On $(n_1, \cdots, n_m)$-hyponormal tuples of Hilbert space operators

1.   Introduction

Throughout this work, we will denote by $\mathcal{B}(\mathcal{K})$ the algebra of bounded linear operators acting on a complex Hilbert space $\mathcal{K}$. For $\mathcal{Q}\in \mathcal{B}(\mathcal{K})$, we denote by $\ker(\mathcal{Q})$ and $\mathcal{Q}^*$ for the null space and the operator adjoint of $\mathcal{Q}$, respectively. An operator $\mathcal{Q}\in \mathcal{B}(\mathcal{K})$ is said to be normal if $\mathcal{Q}^*\mathcal{Q} = \mathcal{Q}\mathcal{Q}^*$ ^[10,18,20], hyponormal if $\left[\mathcal{Q}^*, \mathcal{Q} \right]: = \mathcal{Q}^*\mathcal{Q}-\mathcal{Q}\mathcal{Q}^*\geq0\; \big(\|\mathcal{Q}\omega\|$$\geq \|\mathcal{Q}^*\omega\| \; \forall\; \omega \in \mathcal{K}$) ^[7,22]). Note that

The authors in ^[14] have introduced the concept of $n$-hyponormality for some positive integer $n$ as follows: an operator $\mathcal{Q}$ is said to be $n$-hyponormal if $\left[ \mathcal{Q}^{*n}, \mathcal{Q}^n\right]\geq0$, or equivalently $\big(\|\mathcal{Q}^n\omega\| \geq \|\mathcal{Q}^{*n}\omega\| \; \forall\; \omega \in \mathcal{K}\big).$ Note that

We invite the reader to reading ^[14,15,24] for more details on this topic.

In recent years, the study of some concepts of operators theory in several variables has been studied at several levels by many authors, based on studies carried out on the theory of operators in one variable (see ^[1,2,3,8,9]). We mention here the following concepts related to our study, namely, joint normality, joint hyponormality and joint quasihyponormality. A tuple $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m) \in\mathcal{B}(\mathcal{K})^m$, is said to be joint normal ^[4,5,6] if $\mathcal{Q}$ satisfies the following conditions:

However, $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m)$ is said to be a joint hyponormal (^[4]) if

Note that ${ \mathcal{Q}}^*: = (\mathcal{Q}_1^*, \cdots, \mathcal{Q}_m^*).$

Recently, Sid Ahmed et al.^[17] have introduced the concept of joint $m$-quasihyponormal as follows: An tuple $\mathcal{Q} = (\mathcal{Q}_1, \; \cdots, \mathcal{Q}_m)\in {\mathcal B}({\mathcal K})^m$ is said to be a joint $m$-quasihyponormal if $\mathcal{Q}$ satisfies

In this work, we present natural generalizations of joint hyponormalty to $(n_1, \cdots, n_m)$-hyponrmality and joint quasihyponormality to $(q_1, \cdots, q_m)$-quasi-$(n_1, \cdots, n_m)$-hyponrmality. $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m)$ is said to be an $(n_1, \cdots, n_m)$-hyponormal if

for some $(n_1, \cdots, n_m)\in \mathbb{N}^m$, and it is said to be $(q_1, \cdots, q_m)$ quasi-$(n_1, \cdots, n_m)$-hyponrmal if

for some $(n_1, \cdots, n_m)\in \mathbb{N}^m$ and $(q_1, \cdots, q_m)\in \mathbb{N}^m$.

Additional references regarding tuples of operators are cited here ^{[12,16,19,21,23]}.

2.   Class of $(n_1, \cdots, n_m)$-Hyponormality of operators

In this section, the definition and properties corresponding to the $(n_1, \cdots, n_m)$-hyponormal tuples of operators are introduced.

Definition 2.1. Let $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m) \in {\mathcal B} ({\mathcal K})^m$. We say that $\mathcal{Q}$ is an $n = (n_1, \cdots, n_m)$-hyponormal tuple if the operator matrix $\bigg([\mathcal{Q}_k^{*n_k}, \; \mathcal{Q}_l^{n_l}] \bigg)_{1\leq k, \; l\leq m}$=$\bigg(\mathcal{Q}_k^{*n_k} \; \mathcal{Q}_l^{n_l}- \; \mathcal{Q}_l^{n_l}\mathcal{Q}_k^{*n_k} \bigg)_{1\leq k, l\leq m}$ is positive on $\bigoplus\limits_{1\leq k \leq m}{\mathcal K}$ that is

It is clear from this definition that $\mathcal{Q}$ is an $n$-hyponormal tuple if

is positive operator on $\oplus_{1\leq i \leq m}{\mathcal K}: = {\mathcal K}\oplus \cdots \oplus {\mathcal K}$.

Remark 2.1. The following observations can be derived from Definition 2.1.

(ⅰ) When $m = 1$, then $\mathcal{Q}$ is an $n$-hyponormal if and only if $[\mathcal{Q}^{*n}, \; \mathcal{Q}^{n}]\geq0.$ Note that

(ⅱ) If $m = 2$, then $\mathcal{Q} = (\mathcal{Q}_1, \mathcal{Q}_2)$ is $n = (n_1, n_2)$-hyponormal pair if and only if

which can be expressed as:

for all $(\omega_1, \omega_2)\in \mathcal{K}^2.$

Example 2.1. Let $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m)$ such that each $\mathcal{Q}_k$ be $n_k$-hyponomal for $k = 1, \cdots, m$. If $\big[\mathcal{Q}_l^*, \mathcal{Q}_k] = 0$ for $k\not = l$. Then, $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m)$ is an $n = (n_1, \cdots, n_m)$-hyponormal tuple.

Taking into consideration $\big[\mathcal{Q}_l^*, \mathcal{Q}_k] = 0$ for $k\not = l$ and $\mathcal{Q}_k$ is an $n_k$-hyponprmal, we may rewrite

The following theorem introduces a characterization for the studied class of multivariable operators.

Theorem 2.1. A tuple $\mathcal{Q} = \left(\mathcal{Q}_{1}, \cdots, \mathcal{Q}_{m}\right) \in {\mathcal B}\left({\mathcal{K}}\right) ^{m}$ is an $(n_1, \cdots, n_m)$-hyponormal tuple if and only if

for every $\omega_1, \cdots, \omega_m \in {\mathcal K}$.

Proof. We observe that

And so it is

Thus, the desired equivalence is obtained. □

Remark 2.2. When you choose $n = (1, \cdots, 1)$, Theorem 2.1 coincides with [4, Remark 1].

Corollary 2.1. Let $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m)\in {\mathcal B}({\mathcal K})^m$ be an $(n_1, \cdots, n_m)$-hyponormal tuple of operators. Then,

Proof. Let $\omega\in \bigcap\limits_{1\leq k\leq m}\ker(\mathcal{Q}_k^{n_k})$ and taking into account Theorem 2.1, we obtain

Hence,

Consequently,

□

Remark 2.3. When $m = 1$, it is well known that if $\mathcal{Q}$ is an $n$-hyponormal single operator, then

Proposition 2.1. Let $\mathcal{Q}\in {\mathcal B}\left({\mathcal K}\right)$ and consider $\widetilde{\mathcal{Q}} = \left(\mathcal{Q}, \cdots, \mathcal{Q}\right)$. Then, $\widetilde{\mathcal{Q}}$ is an $(n, \cdots, n)$-hyponormal tuple if and only if $\mathcal{Q}$ is an $n$-hyponormal.

Proof. We have

□

Lemma 2.1. Let $\mathcal{ Q} = \left(\mathcal{ Q}_{1}, \cdots, \mathcal{ Q}_{m}\right) \in {\mathcal B}\left({\mathcal K}\right)^{m}$, and let $\mu: = (\mu _{1}, \cdots, \mu _{m})\in \mathbb{C}^m$. If $\mathcal{ Q}$ is an $(n_1, \cdots, n_m)$-hyponormal tuple. Then, $\mu\mathcal{ Q}: = \left(\mu _{1}\mathcal{ Q}_{1}, \cdots, \mu _{m}\mathcal{ Q}_{m}\right)$ is an $(n_1, \cdots, n_m)$-hyponormal tuple.

Proof. Using some calculations and taking into account that $\mathcal{ U}$ is an $(n_1, \cdots, n_m)$-hyponormal tuple, we have

□

Remark 2.4. The property of being $(n_1, \cdots, n_m)$-hyponormal for a tuple $\mathcal{Q} = \left(\mathcal{ Q}_{1}, \cdots, \mathcal{ Q}_{m}\right)$ of operators is indeed invariant under permutations of the operators in $\mathcal{ Q}$.

The following proposition describes some properties of $n$-hyponormal $m$ -tuples of operators.

Proposition 2.2. Let $\mathcal{ Q} = \left(\mathcal{ Q}_{1}, \cdots, \mathcal{ Q}_{m}\right) \in {\mathcal B}\left({\mathcal K}\right)^{m}$ be an $(n_1, \cdots, n_m)$-hyponormal tuple. The following properties hold:

1) If $\mathcal{N} \in {\mathcal B}({\mathcal H})$ is a normal operator such that $\mathcal{N}$ commutes with each $\mathcal{ Q}_{k}$, then, $\mathcal{N}\mathcal{ Q}: = \left(\mathcal{ N}\mathcal{ Q}_{K}, \cdots, \mathcal{N}\mathcal{ Q}_{m}\right)$ is $(n_1, \cdots, n_m)$-hyponormal tuple.

2) For any unitary operator $\mathcal{W}\in {\mathcal B}\left({\mathcal K}\right)$, the tuples $\mathcal{ W} \mathcal{Q}\mathcal{W}^{*}: = \left(\mathcal{W}\mathcal{Q}_{1}\mathcal{W}^{*}, \cdots, \mathcal{W}\mathcal{Q}_{m}\mathcal{W}^{*}\right)$ is $(n_1, \cdots, n_m)$-hyponormal tuple.

Proof. 1) Given that $\mathcal{N}$ is a normal operator for which $\mathcal{N}\mathcal{Q}_k = \mathcal{Q}_k\mathcal{N}$ for $k = 1, \cdots, m$, and referring to Fuglede-Putnam theorem ^[11], we obtain $\mathcal{N}^*\mathcal{Q}_k = \mathcal{Q}_k\mathcal{N}^*$. Based on these statements, we can obtain the relationships

Therefore, $\mathcal{N}\mathcal{Q}: = \left(\mathcal{ N}\mathcal{Q}_{1}, \cdots, \mathcal{N}\mathcal{Q}_{m}\right)$ is $n$-hyponormal tuple. From it, the desired results are produced.

2) Suppose any unitary operator $\mathcal{V}\in {\mathcal B}\left({\mathcal K}\right)$ such that,

Hence, for each collection $\omega_1, \cdots, \omega_m \in {\mathcal K}$, we have

Which is ends of the proof □

The following theorem generalizes the statement $(1)$ of Proposition 2.2.

Theorem 2.2. Let $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m) \in {\mathcal B}({\mathcal K})^m$ and $\mathcal{W} = (\mathcal{W}_1, \cdots, \mathcal{W}_m) \in {\mathcal B}({\mathcal K})^m$ for which the following conditions are satisfied

If $\mathcal{Q}$ is an $(n_1, \cdots, n_m)$-hyponormal tuple, then $\mathcal{W}\mathcal{Q}: = (\mathcal{W}_1\mathcal{Q}_1, \cdots, \mathcal{W}_m\mathcal{Q}_m)$ is too.

Proof. Let $\omega_1, \cdots, \omega_m \in {\mathcal K}$, and taking into account (2.3), we may write

We have

where $\psi_k = \mathcal{W}_k^{*n_k} \omega_i$ for $i = 1, \cdots, m$ and $\omega_1, \cdots, \omega_m \in {\mathcal K}$.

In view of the fact that $\mathcal{Q}$ is an $(n_1, \cdots, n_m)$-hyponorml tuple, we can obtain

This completes the proof. □

Theorem 2.3. Let $\mathcal{N}\in {\mathcal B}\big(\mathcal{K}\big)$ be an invertible operator and ${ \mathcal{Q}} = (\mathcal{Q}_1\cdots, \mathcal{Q}_m) \in \mathcal{B}(\mathcal{H})^m$ be a tuple of operators such that each $\mathcal{Q}_k$ commutes with ${\mathcal{N}}^*{\mathcal{N}}$ for $k = 1, \cdots, m$. Then, ${\mathcal{Q}} = (\mathcal{Q}_1\cdots, \mathcal{Q}_m)$ is an $(n_1, \cdots, n_m)$-hyponormal tuple if and only if

${\mathcal{N}}{ \mathcal{Q}}{\mathcal{N}}^{-1}: = \big({\mathcal{N}}\mathcal{Q}_1{\mathcal{N}}^{-1}, \cdots, {\mathcal{N}}\mathcal{Q}_m{\mathcal{N}}^{-1}\big)$ is an $(n_1, \cdots, n_m)$-hyponormal tuple.

Proof. Assume that $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m)$ is an $(n_1, \cdots, n_m)$-hyponormal tuple. We need to show that ${\mathcal{N}}{ \mathcal{Q}}{\mathcal{N}}^{-1}: = \big({\mathcal{N}}\mathcal{Q}_1{\mathcal{N}}^{-1}, \cdots, {\mathcal{N}}\mathcal{Q}_m{\mathcal{N}}^{-1}\big)$ is an $(n_1, \cdots, n_m)$-hyponormal tuple. In fact, let $\omega_1, \cdots, \omega_m \in \mathcal{H}$, we have

Conversely, assume that ${\mathcal{N}}{\mathcal{Q}}{\mathcal{N}}^{-1}: = \big({\mathcal{N}}\mathcal{Q}_1{\mathcal{N}}^{-1}, \cdots, {\mathcal{N}}\mathcal{Q}_m{\mathcal{N}}^{-1}\big)$ is an $(n_1, \cdots, n_m)$-hyponormal tuple. Set $\mathcal{Q}_k = \mathcal{N}\mathcal{Q}_k\mathcal{N}^{-1}$ for $k = 1, \cdots, m$. We can check that each $\mathcal{Q}_k$ commutes with $\big(\mathcal{N}^{-1}\big)^*\mathcal{N}^{-1}$, and moreover

Based on the first statement, we have $\big(\mathcal{N}^{-1}\mathcal{Q}_1\mathcal{N }, \cdots, \mathcal{N}^{-1}\mathcal{Q}_m\mathcal{N}\big)$ is an $(n_1, \cdots, n_m)$-hyponormal tuple, and so it shall be $\big(\mathcal{Q}_1, \cdots, \mathcal{Q}_m)$ is an $(n_1, \cdots, n_m)$-hyponormal tuple. □

Definition 2.2. (^[5]) An operator $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m) \in \mathcal{B}(\mathcal{H})^m$ is said to be $(n_1, \cdots, n_m)$-normal tuple if

Theorem 2.4. Let $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m) \in\mathcal{B}(\mathcal{K})^m$ and $n = (n_1, \cdots, n_m) \in \mathbb{N}^m$. The following statements hold:

1) If $\mathcal{Q}$ is an $(n_1, \cdots, n_m)$-hyponormal tuple, then $\mathcal{Q}^{*}$ is an $(n_1, \cdots, n_m)$-hyponormal tuple if and only if

2) Assume that $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m)$ be commuting tuple of operators. If $\mathcal{Q}$ and ${\mathcal{Q}^{*}}$ are $(n_1, \cdots, n_m)$-hyponormal tuple, then $\mathcal{Q}$ is an $(n_1, \cdots, n_m)$-normal tuple.

Proof. 1) Let $k, l\in \{1, 2, \cdots, m\}$, we observe that

Assume that $\mathcal{Q}$ and $\mathcal{Q}^{*}$ are $(n_1, \cdots, n_m)$-hyponormal tuples. It follows that for each finite collections $\omega_1, \; \cdots, \omega_m \in {\mathcal K}$, we have

Or equivalently,

For fixed couple $(k_0, l_0)$, let $\omega_1, \; \cdots, \omega_m \in {\mathcal K}$ be chosen so that $\omega_p = 0$ for $p \notin \{k_0, l_0\}$. Applying the first inequality and the second inequality above to the tuple $\big(\omega_1, \; \cdots, \omega_m\big)$, we obtain respectively,

and

Thus, we have

for each $\omega \in {\mathcal K}, k, l = 1, \cdots, m.$

Letting $\omega_{k_0} = \omega_{l_0} = \omega \in {\mathcal K}$, we obtain

for each$\omega \in {\mathcal K}, \; \; k, l = 1, \cdots, m.$ Letting $\omega_{k_0} = \omega$ and $\omega_{l_0} = -\omega \in {\mathcal K}$, we obtain

for each $\omega \in {\mathcal K}, k, l = 1, \cdots, m.$

By combining (2.4) and (2.5), we obtain:

Hence,

Observing that

this implying that

Therefore, $\left[\mathcal{Q}_{l}^{*n_{l}}, \; \mathcal{Q}_{{k}}^{n_{k}}\right]$ is purely imaginary.

Assume that $\left[\mathcal{Q}_{l}^{*n_{l}}, \; \mathcal{Q}_{{k}}^{n_{k}}\right]$ is purely imaginary for all $k, l = 1.\cdots, m.$ Thus,

This means that

Let $\omega_1, \; \cdots, \omega_m \in {\mathcal K}$, and taking into account that

However,

The above simplification shows:

Which prove that $\mathcal{Q}^{*}$ is an $(n_1, \cdots, n_m)$-hyponormal tuple.

2) Obviously that

Given that both $\mathcal{Q}$ and $\mathcal{Q}^*$ are $(n_1, \cdots, n_m)$-hyponormal, it follows that

In particular,

□

Lemma 2.2. (^[5]) Let $\mathcal{T}, \mathcal{S}\in {\mathcal B}({\mathcal K})$ and $p, q \in \mathbb{N}.$ Then,

Proposition 2.3. Let $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m) \in\mathcal{B}(\mathcal{K})^m$ and $n = (n_1, \cdots, n_m) \in \mathbb{N}^m$. $\mathcal{Q} = (\mathcal{Q}_1, \cdots, \mathcal{Q}_m)$ is an $(n_1, \cdots, n_m)$-hyponormal tuple, then $\mathcal{Q}^r = (\mathcal{Q}_1^{r_1}, \cdots, \mathcal{Q}_m^{r_m})$ is too.

Proof. If $q_r\in\{0, 1\}$ for all $k \in \{1, \cdots, m\}$, then $[(\mathcal{Q}_k^{n_k})^{r_k}, (\mathcal{Q}_l^{n_l})^{r_l}] = 0$ for all $k, l = 1, \cdots, m$.

Assume that $r_k > 1$ for all $k \in \{1, \cdots, m\}$, and taking into account Lemma 2.2

We infer that

Using the $(n_1, \cdots, n_m)$-hyponormality of $\mathcal{Q}$, we obtain for all $(\omega_k)_{1\leq k\leq m} \in \mathcal{K}^m$

which implies that

for all $(\omega_k)_{1\leq k\leq m} \in \mathcal{K}^m$. Therefore, $\mathcal{Q}^r$ is an $(n_1, \cdots, n_m)$-hyponormal tuple. □

3.   Conclusions

This paper introduces and explores the concept of $(n_1, \cdots, n_m)$-hyponormal tuples, a new class of multivariable operators that integrates the notions of joint normal and joint hyponormal operators. The study demonstrates that $(n_1, \cdots, n_m)$-hyponormal tuples inherit several important properties from joint hyponormal operators. This new class of operators not only enriches the theory of multivariable operators but also provides a framework for further exploration and analysis of operator tuples.

Author contributions

Sid Ahmed Ould Beinane and Sid Ahmed Ould Ahmed Mahmoud: conceptualization, validation, formal analysis, supervision, writing-review and editing. All authors contributed equally to the writing of this article. All authors have read and approved the final version of the manuscript for publication.

Acknowledgements

This work was funded by the Deanship of Scientific Research at Jouf University under grant (No. DSR-2021-03-03123). The authors would like to express their sincere gratitude to the anonymous reviewers for their thoughtful reading and valuable comments on this work.

Conflict of interest

The authors affirm that they have no conflicts of interest to disclose.