, T_{2}\in B(H)$. Then, there are the following equivalent statements:

$(i)$ For some $\alpha > 0$, $T_{1}T_{1}^{*}\leq \alpha^{2}T_{2}T_{2}^{*}$;

$(ii)$ $R(T_{1})\subseteq R(T_{2})$;

$(iii)$ $T_{1} = T_{2}W$ for some $W\in B(H)$.

Lemma 2.2. ^[15]. Let $H_1, H_2$ be two Hilbert spaces and $T_{1}\in B(H_1, H_2)$, where $R({T_{1}})$ is closed. Then, there exists $T_{1}^{+}: H_2\rightarrow H_1$, the pseudo-inverse of $T_{1}$, such that $T_{1}T_{1}^{+}x = x$, $\forall x\in R({T_{1}})$.

Definition 2.1. ^[8] Let $n\in \mathbb{N}$ and $X$ be a linear space of dimensions $d\geq n$. let $\Vert \cdot, \cdots, \cdot\Vert : X^n\rightarrow \mathbb{R}$ be a function such that for every $v, w_1, w_2, \cdots, w_n \in X$ and $\alpha\in \mathbb{R}$, there is

(i) $\Vert w_1, w_2, \cdots, w_n\Vert = 0$ if and only if $w_1, w_2, \cdots, w_n$ are linearly dependent;

(ii) $\Vert w_1, w_2, \cdots, w_n\Vert$ is invariant under any permutations of $w_1, w_2, \cdots, w_n$;

(iii) $\Vert \alpha w_1, w_2, \cdots, w_n\Vert = |\alpha|\Vert w_1, w_2, \cdots, w_n\Vert$, $\alpha\in \mathbb{R}$;

(iv) $\Vert w_1+v, w_2, \cdots, w_n\Vert\leq\Vert w_1, w_2, \cdots, w_n\Vert+\Vert v, w_2, \cdots, w_n\Vert$.

The function $\Vert \cdot, \cdots, \cdot\Vert : X^n\rightarrow \mathbb{R}$ is called an $n$-norm on $X$, and the pair (X, $\Vert \cdot, \cdots, \cdot\Vert)$ is called an (real) $n$-normed space.

Remark 2.1. Gähler introduced the concept of $n$-norm to generalize the notion of length, area, and volume in a real vector space (see ^[6]).

Definition 2.2. ^[9] Let $n\in \mathbb{N}$ and $X$ be a linear space of dimensions $d\geq n$ and let $\langle \cdot, \cdot|\cdot, \cdots, \cdot\rangle: X^{n+1}\rightarrow \mathbb{R}$ be a function such that for every $w, v, w_1, w_2, \cdots, w_n \in X$ and $\alpha\in \mathbb{R}$, we have

(i) $\langle w_1, w_1|w_2, \cdots, w_n\rangle\geq 0$ and $\langle w_1, w_1|w_2, \cdots, w_n\rangle = 0$ if and only if $w_1, w_2, \cdots, w_n$ are linearly dependent;

(ii) $\langle w, v|w_2, \cdots, w_n\rangle = \langle w, v|w_{i_2}, \cdots, w_{i_n}\rangle$ for every permutation $(i_2, \cdots, i_n)$ of $(2, \cdots, n)$;

(iii) $\langle w, v|w_2, \cdots, w_n\rangle = \langle v, w|w_2, \cdots, w_n\rangle$;

(iv) $\langle \alpha w, v|w_2, \cdots, w_n\rangle = \alpha\langle w, v|w_2, \cdots, w_n\rangle, \quad for\, every\, \, \alpha\in \mathbb{R}$;

(v) $\langle w+v, w_1|w_2, \cdots, w_n\rangle = \langle w, w_1|w_2, \cdots, w_n\rangle+\langle v, w_1|w_2, \cdots, w_n\rangle$.

The function $\langle \cdot, \cdot|\cdot, \cdots, \cdot\rangle: X^{n+1}\rightarrow \mathbb{R}$ is called an $n$-inner product. Here, the pair $(X, \langle \cdot, \cdot|\cdot, \cdots, \cdot\rangle)$ is called a (real) $n$-inner product space.

Lemma 2.3. ^[9] Let $X$ be an $n$-inner product space. Then,

defines an n-norm, for which

and

hold for all $w, v, w_1, w_2, \cdots, w_n\in X$.

Remark 2.2. ^[16] Any inner product space $(X, \langle \cdot, \cdot|\cdot, \cdots, \cdot\rangle)$ can be equipped with the standard $n$-inner product

and its induced $n$-norm.

Definition 2.3. ^[17] Let $(X, \langle \cdot, \cdot|\cdot, \cdots, \cdot\rangle)$ be an $n$-inner product space and $\{e_i\}_{i = 1}^{n}$ be linearly independent vectors in $X$. Then, for a given set $F = \{a_2, \cdots, a_n\}\subset X$, if $\langle e_i, e_j|a_2, \cdots, a_n\rangle = \delta_{i, j}, i, j\in\{1, 2, \cdots, n\}$, where

the family $\{e_i\}_{i = 1}^{n}$ is said to be $F$-orthogonal. If an $F$-orthogonal set is countable, we can arrange it in the form of a sequence $\{e_i\}$ and call it an $F$-orthogonal sequence.

Remark 2.3. It was shown in ^[18] that $\ell^{2}(\mathbb{N})$ has its natural $n$-norm, which can be viewed as a generalization of its usual norm. It was proven in ^[17] that $\ell^{2}(\mathbb{N})$ has an $F$-orthonormal basis $\{ e_{j} \}_{j = 1}^{\infty}$.

Definition 2.4. ^[8] A sequence $\{x_k\}$ in a linear n-normed space $X$ is said to be convergent to some $x\in X$ if for every $c_2, \cdots, c_n\in X$, $\lim\limits_{k \to \infty}\Vert x_k-x, c_2, \cdots, c_n\Vert = 0$, and it is called a Cauchy sequence if $\lim\limits_{l, k \to \infty}\Vert x_l-x_k, c_2, \cdots, c_n\Vert = 0$ for every $c_2, \cdots, c_n\in X$. The space $X$ is said to be complete if every Cauchy sequence in this space is convergent with $X$. An $n$-inner product space is called an $n$-Hilbert space if it is complete with respect to its induced norm.

In order to construct the workspace for our discussion, let $H$ be an $n$-Hilbert space; consider $C = \{c_2, c_3, \cdots, c_n\}$, where $c_2, c_3, \cdots, c_n$ are fixed elements in $H$. Let $L_C$ be the linear subspace of $H$ spanned by the non-empty finite set $C$. Then, the quotient space $H/L_C$ is a normed linear space with respect to the norm

Let $M_C$ be the orthogonal complement of $L_C$, that is, $H = L_C\oplus M_C$. Define $\langle f, g\rangle_C = \langle f, g\left|c_2, \cdots, c_n\right.\rangle$ on $H$. Then, $\langle\cdot, \cdot\rangle_C$ is a semi-inner product for $H$, and this semi-inner product induces an inner product on the quotient space $H/L_C$, which is given by

Now, by identifying $H/L_C$ with $M_C$ in an obvious way, we obtain an inner product on $M_C$. Now, for every $f\in M_C$, we define $\|f\|_{C} = \sqrt{\langle f, f \rangle_{C} }$, and $(M_C, \Vert\cdot\Vert_C)$ is a norm space. Let $H_C$ be the completion of the inner product space $M_C$.

Remark 2.4. In fact, when given an inner product space $(V, \langle\cdot, \cdot\rangle)$ and a linear independent set $\{c_1, c_2, \cdots, c_n\}$ in $V$, we can, in general, derive a new inner product of $\langle\cdot, \cdot\rangle^*$ from the given inner product $\langle\cdot, \cdot\rangle$ by first defining an $n$-inner product on $V$ and then defining the new inner product $\langle\cdot, \cdot\rangle^*$ on $V$ with respect to $\{c_1, c_2, \cdots, c_n\}$ (see ^[19]).

Remark 2.5. For any $n$-inner product space with $n\geq 2$, we can derive an inner product from the $n$-inner product so that one can develop the notion of orthogonality and the Fourier series theory in an $n$-inner product space just as in an inner product space (see ^[20]).

Definition 2.5. ^[11] Let $H$ be an $n$-Hilbert space and $c_2, \cdots, c_n \in H$. If there exists a constant $0 < A\le B < \infty$ such that

then $\{f_{i}\}_{i = 1}^{\infty}$ in $H$ is said to be an $(A, \, B)$ frame associated with $\left(c_2, \cdots, c_n\right)$ for $H$, with lower- and upper-frame bounds of $A$ and $\, B$.

If $\left\{f_i\right\}_{i = 1}^\infty$ only satisfies the right-hand side of the inequality, then $\{f_{i}\}_{i = 1}^{\infty}$ is called a Bessel sequence associated with $\left(c_2, \cdots, c_n\right)$ for $H$.

Let $\{f_{i}\}_{i = 1}^{\infty}$ be an $(A, B)$ frame associated with $\left(c_2, \cdots, c_n\right)$ for $H$, with the frame bounds $A, \, B$. Then, the preframe operator for $\left\{f_i\right\}_{i = 1}^\infty$ is

The analysis operator for $\left\{f_i\right\}_{i = 1}^\infty$ is

and the frame operator $S_C$ for $\left\{f_i\right\}_{i = 1}^\infty$ is

for all $f\in H_C$.

It is easy to prove the following fact.

Let $H$ be a $n$-Hilbert space. $c_2, \cdots, c_n \in H$, we say $\{f_{i}\}_{i = 1}^{\infty}$ in $H$ is an $(A, \, B)$ frame associated with $\left(c_2, \cdots, c_n\right)$ for $H$, with lower- and upper-frame bounds of $A$ and $\, B$ if and only if it is an $(A, \, B)$ frame associated with $\left(c_2, \cdots, c_n\right)$ for $H_C$, with lower- and upper-frame bounds of $A$ and $\, B$.

In what follows, we use $\left(H, \langle\cdot, \cdot\left|\cdot, \cdots, \cdot\right.\rangle\right)$ to denote an $n$-Hilbert space and $I_{H}$ to denote the identity operator on $H$. Let $B\left(H_C\right)$ be the space of all bounded linear operators on $H_C$.

3.   Weaving $K$-frames in $n$-Hilbert spaces

Definition 3.1. ^[12] Let $K \in B(H_C)$. A sequence $\{f_i\}_{i = 1}^{\infty}\subseteq H$ is said to be an $(A, \, B)$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$} if there are constants $0 < A\leq B < +\infty$ such that

for all $f\in H_C$, where $K^*$ denotes the adjoint operator of $K$.

Definition 3.2. Let $\left\{\left\{f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ be a collection of $K$-frames associated with $(c_2, \cdots, c_n)$ for $H$. $\left\{\left\{f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ is said to be $(A_F, \, B_F)$ $K$-woven frame associated with $(c_2, \cdots, c_n)$ for $H$ if there are constants of $A_F$ and $B_F$ such that for every partition $\left\{\sigma_1, \sigma_2, \cdots, \sigma_m\right\}$ of $\mathbb{N}$, $\left\{\left\{f_{1j}\right\}_{j\in \sigma_1}, \ldots, \left\{f_{mj}\right\}_{j\in \sigma_m}\right\}$ is an $(A_F, \, B_F)$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$ with lower- and upper-$K$-frame bounds of $A_F$ and $B_F$, respectively. Each collection $\left\{\left\{f_{1j}\right\}_{j\in \sigma_1}, \ldots, \left\{f_{mj}\right\}_{j\in \sigma_m}\right\}$ is called a weaving associated with $(c_2, \cdots, c_n)$ for $H$.

$\left\{\left\{f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ is said to be $A_F$ tight $K$-woven frame associated with $(c_2, \cdots, c_n)$ for $H$, if $A_F = B_F$, and is said to be Parseval $K$-woven frame associated with $(c_2, \cdots, c_n)$ for $H$, if $A_F = B_F = 1$.

Example 3.1. We consider the simple case when $K = I_H$ in Definitions 3.1 and 3.2. When given the subset $E$ of $\mathbb{N}$, a family $\mathcal{F} = \left\{\left\{f_{ij}\right\}_{j = 1}^{\infty}\right\}_{i\in E}$ of sequences $f_i(i\in E)$ in an $n$-Hilbert space $H$ is present, and for every partition $\sigma = \left\{\sigma_i\right\}_{i\in E}$ of $\mathbb{N}$, let $\Delta$ be a bijection from $\Sigma(\sigma): = \cup_{i\in E}\{(i, j):j\in{\sigma_i}\}$ onto $\mathbb{N}$. According to this bijection $\Delta$, for each $n\in\mathbb{N}$, there exists a unique element $(i, j)$ of $\Sigma(\sigma)$ such that $j\in\sigma_i$ and $n = \Delta(i, j)$. We define $f^{\sigma, \Delta}_n = f_{ij}$ and then obtain the sequence $\mathcal{F}^{\sigma, \Delta} = \{f^{\sigma, \Delta}_n\}_{n = 1}^\infty$, denoted by $\left\{f_{ij}\right\}_{j\in\sigma_i, i\in E}$ (or $\cup_{i\in E}\left\{f_{ij}\right\}_{j\in\sigma_i}$ for short.) We call the sequence $\mathcal{F}^{\sigma, \Delta}$ a woven sequence when $K = I_H$ of the family $\mathcal{F}$ with respect to the partition $\sigma$ and the bijection $\Delta$.

For example, when $E = \mathbb{N}, \, \, \sigma_i = \{2i-1, 2i\}(i\in \mathbb{N})$, we obtain a partition $\sigma = \left\{\sigma_i\right\}_{i\in E}$ of $\mathbb{N}$. By listing the elements of $\Sigma(\sigma) = \cup_{i\in \mathbb{N}}\{(i, 2i-1), (i, 2i)\}$ according to the direction shown in Figure 1, we obtain a bijection $\Delta$ from $\Sigma(\sigma)$ onto $\mathbb{N}$. In this case, the woven sequence $\mathcal{F}^{\sigma, \Delta}$ of the family $\mathcal{F}$ with respect to the partition $\sigma$ and the bijection $\Delta$ is as follows:

where $f_{i, j} = f_{ij}.$

Clearly, for any two bijections $\Delta_k(k = 1, 2)$ from $\Sigma(\sigma)$ onto $\mathbb{N}$, $\mathcal{F}^{\sigma, \Delta_1}$ is a frame (or Bessel sequence) associated to $\left(c_2, \cdots, c_n\right)$ for $H$ if and only if $\mathcal{F}^{\sigma, \Delta_2}$ is a frame (or Bessel sequence) associated with $\left(c_2, \cdots, c_n\right)$ for $H$ too.

4.   Some characterizations of weaving $K$-frames in $n$-Hilbert spaces

Theorem 4.1. For every $i\in [m]$, let $\{f_{ij}\}_{j\in 1}^{\infty}$ be $(A_{i}\, \, , B_{i})$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$. Then, for every partition $\left\{\sigma_1, \sigma_2, \cdots, \sigma_m\right\}$ of $\mathbb{N}$, $\cup_{{i\in [m]}}\{f_{ij}\}_{j\in \sigma_i}$ is a $\left(\sum\limits_{i\in [m]}B_i\right)$ Bessel sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. For every partition $\left\{\sigma_1, \sigma_2, \cdots, \sigma_m\right\}$ of $\mathbb{N}$, according to Definitions 3.2 and 3.1, we have

for all $f\in H_C$. □

Theorem 4.2. For every $i\in [m]$, let $\{f_{ij}\}_{j = 1}^{\infty}$ be $(A_{i}\, \, ,B_{i})$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$. Then, there are the following equivalent statements:

($i$) For every partition $\left\{\sigma_1, \sigma_2, \cdots, \sigma_m\right\}$ of $\mathbb{N}$, let $L_{\sigma}\in B(\ell^{2}(\mathbb{N}), H_C)$ be defined by $L_{\sigma}(e_{j}) = f_{ij}$ if $j\in \sigma_i\, \, (i = 1\cdots m)$, and there is $A_F > 0$, so that for every partition $\{\sigma_{i}\}_{i\in [m]}$ of $\mathbb{N}$,

holds true, where $\{ e_{j} \}_{j = 1}^{\infty}$ is an $F$-orthonormal basis for $\ell^{2}(\mathbb{N})$.

($ii$) $\left\{\left\{f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ is $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. $(ii)\Rightarrow (i)$: For every partition $\left\{\sigma_1, \sigma_2, \cdots, \sigma_m\right\}$ of $\mathbb{N}$, let $T_\sigma$ be the preframe operator of $\cup_{i\in [m]}\{f_{ij}\}_{j\in \sigma_{i}}$. Let $L_{\sigma} = T_{\sigma}$. Then, there is $L_{\sigma}(e_{i}) = T_{\sigma}(e_{i}) = f_{ij}$ for every $j\in \sigma(i\in [m])$. Let $A_F$ be the lower $K$-frame bound for $\{ \{ f_{ij}\}_{j\in 1}^{\infty}: i\in [m] \}$.

Then, we have

$A_F\langle KK^{*}f, f|c_2, \cdots, c_n \rangle$

$\leq \sum\limits_{i\in [m]}\sum\limits_{j\in\sigma_{i}}|\langle f, f_{ij}|c_2, \cdots, c_n\rangle |^2$

$= \langle L_{\sigma}L_{\sigma}^{*}f, f|c_2, \cdots, c_n \rangle.$

For every $f\in H_C$.

Then, there is $A_FKK^{*}\leq L_{\sigma}L_{\sigma}^{*}$.

$(i)\Rightarrow (ii)$: According to Theorem 4.1, the positive number $\sum\limits_{i\in[m]}B_{i}$ is an upper $K$-frame bound.

For every partition $\left\{\sigma_1, \sigma_2, \cdots, \sigma_m\right\}$ of $\mathbb{N}$, then, by virtue of Eq (4.1) and the definition of $L_{\sigma}$ in $(i)$, we have

$f\in H_C$.

So, we obtain the lower $K$-frame inequality. Then, $\left\{\left\{f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ is a $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$. □

Theorem 4.3. Let $\left\{\left\{f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ be a collection of $K$-frames associated with $(c_2, \cdots, c_n)$ for $H$. Then, the following statements are equivalent:

(i) For all $T_{1}\in B(H_C)$, $\left\{ \{ T_{1}(f_{1j}) \}_{j = 1}^{\infty}, \cdots, \{ T_{1}(f_{mj}) \}_{j = 1}^{\infty}\right\}$ is a $T_{1}K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

$(ii)$ $\left\{\left\{f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ is a $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. $(ii)\Rightarrow (i)$: Let the $K$-frame bounds for $\left\{\left\{f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ be $(A_F, \, B_F)$.

For every partition $\left\{\sigma_1, \sigma_2, \cdots, \sigma_m\right\}$ of $\mathbb{N}$, according to Definitions 3.2 and 3.1, there is

Similarly, we have

for all $f\in H_C$.

It follows that, $\left\{ \{ T_{1}(f_{1j}) \}_{j = 1}^{\infty}, \cdots, \{ T_{1}(f_{mj}) \}_{j = 1}^{\infty}\right\}$ is a $\left(A_F, \, B_F\Vert T_{1}\Vert^{2} \right)$ $T_{1}K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

$(i)\Rightarrow (ii)$: Let $T_{1} = I_H$. Then, $\{ \{ f_{ij} \}_{j\in l}^{\infty}: i\in [m] \}$ is $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$. □

5.   Construction of weaving $K$-frames in $n$-Hilbert spaces

It was shown in ^[18] that $\ell^{2}(\mathbb{(N)}$ has its natural $n$-norm, which can be viewed as a generalization of its usual norm. It was proven in ^[17] that $\ell^{2}(\mathbb{N})$ has an $F$-orthonormal basis $\{ e_{j} \}_{j = 1}^{\infty}$.

Theorem 5.1. Taking $p, q\in [m]$, where $p, q$ are fixed elements in $[m]$, the following statements are equivalent:

$(i)$ There exists a Bessel sequence $\left\{g_j\right\}_{j\in N}$ associated with $(c_2, \cdots, c_n)$ for $H$ such that for all $\sigma\subset \mathbb{N}$, there is

$(ii)$ Two $K$-frames $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}\right\}_{j\in \mathbb{N}}$ are $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. $(ii)\Rightarrow (i)$. Let $A_F$ be a lower $K$-frame bound of $\left\{f_{pj}\right\}_{j\in N}\cup\left\{f_{qj}\right\}_{j\in N}$. For all $\sigma\subset \mathbb{N}$, let $T_F$ be the preframe operator of the Bessel sequence $\left\{f_{pj}\right\}_{j\in\sigma}\cup\left\{f_{qj}\right\}_{j\in\sigma^c}$. Then, $T_F\left(e_j\right) = f_{pj}$ if $j\in\sigma$, and $T_F\left(e_j\right) = f_{qj}$ if $j\in\sigma^c$, where $\left\{e_j\right\}_{j\in \mathbb{N}}$ is the $F$-orthonormal basis of $\ell^2\left(\mathbb{N}\right)$ (see Definition 2.3 and Remark 2.4).

Since

then we have $KK^\ast\le\frac{1}{A_F}T_F{T_F}^\ast$ According to Lemma 2.1, there exists $W\in B(H_C, \ell^2(\mathbb{N}))$ such that $K = T_FW$. By taking $g_j = W^\ast e_j$ for $j\in \mathbb{N},$ then $\left\{g_j\right\}_{j\in N}$ is a Bessel sequence associated with $(c_2, \cdots, c_n)$ for $H$. Then, we have

for all $f\in H_C$.

$(i)\Rightarrow (ii)$. Let $B_{2}$ be the Bessel bound of $\left\{g_j\right\}_{j\in \mathbb{N}}$. By virtue of Eq (5.1), we have

for all $f, g\in H_{C}$.

It follows that, $K^\ast f = \sum\limits_{j\in\sigma}\langle f, f_{pj}|c_2, \cdots, c_n\rangle g_j+\sum\limits_{j\in\sigma^c}\langle f, f_{qj}|c_2, \cdots, c_n\rangle g_j.$

Thus,

□

Let $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}\right\}_{j\in \mathbb{N}}$ be two Bessel sequences associated with $(c_2, \cdots, c_n)$ for $H$. For every $\sigma\subset \mathbb{N}$, define $S_{F_p, F_q}^{\sigma}:H_C \rightarrow H_C$ by

Then, $S_{F_p, F_q}^{\sigma}$ is a positive and self-adjoint operator.

Theorem 5.2. Taking $p, q\in [m]$, where $p, q$ are fixed elements in $[m]$, let $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ be an $\left(A_p, B_p\right)$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$, and let $\left\{f_{qj}\right\}_{j\in \mathbb{N}}$ be an $\left(A_q, B_q\right)$K-frame associated with $(c_2, \cdots, c_n)$ for $H$. If there are constants $\lambda, \mu \in [0, 1)$ such that $\Vert S_{F_{p}, F_{q}}^{\sigma}f-K^*f, c_2, \cdots, c_n \Vert\leq\lambda\Vert S_{F_{p}, F_{q}}^{\sigma}f, c_2, \cdots, c_n\Vert+\mu\Vert K^*f, c_2, \cdots, c_n\Vert$ holds for all $f\in H_C$, then $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}\right\}_{j\in \mathbb{N}}$ are $\left(\frac{(1-\mu)^{2}}{(1+\lambda)^{2}\Vert T_F \Vert^{2}}, \, \, B_{p}+B_{q}\right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$, where $T_F$ is the preframe operator of the Bessel sequence $\left\{f_{pj}\right\}_{j\in\sigma}\cup \left\{f_{qj}\right\}_{j\in\sigma^c}$.

Proof. For all $f\in H_C$, there is

So, $\frac{1-\mu}{1+\lambda}\Vert K^{*}f, c_2, \cdots, c_n \Vert\leq \Vert S_{F_p, F_q}^{\sigma}f, c_2, \cdots, c_n \Vert$; then, by Eq (5.2) and the definition of $T_F$, we have

□

Theorem 5.3. Taking $i = p, q, r\in [m]$, where $p, q, r$ are fixed elements in $[m]$, let $\left\{f_{ij}\right\}_{j\in \mathbb{N}}$ be an $\left(A_i, B_i\right)$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$, and let $T_i$ be the preframe operator. Let $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}\right\}_{j\in \mathbb{N}}$ be $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$, with lower $K$-frame bounds of $A^{pq}$, and let $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{rj}\right\}_{j\in \mathbb{N}}$ be $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$ with lower $K$-frame bounds of $A^{pr}$. If $K\in B\left(H_{C}\right)$ is a positive and closed range operator, and if $Kf = \sum\limits_{j\in N}{\langle f, f_{qj}|c_2, \cdots, c_n\rangle f_{rj}}$ holds for all $f\in H_C$ and there is $A^{pq}+A^{pr} > \left(B_p+2\sqrt{B_qB_r}\right)\Vert K^{+}\Vert^2$, then $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}+f_{rj}\right\}_{j\in \mathbb{N}}$ are $\left(A^{pq}+A^{pr}-B_p+2\sqrt{B_qB_r}\Vert K^+\Vert^2, \, \, B_p+2\left(B_q+B_r\right)\right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $R(K)$.

Proof. For all $\sigma\subset \mathbb{N}$, let $T_{f_q}^\sigma\left(\left\{a_j\right\}_{j\in N}\right) = \sum\limits_{j\in\sigma}{a_jf_{qj}}$ and $T_{f_r}^\sigma\left(\left\{a_j\right\}_{j\in N}\right) = \sum\limits_{j\in\sigma}{a_jf_{rj}}$ for any $\left\{a_j\right\}_{j\in N}\in \ell^2\left(\mathbb{N}\right)$. Then, there is $\Vert T_{f_{q}}^{\sigma}\Vert\leq\Vert T_{q}\Vert$ and $\Vert T_{f_{r}}^{\sigma}\Vert \leq\Vert T_{r}\Vert$.

Hence,

for all $f\in R(K)$.

The proof for concluding that the upper bound is $B_p+2\left(B_q+B_r\right)$, which is similar. □

Theorem 5.4. Taking $p, q\in [m]$, where $p, q$ are fixed elements in $[m]$, let $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ be an $(A_{p}, B_p)$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$, and let $\left\{f_{qj}\right\}_{j\in \mathbb{N}}$ be an $(A_{q}, B_{q})$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$, if there are constants of $0\leq\lambda, \mu < 1$ such that

for all $\mathbb{I}\subset \mathbb{N}$.

Then, $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}\right\}_{j\in \mathbb{N}}$ are $\left(\left(A_p\, \min{\left\{1, (\frac{1-\lambda}{1-\mu})^2\right\}}\right), \, \, B_p+B_q \right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. For all $\sigma\subset \mathbb{N}$, according to Eq (5.3), we have

for all $f\in H_{C}$.

Then, we have

Hence,

□

Theorem 5.5. Taking $p, q\in [m]$, where $p, q$ are fixed elements in $[m]$, let $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}\right\}_{j\in \mathbb{N} }$ be $(A_F, \, B_F)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$, and let $\left\{g_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{g_{qj}\right\}_{j\in \mathbb{N}}$be$(A_G, \, B_G)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$. For all $\sigma\subset \mathbb{N}$, let $T_F^\sigma$ be the preframe operators of $\left\{f_{pj}\right\}_{j\in\sigma}\cup\left\{f_{qj}\right\}_{j\in\sigma^c}$ and let $T_G^\sigma$ be the preframe operators of $\left\{g_{pj}\right\}_{j\in\sigma}\cup\left\{g_{qj}\right\}_{j\in\sigma^c}$. If there are constants of $0\le\lambda, \mu < 2$ such that

for all $\sigma\subset \mathbb{N}$ and for all $f\in H_{C}$, then $\left\{f_{pj}+g_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}+g_{qj}\right\}_{j\in \mathbb{N}}$ are $\left(((2-\lambda)A_F+(2-\mu)A_G, 2(B_F+B_G)\right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. According to the assumption, for all $\sigma\subset \mathbb{N}$, we have $\Vert T_{F}^{\sigma}f-T_{G}^{\sigma}f, c_2, \cdots, c_n\Vert^{2}\leq\lambda\Vert T_{F}^{\sigma}f, c_2, \cdots, c_n \Vert^{2}+\mu\Vert T_{G}^{\sigma}f, c_2, \cdots, c_n \Vert^{2}$ for all $f\in H_{C}$; thus,

□

6.   Some stability and perturbation results on weaving $K$-frames in $n$-Hilbert spaces

Theorem 6.1. For all $i\in\left[m\right]$, let $\left\{f_{ij}\right\}_{j\in \mathbb{N}}$ be an $\left(A_i, B_i\right)$ $K$-frame associated to $(c_2, \cdots, c_n)$ for $H$, and let $T_{f_i}$ be the preframe operator. For all $\sigma\subset \mathbb{N}$, let $T_{f_i}^\sigma$ be $T_{f_i}^\sigma(\{a_j\}_{j\in N}) = \sum\limits_{j\in \sigma} a_j f_{i, j}$, and $R\left(T_{f_i}^\sigma\right)\subset R\left(K\right)\left(i\in\left[m\right]\right)$. Suppose that $R\left(T_{f_i}^\sigma\right)\subset R\left(K\right)\left(i\in\left[m\right]\right)$. If there are the constants $\alpha_i, \beta_i, \gamma_i\geq0\left(i\in\left[m\right]\right)$ such that

and

for some fixed $n\in\left[m\right]$ and for any sequence of scalars $\left\{a_j\right\}_{c\in \mathbb{N}}\in \ell^2\left(\mathbb{N}\right)$, then $\left\{ \left\{f_{1j}\right\}_{j\in \mathbb{N}}, \cdots, \left\{f_{mj}\right\}_{j\in \mathbb{N}}\right\}$ is $\left(\left(A_{n}-\sum_{i\in\left[m\right]\backslash\left\{n\right\}}(\sqrt{B_{n}+B_{i}})(\lambda_{i}\sqrt{B_{n}}+\mu_{i}\sqrt{B_{i}}+\gamma_{i})\Vert \widetilde{ K^*}^{-1}\Vert^{2} \right)^2, \, \, \, \, \, \sum\limits_{i\in\left[m\right]} B_i \right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$, where $\widetilde{K^\ast}:Ker^\bot\left(K^\ast\right)\rightarrow R\left(K^\ast\right)$ is the restriction of $K^\ast$ on $Ker^\bot\left(K^\ast\right)$.

Proof. There is $\Vert T_{f_{n}} \Vert\leq\sqrt{B_{n}}$ and $\Vert T_{f_{i}} \Vert\leq\sqrt{B_{i}}$ for a fixed $n\in\left[m\right]$ and $i\in\left[m\right]\backslash\left\{n\right\}$. By using Eq (6.1), we have

for all $\sigma\subset \mathbb{N}$, $\left\{a_j\right\}_{j\in N}\in \ell^2\left(\mathbb{N}\right)$, and for all $i\in\left[m\right]\backslash\left\{n\right\}$.

Therefore, $\Vert T_{f_{n}}-T_{f_{i}} \Vert\leq\lambda_{i}\sqrt{B_{n}}+\mu_{i}\sqrt{B_{n}}+\gamma_{i}$. Then, for every partition $\left\{\sigma_i\right\}_{i\in\left[m\right]}$ of $N$, we have

It follows that

for all $g\in Ker^\bot\left(K^\ast\right)$.

Then, for all $f\in H_{C}$, $f = f_{k_1}+f_{k_2}$ holds, where $f_{k_1}\in Ker\left(K^\ast\right)$, and $f_{k_2}\in Ker^\bot\left(K^\ast\right)$ and $for\, \, all \, \, \sigma\subset N$ and $for\, \, all \, \, i\in\left[m\right]$, $R\left(T_{f_n}^{\sigma_i}\right)\subset R\left(K\right)$ holds. So, we have

and the upper $K$-frame bound $\sum\limits_{i\in\left[m\right]} B_i$ of $\left\{f_{ij}\right\}_{j\in N.i\in\left[m\right]}$ is obvious. This completes the proof. □

Theorem 6.2. Let $\left\{\left\{f_{1j}\right\}_{j\in \sigma_1}, \ldots, \left\{f_{mj}\right\}_{j\in \sigma_m}\right\}$ be $\left(A_F, \, B_F\right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$, and let $T_{1}\in B(H_C)$, $T_{1}K = KT_{1}$, and $T_{1}$ have a closed range. If $R(K^{*})\subset R(T_{1})$, then $\left\{ \{ T_{1}(f_{1j}) \}_{j\in \sigma_1}, \cdots, \{ T_{1}(f_{mj}) \}_{j\in \sigma_m}\right\}$ is $\left(A_F\Vert T^{+}\Vert^{-2}, \, \, \, \, \, B_F\Vert T_{1}\Vert^{2}\right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. For every partition $\left\{\sigma_1, \sigma_2, \cdots, \sigma_m\right\}$ of $\mathbb{N}$, then

holds for all $f\in H_C$.

Since $T_{1}K = KT_{1}$, then $K^{*}T_{1}^{*} = T_{1}^{*}K^{*}$ holds. By Lemma 2.2 and the facts that $T_{1}$ has a closed range and $R(K^{*})\subset R(T_{1})$,

holds true. Thus, we have

□

Theorem 6.3. Let $K\in B(H_C)$ have a closed range. Let $\left\{\left\{f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ be $\left(A_F, \, \, B_F\right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$. Then, $\left\{\left\{K^*f_{1j}\right\}_{j = 1}^{+\infty}, \ldots, \left\{K^*f_{mj}\right\}_{j = 1}^{+\infty}\right\}$ is $\left(A_F\Vert K^{+}\Vert^{-2}, \, \, \, B_F\Vert K\Vert^{-2}\right)$ $K^{*}$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. According to Lemma 2.2,

holds true. For every partition $\left\{\sigma_1, \sigma_2, \cdots, \sigma_m\right\}$ of $\mathbb{N}$, we have

□

Theorem 6.4. Taking $p, q\in [m]$, where $p, q$ are fixed elements in $[m]$, let $\{ f_{pj} \}_{j\in \mathbb{N}}$ be an $(A_{p}, B_{p})$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$, let $\{ f_{qj} \}_{j\in \mathbb{N}}$ be an $(A_{q}, B_{q})$ $K$-frame associated with $(c_2, \cdots, c_n)$ for $H$, and let them be $(A_F, \, \, B_F)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$. Let $T_{i}\in B(H_{C})$ be surjective, and $T_{i}K = KT_{i}(i = p, q)$. If $Ker(K^{*})\subset Ker(T_{i}^{*})$ for $i = p, q$, and $\Vert T_{p}^{+}\Vert\Vert T_{q}-T_{p}\Vert\Vert \widetilde{K^{*}}^{-1}\Vert\leq \sqrt{A_F/D_{q}}$, then $\{ T_{p}f_{pj} \}_{j\in \mathbb{N}}$ and $\{ T_{q}f_{qj} \}_{j\in \mathbb{N}}$ are $\left(\left(\sqrt{A_F}\Vert U_{p}^{+}\Vert^{-1}-\sqrt{D_{q}}\Vert U_{q}-U_{p}\Vert^{-1}\Vert \widetilde{K^{*}}^{-1}\Vert \right)^{2}, \, \, \, \, \, D_{p}\Vert T_{p}\vert^2+D_{q}\Vert T_{q}\Vert^{2}\right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$, where $\widetilde{K^{*}}: Ker^{\perp}(K^{*})\rightarrow R(K^{*})$ is the restriction of $K^{*}$ on $Ker^{\perp}(K^{*}).$

Proof. For all $\sigma\subset \mathbb{N}$, we have

for all $g\in Ker^\bot\left(K^\ast\right)$.

For for all $f\in H_C$, we have $f = f_{k_1}+f_{k_2}$, where $f_{k_1}\in Ker\left(K^\ast\right)$ and $f_{k_2}\in Ker^\bot\left(K^\ast\right)$, and since $Ker\left(K^\ast\right)\subset Ker^\bot\left(U_i^\ast\right)\left(i = p, q\right)$, then

and

holds true. □

Theorem 6.5. Taking $p, q\in [m]$, where $p, q$ are fixed elements in $[m]$, suppose that two $K$-frames $\left\{f_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}\right\}_{j\in \mathbb{N}}$ are $(C_F, \, \, D_F)$ $K$-woven sequence, let $S_F^\sigma$ be the frame operator of $\left\{f_{pj}\right\}_{j\in\sigma}\cup\left\{f_{qj}\right\}_{j\in\sigma^c}$, and let $U\in B\left(H_C\right)$ be a positive operator. If $US_F^\sigma = S_F^\sigma U$, then $\left\{f_{pj}+Uf_{pj}\right\}_{j\in \mathbb{N}}$ and $\left\{f_{qj}+Uf_{qj}\right\}_{j\in \mathbb{N}}$ is $\left(C_F, D_{F}\Vert I_{H}+U\Vert^{2}\right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. For all $\sigma\subset \mathbb{N}$,

holds true for all $f\in H_C$.

Since $U$ is a positive operator with $US_F^\sigma = S_F^\sigma U$, we can prove that $US_F^\sigma\geq 0$ and $S_F^\sigma U^\ast\geq 0$. Then,

for every $f\in H_{C}$. Then,

holds true. □

Theorem 6.6. Taking $p, q\in [m]$, where $p, q$ are fixed elements in $[m]$, let $\{f_{pj}\}_{j\in \mathbb{N}}$ and $\{f_{qj}\}_{j\in \mathbb{N}}$ be $(C_F, \, \, \, D_F)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$, and let $\{g_{pj}\}_{j\in \mathbb{N}}$ and $\{g_{qj}\}_{j\in \mathbb{N}}$ be $(C_G, \, \, D_G)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$. For $\forall\sigma\subset \mathbb{N}$, let $T_F^\sigma$ be the preframe operators of $F = \left\{f_{pj}\right\}_{j\in\sigma}\cup\left\{f_{qj}\right\}_{j\in\sigma^c}$, and let $T_G^\sigma$ be the preframe operators of $G = \left\{g_{pj}\right\}_{j\in\sigma}\cup\left\{g_{qj}\right\}_{j\in\sigma^c}$. Let $U_p, U_q\in B\left(H_{C}\right)$ be co-isometrics $KU_i = U_iK$, where $i = p, q$, if $T_F^{\sigma^\ast}T_G^\sigma = 0$. Then, $\{ U_{p}f_{pj}+U_{q}g_{pj} \}_{j\in \mathbb{N}}$ and $\{ U_{p}f_{qj}+U_{q}g_{qj} \}_{j\in \mathbb{N}}$ are $\left(C_F+C_G, \, \, \, \, \, 2\left(D_{F}\Vert U_{p}\Vert^{2}+D_{G}\Vert U_{q}\Vert^{2} \right)\right)$ $K$-woven sequence associated with $(c_2, \cdots, c_n)$ for $H$.

Proof.

for all $f\in H_C$.

By virtue of $T_\mathrm{G}^{\sigma^\ast}T_\mathrm{F}^\sigma = T_\mathrm{F}^{\sigma^\ast}T_\mathrm{G}^\sigma = 0$, then

for all $f\in H_C$ holds true. □

7.   Applications

Based on the notion and results of weaving $K$-frames in $n$-Hilbert space, as an application, we now establish new inequalities on weaving $K$-frames in $n$-Hilbert space.

Taking $p, q\in [m]$, where $p, q$ are fixed elements in $[m]$, when given a weaving $K$-frame $\{g_{pj}\}_{j\in\sigma}\cup\{g_{qj}\}_{j\in\sigma^c}$ associated to $(c_2, \cdots, c_n)$ for $H$, recall that a Bessel sequence, $F = \{f_j\}_{j\in\mathbb{N}}$, for $H$ is said to be a $K$-dual of $\{g_{pj}\}_{j\in\sigma}\cup\{g_{qj}\}_{j\in\sigma^c}$ associated to $(c_2, \cdots, c_n)$ for $H$ if

For any $\sigma \subset \mathbb{N}$, for all $\{a_j\}_{j\in\mathbb{N}}\in\ell^\infty(\mathbb{N})$, and for all $f \in H_C$, we define two bounded linear operators, $T_1, \, T_2\in B(H_C)$, as follows:

Theorem 7.1. Taking $p, q\in [m]$, where $p, q$ are fixed elements in $[m]$, suppose that two $K$-frames $\{g_{pj}\}_{j\in\mathbb{N}}$ and $\{g_{qj}\}_{j\in\mathbb{N}}$ are $K$-woven sequence associated to $(c_2, \cdots, c_n)$ for $H$. Then, for any $\sigma \subset \mathbb{N}$, for all $\{a_j\}_{j\in\mathbb{N}}\in\ell^\infty(\mathbb{N})$, and for all $f \in H_C$, we have

where $T_1$ and $T_2$ are given in Eq (7.1), and $\{f_j\}_{j\in\mathbb{N}}$ is a $K$-dual of $\{g_{pj}\}_{j\in\sigma}\cup\{g_{qj}\}_{j\in\sigma^c}$ associated with $(c_2, \cdots, c_n)$ for $H$.

Proof. The proof is divided into three steps:

Step 1.

Suppose that $P, \, Q, \, K\in B(H)$ and $P + Q = K$. Then, for each $f\in H$,

Step 2.

For any $\sigma \subset \mathbb{N}$, for all $\{a_j\}_{j\in\mathbb{N}}\in\ell^\infty(\mathbb{N})$, and for all $f \in H_C$, it is easy to check that $T_1 + T_2 = K$. By virtue of Step 1, we obtain

Step 3.

and the proof is complete. □

Conclusions

In this paper, we develop the idea of weaving $K$-frames in $n$-Hilbert spaces and established some properties of these frames.

This work first introduces and discusses the concept of weaving $K$-frames in $n$-Hilbert spaces (Definitions 3.1 and 3.2) and gives examples (Example 3.1). Then, some characterization conditions of weaving $K$-frames in $n$-Hilbert space are proved by virtue of auxiliary operators, such as the preframe operator, analysis operator, and frame operator (Theorems 4.1–4.3). Then, several constructions of weaving $K$-frames in $n$-Hilbert spaces are offered by the same auxiliary operators, such as the preframe operator, analysis operator, and frame operator (Theorems 5.1–5.5). Finally, the perturbation and stability theorems of weaving $K$-frames in $n$-Hilbert spaces are discussed by virtue of the same auxiliary operators (Theorems 6.1–6.6). As applications, new inequalities on weaving $K$-frames in $n$-Hilbert spaces are established (Theorem 7.1). The obtained results further enriched the frame theory in $n$-Hilbert spaces.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to thank to the reviewers for the pertinent remarks, which led to an improvement of the paper.

Conflict of interest

The author declares that there are no conflicts of interest.