M-hyponormality in several variables operator theory

Abstract

In recent years, the study of bounded linear operators in several variables has received great interest from many authors, including the second author’s previous contributions. In our present work, we define a new class of multivariable operator theory, which we have called M-hyponormal tuple. We present some algebraic and spectral properties associated with them.

1 Introduction and preliminaries

Let $\mathcal{{H}}$ be a separable infinite dimensional complex Hilbert space with inner product $\langle .\;| \;.\rangle$, $\mathcal{B}((\mathcal{H})$ be the set of all bounded linear operators on $\mathcal{H}$. An operator $\mathcal{U}\in {\mathcal {B}}({\mathcal {H}})$ is said to be normal if $\mathcal{U}^{*}\mathcal{U}-\mathcal{U}\mathcal{U}^{*}=0$ $\big([\mathcal{U}^{*},\mathcal{U}]=0\big)$ [12, 17, 19], hyponormal if $\mathcal{U}^{*}\mathcal{U}-\mathcal{U}\mathcal{U}^{*}\geq 0$ $\big([\mathcal{U}^{*},\mathcal{U}] \geq 0\big)$ [9, 21]) and M-hyponormal if $M\|\big(\mathcal{U}-\gamma \big)\omega \| \geq \|\big(\mathcal{U}- \gamma \big)^{*}\omega \|\geq 0$ for all $\gamma \in {\mathbb{C}}$, $\omega \in \mathcal{H}$ and $M>0$ ([20, 22]). The concepts of multivariable operators have attracted the attention of many authors. Recently, many extensions of some concepts of single operators to tuple of operators on Hilbert and Banach spaces have been immensely studied in several papers (see [1, 3–7, 10, 11, 13, 14, 16, 18].

An m-tuple ${\mathbf{U}}=(\mathcal{U}_{1},\ldots ,\mathcal{U}_{m})\in { \mathcal {B}}({\mathcal {H}})^{m}$ is said to be joint normal if $\mathcal{U}_{l}\mathcal{U}_{k}=\mathcal{U}_{k}\mathcal{U}_{l}$ for all $(l,\;k)\in \{1,\ldots ,m\}^{2}$ and each $\mathcal{U}_{k}$ is a normal. However, ${\mathbf{U}}=(\mathcal{U}_{1},\ldots ,\mathcal{U}_{m})$ is said a joint hyponormal if the matrix operator

$$ [ {\mathbf{U^{*}}},{\mathbf{U}}]=\bigg(\big[\mathcal{U}_{k}^{*}, \; \mathcal{U}_{l}\big]\bigg)_{l,\;k=1}^{m}=\left ( \textstyle\begin{array}{cccc} [\mathcal{U}_{1}^{*}, \mathcal{U}_{1}] & [\mathcal{U}_{2}^{*}, \mathcal{U}_{1}] &\cdots & [\mathcal{U}_{m}^{*},\mathcal{U}_{1}] \cr [\mathcal{U}_{1}^{*}, \mathcal{U}_{2} ]& [\mathcal{U}_{2}^{*}, \mathcal{U}_{2}] &\cdots & [\mathcal{U}_{m}^{*},\mathcal{U}_{2}] \cr \vdots & \vdots & \vdots & \vdots \cr [\mathcal{U}_{1}^{*},\mathcal{U}_{m}] & [\mathcal{U}_{2}^{*}, \mathcal{U}_{m}] &\cdots & [\mathcal{U}_{m}^{*}, \mathcal{U}_{m}] \cr \end{array}\displaystyle \right ) $$

is positive on $\displaystyle \bigoplus _{k=1}^{m}{\mathcal {H}}$, that is $\displaystyle \sum _{1\leq l,\;k\;\leq m}\left \langle [\mathcal{U}_{k}^{*}, \;\mathcal{U}_{l}] \omega _{k}\;|\;\omega _{l}\right \rangle \geq 0$, for each finite collection $\omega _{1}, \ldots ,\omega _{m}$ of ${\mathcal {H}}$ (see [5]). Note that ${\mathbf{U}}^{*}:=(\mathcal{U}_{1}^{*},\ldots ,\mathcal{U}_{m}^{*})$.

Recently, Sid Ahmed et al. [2] has introduced the notion of joint m-quasihyponormal as follows: ${\mathbf{U}}=(\mathcal{U}_{1},\;\ldots ,\mathcal{U}_{m})\in { \mathcal {B}}({\mathcal {H}})^{m}$ is said to be joint m-quasihyponormal if U satisfies

$$ \displaystyle \sum _{1\leq l,\;k\;\leq m}\big\langle \mathcal{U}_{k}^{*} \big[\mathcal{U}_{k}^{*},\;\; \mathcal{U}_{l}\big]\mathcal{U}_{l} \omega _{k}\;|\;\omega _{l}\big\rangle \geq 0$$

(1.1)

for each finite collections $(\omega _{l})_{1\leq l\leq m}\in {\mathcal {H}}^{m}$.

Equivalently, ${\mathbf{U}}=(\mathcal{U}_{1},\ldots ,\mathcal{U}_{m})\in { \mathcal {B}}({\mathcal {H}})^{m}$ is joint m-quasihyponormal tuple if the operator matrix,

$$\begin{aligned} {\mathbf{U^{*}}}[ {\mathbf{U^{*}}},{\mathbf{U}}] {\mathbf{U}} &= \bigg(\mathcal{U}_{k}^{*}\big[\mathcal{U}_{k}^{*},\; \mathcal{U}_{l} \big]\mathcal{U}_{l}\bigg)_{l,\;k=1}^{m}\\&=\left ( \textstyle\begin{array}{cccc} \mathcal{U}_{1}^{*} [\mathcal{U}_{1}^{*}, \mathcal{U}_{1}]\mathcal{U}_{1} & \mathcal{U}_{2}^{*}[\mathcal{U}_{2}^{*},\mathcal{U}_{1}]\mathcal{U}_{1} &\cdots & \mathcal{U}_{m}^{*}[\mathcal{U}_{m}^{*},\mathcal{U}_{1}] \mathcal{U}_{1} \cr \mathcal{U}_{1}^{*} [\mathcal{U}_{1}^{*}, \mathcal{U}_{2} ] \mathcal{U}_{2}&\mathcal{U}_{2}^{*} [\mathcal{U}_{2}^{*}, \mathcal{U}_{2}] \mathcal{U}_{2} &\cdots &\mathcal{U}_{m}^{*} [\mathcal{U}_{m}^{*}, \mathcal{U}_{2}]\mathcal{U}_{2} \cr \vdots & \vdots & \vdots & \vdots \cr \mathcal{U}_{1}^{*} [\mathcal{U}_{1}^{*},\mathcal{U}_{m}] \mathcal{U}_{m}& \mathcal{U}_{2}^{*}[\mathcal{U}_{2}^{*}, S_{m}]\mathcal{U}_{m} & \cdots & \mathcal{U}_{m}^{*} [\mathcal{U}_{m}^{*}, \mathcal{U}_{m}] \mathcal{U}_{m} \cr \end{array}\displaystyle \right ) \end{aligned}$$

is positive on $\displaystyle \bigoplus _{k=1}^{m}{\mathcal {H}}$.

The aim of this work is to present a concept of M-hyponormality for tuple of operators. A tuple $\mathcal{U}=(\mathcal{U}_{1},\ldots .\mathcal{U}_{m}) \in { \mathcal {B}}({\mathcal {H}})^{m}$ is said to be M-hyponormal if there exists $M> 0$ for which

$$ \sum _{1\leq k,l\leq m}\left \langle \bigg(M^{2}\big(\mathcal{U}_{k}- \gamma _{k}\big)^{*}\big(\mathcal{U}_{l}-\gamma _{l}\big)-\big( \mathcal{U}_{l}-\gamma _{l}\big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \bigg)\omega _{k}\mid \omega _{l}\right \rangle \geq 0, $$

(1.2)

for all $\omega =(\omega _{k})_{1\leq k \leq m}\in \mathcal{H}^{m}$ and $\gamma =(\gamma _{1},\ldots ,\gamma _{m})\in \mathbb{C}^{m}$.

When $m=1$ (1.2) coincides with

$$ \left \langle \bigg(M^{2}\big(\mathcal{U}-\gamma \big)^{*}\big( \mathcal{U}-\gamma \big)-\big(\mathcal{U}-\gamma \big)\big( \mathcal{U}-\gamma \big)^{*}\bigg)\omega \mid \omega \right \rangle \geq 0, $$

(1.3)

$\forall \;\;\gamma \in \mathbb{C}$, $\omega \in {\mathcal{H}}$ or equivalently

$$ M^{2}\big(\mathcal{U}-\gamma \big)^{*}\big(\mathcal{U}-\gamma \big)- \big(\mathcal{U}-\gamma \big)\big(\mathcal{U}-\gamma \big)^{*}\geq 0, $$

that is $\mathcal{U}$ is M-hyponormal.

When $m=2$ (1.2) coincides with

$$\begin{aligned} &\left \langle \bigg(M^{2}\big(\mathcal{U}_{1}-\gamma _{1}\big)^{*} \big(\mathcal{U}_{1}-\gamma _{1}\big)-\big(\mathcal{U}_{1}-\gamma _{1} \big)\big(\mathcal{U}-\gamma _{1}\big)^{*}\bigg)\omega _{1}\mid \omega _{1}\right \rangle \\ &+\left \langle \bigg(M^{2}\big(\mathcal{U}_{1}-\gamma _{1}\big)^{*} \big(\mathcal{U}_{2}-\gamma _{2}\big)-\big(\mathcal{U}_{2}-\gamma _{2} \big)\big(\mathcal{U}_{1}-\gamma _{1}\big)^{*}\bigg)\omega _{1}\mid \omega _{2}\right \rangle \\ &+ \left \langle \bigg(M^{2}\big(\mathcal{U}_{2}-\gamma _{2}\big)^{*} \big(\mathcal{U}_{1}-\gamma _{1}\big)-\big(\mathcal{U}_{1}-\gamma _{1} \big)\big(\mathcal{U}_{2}-\gamma _{2}\big)^{*}\bigg)\omega _{2}\mid \omega _{1}\right \rangle \\ &+ \left \langle \bigg(M^{2}\big(\mathcal{U}_{2}-\gamma _{2}\big)^{*} \big(\mathcal{U}_{2}-\gamma _{2}\big)-\big(\mathcal{U}_{-}\gamma _{2} \big)\big(\mathcal{U}_{2}-\gamma _{2}\big)^{*}\bigg)\omega _{2}\mid \omega _{2}\right \rangle \\ \geq &0. \end{aligned}$$

2 Main results

Our interest in this section is to provide most important results obtained for this new class of multivariable operators.

Theorem 2.1

Let ${\mathbf{U}}=\left ( \mathcal{U}_{1},\ldots ,\mathcal{U}_{m} \right ) \in {\mathcal {B}}\left ( {\mathcal {H}}\right ) ^{m}$, then U is M-hyponormal tuple if and only if

$$ \sum _{1\leq k,l\leq m}\left \langle M^{2} \big(\mathcal{U}_{l}- \gamma _{l})\big)\omega _{k}\mid \big(\mathcal{U}_{k}-\gamma _{k} \big)\omega _{l}\right \rangle - \bigg\| \sum _{1\leq k\leq m} \big( \mathcal{U}_{k}-\gamma _{k})^{*}\omega _{k}\bigg\| ^{2}\geq 0, $$

(2.1)

for every $\omega _{1},\ldots ,\omega _{m} \in {\mathcal {H}}$ and $\gamma =(\gamma _{1},\ldots ,\gamma _{m})\in \mathbb{C}^{m}$.

Proof

For $M>0$, we have

$$\begin{aligned} &\sum _{1\leq k,l\leq m}\left \langle \bigg(M^{2}\big(\mathcal{U}_{k}- \gamma _{k}\big)^{*}\big(\mathcal{U}_{l}-\gamma _{l}\big)-\big( \mathcal{U}_{l}-\gamma _{l}\big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \bigg)\omega _{k}\mid \omega _{l}\right \rangle \geq 0 \\ \Leftrightarrow & \sum _{1\leq k,l\leq m}\left \langle \bigg(M^{2} \big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\big(\mathcal{U}_{l}-\gamma _{l} \big)\omega _{k}\mid \omega _{l}\right \rangle -\left \langle \big( \mathcal{U}_{l}-\gamma _{l}\big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \bigg)\omega _{k}\mid \omega _{l}\right \rangle \geq 0 \\ \Leftrightarrow & \sum _{1\leq k,l\leq m}\left \langle \bigg(M^{2} \big(\mathcal{U}_{l}-\gamma _{l}\big)\omega _{k}\mid \big(\mathcal{U}_{k}- \gamma _{k}\big)\omega _{l}\right \rangle - \sum _{1\leq k,l\leq m} \left \langle \big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega _{k} \mid \big(\mathcal{U}_{l}-\gamma _{l}\big)^{*} \omega _{l}\right \rangle \geq 0 \\ \Leftrightarrow &\sum _{1\leq k,l\leq m}\left \langle \bigg(M^{2} \big(\mathcal{U}_{l}-\gamma _{l}\big)\omega _{k}\mid \big(\mathcal{U}_{k}- \gamma _{k}\big)\omega _{l}\right \rangle - \bigg\| \sum _{1\leq k \leq m} \big(\mathcal{U}_{k}-\gamma _{k})^{*}\omega _{k}\bigg\| ^{2} \geq 0. \end{aligned}$$

Therefore, U is M-hyponormal tuple if and only if U satisfies (2.1). □

Remark 2.1

If $m=1$ (2.1) coincides with

$$ \|\big(\mathcal{U}-\gamma \big)\omega \| \geq \|\big(\mathcal{U}- \gamma \big)^{*}\omega \|\quad \forall \; \omega \in \mathcal{H},\; \gamma \in \mathbb{C}. $$

Set

$$ [0]= \{\alpha :=(\alpha _{1},\ldots ,\alpha _{m})\in \mathbb{C}^{m}/ \; \displaystyle \prod _{1\leq k \leq m} \alpha _{k}=0 \}. $$

Proposition 2.1

Let $\mathbf{U}=\left ( \mathcal{U}_{1},\ldots ,\mathcal{U}_{m}\right ) \in {\mathcal {B}}\left ( {\mathcal {H}}\right )^{m}$ be an M-hyponormal tuple and let $\alpha :=(\alpha _{1},\ldots ,\alpha _{m})\in \mathbb{C}^{m}$. The following properties hold:

$(1)$ $\alpha \mathbf{U}:=\left ( \alpha _{1}\mathcal{U}_{1},\ldots , \alpha _{m}\mathcal{U}_{m}\right )$ is an M-hyponormal tuple for $\alpha \notin [0]$.

$(2)$ ${\mathbf{U}}-\alpha I:=\left ( \mathcal{U}_{1}-\alpha _{1}I, \ldots ,\mathcal{U}_{m}-\alpha _{m}I\right )$ is an M-hyponormal tuple.

Proof

$(1)$ By taking into account (2.1), we get

$$\begin{aligned} &\sum _{1\leq i,j\leq m}\left \langle M^{2} \big(\alpha _{l} \mathcal{U}_{l}-\gamma _{l})\big)\omega _{k}\mid \big(\alpha _{k} \mathcal{U}_{k}-\gamma _{k} \big)\omega _{l}\right \rangle - \bigg\| \sum _{1\leq k\leq m} \big(\alpha _{k}\mathcal{U}_{k}-\gamma _{k})^{*} \omega _{k}\bigg\| ^{2} \\ =& \sum _{1\leq i,j\leq m}\left \langle M^{2} \big(\mathcal{U}_{l}- \frac{\gamma _{l}}{\alpha _{l}}\big)\overline{\alpha _{k}}\omega _{k} \mid \big(\mathcal{U}_{k}-\frac{\gamma _{k}}{\alpha _{k}} \big) \overline{\alpha _{l}}\omega _{l}\right \rangle - \bigg\| \sum _{1 \leq k\leq m} \big(\mathcal{U}_{k}-\frac{\gamma _{k}}{\alpha _{k}})^{*} \overline{\alpha _{k}}\omega _{k}\bigg\| ^{2} \\ \geq &0. \end{aligned}$$

$(2)$ The proof of the statement $(2)$ follows from (1.2). □

Theorem 2.2

Let ${\mathbf{U}}=(\mathcal{U}_{1},\ldots ,\mathcal{U}_{m}) \in \mathcal{B}(\mathcal{H})^{m}$ be a joint M-hyponormal tuple, then ${\mathbf{U}}^{*}$ is an $\displaystyle \frac{1}{M}$-hyponormal tuple if and only if

$$ \left \langle Re \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega \mid \omega \right \rangle =0,\;\;\forall \;\omega \in {\mathcal {H}},\;k,l =1, \ldots ,m. $$

Proof

Assume that ${\mathbf{U}}^{*}$ is an $\frac{1}{M}$-hyponormal tuple.

From the fact that U is an M-hyponormal tuple, we may write

$$ \sum _{1\leq k,l\leq m}\left \langle \bigg(M^{2}\big(\mathcal{U}_{k}- \gamma _{k}\big)^{*}\big(\mathcal{U}_{l}-\gamma _{l}\big)-\big( \mathcal{U}_{l}-\gamma _{l}\big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \bigg)\omega _{k}\mid \omega _{l}\right \rangle \geq 0. $$

(2.2)

However, ${\mathbf{U}}^{*}$ is $\frac{1}{M}$-hyponormal, thus we may write without loss of generality

$$ \sum _{1\leq k,l\leq m}\left \langle \bigg(\frac{1}{M^{2}}\big( \mathcal{U}_{k}^{*}-\overline{\gamma _{k}}\big)^{*}\big(\mathcal{U}_{l}^{*}- \overline{\gamma _{l}}\big)-\big(\mathcal{U}_{l}^{*}- \overline{\gamma _{l}}\big)\big(\mathcal{U}_{k}^{*}- \overline{\gamma _{k}}\big)^{*}\bigg)\omega _{k}\mid \omega _{l} \right \rangle \geq 0. $$

(2.3)

Equation (2.3) can be written as

$$ \sum _{1\leq k,l\leq m}\left \langle \bigg(\frac{1}{M^{2}}\big( \mathcal{U}_{k}-\gamma _{k}\big)\big(\mathcal{U}_{l}-{\gamma _{l}} \big)^{*}-\big(\mathcal{U}_{l}-{\gamma _{l}}\big)^{*}\big(\mathcal{U}_{k}-{ \gamma _{k}}\big)\bigg)\omega _{k}\mid \omega _{l}\right \rangle \geq 0, $$

(2.4)

or

$$ \sum _{1\leq k,l\leq m}\left \langle \bigg({M^{2}}\big(\big( \mathcal{U}_{l}-{\gamma _{l}}\big)^{*}\big(\mathcal{U}_{k}-{\gamma _{k}} \big)-\big(\mathcal{U}_{k}-\gamma _{k}\big)\big(\mathcal{U}_{l}-{ \gamma _{l}}\big)^{*}\bigg)\omega _{k}\mid \omega _{l}\right \rangle \leq 0. $$

(2.5)

Choosing $\omega _{j}=0$ for $j\notin \{l,k\}$, we can get from (2.2) and (2.5) that

$$\begin{aligned} &\left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega _{k}\mid \omega _{k}\right \rangle \\ +&\left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega _{l}\mid \omega _{l}\right \rangle \\ +& \left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega _{k}\mid \omega _{l}\right \rangle \\ +& \left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega _{l}\mid \omega _{k}\right \rangle \geq 0 \end{aligned}$$

and

$$\begin{aligned} &\left \langle \bigg({M^{2}}\big(\big(\mathcal{U}_{l}-{\gamma _{l}} \big)^{*}\big(\mathcal{U}_{l}-{\gamma _{l}}\big)-\big(\mathcal{U}_{l}- \gamma _{l}\big)\big(\mathcal{U}_{l}-{\gamma _{l}}\big)^{*}\bigg) \omega _{l}\mid \omega _{l}\right \rangle \\ +& \left \langle \bigg({M^{2}}\big(\big(\mathcal{U}_{k}-{\gamma _{k}} \big)^{*}\big(\mathcal{U}_{k}-{\gamma _{k}}\big)-\big(\mathcal{U}_{k}- \gamma _{k}\big)\big(\mathcal{U}_{k}-{\gamma _{k}}\big)^{*}\bigg) \omega _{k}\mid \omega _{k}\right \rangle \\ +& \left \langle \bigg({M^{2}}\big(\big(\mathcal{U}_{l}-{\gamma _{l}} \big)^{*}\big(\mathcal{U}_{k}-{\gamma _{k}}\big)-\big(\mathcal{U}_{k}- \gamma _{k}\big)\big(\mathcal{U}_{l}-{\gamma _{l}}\big)^{*}\bigg) \omega _{l}\mid \omega _{k}\right \rangle \\ +& \left \langle \bigg({M^{2}}\big(\big(\mathcal{U}_{k}-{\gamma _{k}} \big)^{*}\big(\mathcal{U}_{l}-{\gamma _{l}}\big)-\big(\mathcal{U}_{l}- \gamma _{l}\big)\big(\mathcal{U}_{k}-{\gamma _{k}}\big)^{*}\bigg) \omega _{k}\mid \omega _{l}\right \rangle \leq 0, \end{aligned}$$

we find that

$$\begin{aligned} &\left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega _{k}\mid \omega _{k}\right \rangle \\ +&\left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega _{l}\mid \omega _{l}\right \rangle \\ +& \left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega _{k}\mid \omega _{l}\right \rangle \\ +& \left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega _{l}\mid \omega _{k}\right \rangle =0. \end{aligned}$$

Letting $\omega _{k} = \omega _{l} = \omega \in \mathcal{K}$, we then see that

$$\begin{aligned} &\left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega \mid \omega \right \rangle \\ +&\left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega \mid \omega \right \rangle \\ +& \left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega \mid \omega \right \rangle \\ +& \left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega \mid \omega \right \rangle =0. \end{aligned}$$

(2.6)

Letting $\omega _{k} = \omega$ and $\omega _{k}= -\omega \in \mathcal{K,}$ we then see that

$$\begin{aligned} &\left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega \mid \omega \right \rangle \\ +&\left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega \mid \omega \right \rangle \\ -& \left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega \mid \omega \right \rangle \\ -& \left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega \mid \omega \right \rangle =0. \end{aligned}$$

(2.7)

By subtracting (2.6) and (2.7), we obtain

$$\begin{aligned} & \left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega \mid \omega \right \rangle \end{aligned}$$

(2.8)

$$\begin{aligned} +& \left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega \mid \omega \right \rangle =0. \end{aligned}$$

(2.9)

This gives

$$ \left \langle Re\bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega \mid \omega \right \rangle =0\quad \forall \;\omega \in \mathcal{K}. $$

Conversely, assume that

$$ \left \langle Re\bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega \mid \omega \right \rangle =0\quad \forall \;\omega \in \mathcal{K}, k,l=1, \ldots ,m. $$

We deduce that

$$\begin{aligned} & \left \langle \bigg(M^{2}\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \big(\mathcal{U}_{l}-\gamma _{l}\big)-\big(\mathcal{U}_{l}-\gamma _{l} \big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*}\bigg)\omega \mid \omega \right \rangle \\ +& \left \langle \bigg(M^{2}\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*} \big(\mathcal{U}_{k}-\gamma _{k}\big)-\big(\mathcal{U}_{k}-\gamma _{k} \big)\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\bigg)\omega \mid \omega \right \rangle \quad \forall \;\omega \in \mathcal{K}\quad k,l=1, \ldots ,m \\ =& 0. \end{aligned}$$

Let $\omega _{1},\ldots , \omega _{m}\in \mathcal{K}$. We get

$$\begin{aligned} & \sum _{1\leq l,k \leq m}\left \langle \bigg(M^{2}\big(\mathcal{U}_{k}- \gamma _{k}\big)^{*}\big(\mathcal{U}_{l}-\gamma _{l}\big)-\big( \mathcal{U}_{l}-\gamma _{l}\big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \bigg)\omega _{k}\mid \omega _{l}\right \rangle \\ =& -\sum _{1\leq l,k \leq m}\left \langle \bigg(M^{2}\big( \mathcal{U}_{l}-\gamma _{l}\big)^{*}\big(\mathcal{U}_{k}-\gamma _{k} \big)-\big(\mathcal{U}_{k}-\gamma _{k}\big)\big(\mathcal{U}_{l}- \gamma _{l}\big)^{*}\bigg)\omega _{k}\mid \omega _{l}\right \rangle \\ =& M^{2}\sum _{1\leq l,k \leq m}\left \langle \bigg(\frac{1}{M^{2}} \big(\mathcal{U}_{k}-\gamma _{k}\big)\big(\mathcal{U}_{l}-\gamma _{l} \big)^{*}-\big(\mathcal{U}_{l}-\gamma _{l}\big)^{*}\big(\mathcal{U}_{k}- \gamma _{k}\big)\bigg)\omega _{k}\mid \omega _{l}\right \rangle \\ =& M^{2}\sum _{1\leq l,k \leq m}\left \langle \bigg(\frac{1}{M^{2}} \big(\mathcal{U}_{k}^{*}-\overline{\gamma _{k}}\big)^{*}\big( \mathcal{U}_{l}^{*}-\overline{\gamma _{l}}\big)^{*}-\big(\mathcal{U}_{l}^{*}- \overline{\gamma _{l}}\big)\big(\mathcal{U}_{k}^{*}- \overline{\gamma _{k}}\big)^{*}\bigg)\omega _{k}\mid \omega _{l} \right \rangle \\ \geq 0& \quad (\text{since}\; {\mathbf{U}}\;\text{is}\; M-\text{hyponormal tuple}). \end{aligned}$$

We must have

$$ \sum _{1\leq l,k \leq m}\left \langle \bigg(\frac{1}{M^{2}}\big( \mathcal{U}_{k}^{*}-\overline{\gamma _{k}}\big)^{*}\big(\mathcal{U}_{l}^{*}- \overline{\gamma _{l}}\big)^{*}-\big(\mathcal{U}_{l}^{*}- \overline{\gamma _{l}}\big)\big(\mathcal{U}_{k}^{*}- \overline{\gamma _{k}}\big)^{*}\bigg)\omega _{k}\mid \omega _{l} \right \rangle \geq 0 $$

for all $\omega _{,}\cdots ,\omega _{m} \in \mathcal{K}$ and $(\gamma _{1},\ldots ,\gamma _{m})\in \mathbb{C}^{m}$. Therefore, we must have

$$ \sum _{1\leq l,k \leq m}\left \langle \bigg(\frac{1}{M^{2}}\big( \mathcal{U}_{k}^{*}-{\gamma _{k}}\big)^{*}\big(\mathcal{U}_{l}^{*}-{ \gamma _{l}}\big)^{*}-\big(\mathcal{U}_{l}^{*}-{\gamma _{l}}\big) \big(\mathcal{U}_{k}^{*}-{\gamma _{k}}\big)^{*}\bigg)\omega _{k}\mid \omega _{l}\right \rangle \geq 0 $$

for all $\omega _{,}\cdots ,\omega _{m} \in \mathcal{K}$ and $(\gamma _{,}\cdots ,\gamma _{m})\in \mathbb{C}^{m}$. We conclude that ${\mathbf{U}}^{*}$ is $\displaystyle \frac{1}{M}$-hyponormal tuple. □

Theorem 2.3

Let $N\in {\mathcal {B}}\big(\mathcal{K}\big)$ be an invertible operator and ${\mathbf{U}}=(\mathcal{U}_{1}\cdots ,\mathcal{U}_{m}) \in \mathcal{H}(\mathcal{H})^{m}$ be a tuple of operators such that each $\mathcal{U}_{k}$ commutes with ${N}^{*}{N}$ for $k=1,\ldots ,m$. Then ${\mathbf{U}}=(\mathcal{U}_{1}\cdots ,\mathcal{U}_{m})$ is an M-hyponormal tuple if and only if ${N}{\mathbf{U}}{N}^{-1}:=\big( {N}\mathcal{U}_{1}{N}^{-1},\ldots ,{N} \mathcal{U}_{m}{N}^{-1}\big)$ is an M-hyponormal tuple.

Proof

Assume that ${\mathbf{U}}=(\mathcal{U}_{1}, \ldots ,\mathcal{U}_{m})$ is an M-hyponormal tuple. We show that ${N}{\mathbf{U}}{N}^{-1}:=\big( {N}\mathcal{U}_{1}{N}^{-1},\ldots ,{N} \mathcal{U}_{m}{N}^{-1}\big)$ is an M-hyponormal tuple. In fact, let $\omega _{1},\ldots ,\omega _{m} \in \mathcal{H}$ and $\gamma =(\gamma _{1},\ldots ,\gamma _{m})\in \mathbb{C}^{m}$. We have

$$\begin{aligned} &\sum _{1\leq k,l\leq m}\left \langle \bigg(M^{2}\big(N\mathcal{U}_{k}N^{-1}- \gamma _{k}\big)^{*}\big(N\mathcal{U}_{l}N^{-1}-\gamma _{l}\big)\right.\\ &\left.\qquad {}- \big(N\mathcal{U}_{l}N^{-1}-\gamma _{l}\big)\big(N\mathcal{U}_{k}N^{-1}- \gamma _{k}\big)^{*}\bigg)\omega _{k}\mid \omega _{l}\right \rangle \\ =&\sum _{1\leq k,l\leq m}\left \langle N\bigg(M^{2}\big(\mathcal{U}_{k}- \gamma _{k}\big)^{*}\big(\mathcal{U}_{l}-\gamma _{l}\big)-\big( \mathcal{U}_{l}-\gamma _{l}\big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \bigg)N^{-1}\omega _{k}\mid \omega _{l}\right \rangle \\ =& \sum _{1\leq k,l\leq m}\left \langle N\bigg(M^{2}\big(\mathcal{U}_{k}- \gamma _{k}\big)^{*}\big(\mathcal{U}_{l}-\gamma _{l}\big)-\big( \mathcal{U}_{l}-\gamma _{l}\big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \bigg)N^{-1}\omega _{k}\mid NN^{-1}\omega _{l}\right \rangle \\ =& \sum _{1\leq k,l\leq m}\left \langle N^{*}N\bigg(M^{2}\big( \mathcal{U}_{k}-\gamma _{k}\big)^{*}\big(\mathcal{U}_{l}-\gamma _{l} \big)-\big(\mathcal{U}_{l}-\gamma _{l}\big)\big(\mathcal{U}_{k}- \gamma _{k}\big)^{*}\bigg)N^{-1}\omega _{k}\mid N^{-1}\omega _{l} \right \rangle \\ =& \sum _{1\leq k,l\leq m}\left \langle \bigg(M^{2}\big(\mathcal{U}_{k}- \gamma _{k}\big)^{*}\big(\mathcal{U}_{l}-\gamma _{l}\big)\right.\\ &\left.\qquad {}-\big( \mathcal{U}_{l}-\gamma _{l}\big)\big(\mathcal{U}_{k}-\gamma _{k}\big)^{*} \bigg)\sqrt{N^{*}N}N^{-1}\omega _{k}\mid \sqrt{N^{*}N}N^{-1}\omega _{l} \right \rangle \\ \geq &0. \end{aligned}$$

Conversely, assume that ${N}{\mathbf{U}}{N}^{-1}:=\big( {N}\mathcal{U}_{1}{N}^{-1},\ldots ,{N} \mathcal{U}_{m}{N}^{-1}\big)$ is an M-hyponormal tuple. Set $\mathcal{V}_{k}=N\mathcal{U}_{k}N^{-1}$ for $k=1, \ldots ,m$. We can check that each $\mathcal{V}_{k}$ commutes with $\big(N^{-1}\big)^{*}N^{-1}$ and moreover

$$ \big( N^{-1}\mathcal{V}_{1}(N^{-1})^{-1},\ldots , N^{-1}\mathcal{V}_{m}(N^{-1})^{-1} \big)=\big( N^{-1}\mathcal{V}_{1}N,\ldots , N^{-1}\mathcal{V}_{m}N \big)= \big(\mathcal{U}_{1},\ldots ,\mathcal{U}_{m}). $$

Based on the first statement, we have $\big( N^{-1}\mathcal{V}_{1}N,\ldots , N^{-1}\mathcal{V}_{m}N\big)$ is an M-hyponormal tuple and so it shall be $\big(\mathcal{U}_{1},\ldots ,\mathcal{U}_{m})$ is an M-hyponormal tuple. □

For $\mathcal{U} \in \mathcal{B}(\mathcal{K})$, let $\sigma _{p}(\mathcal{U})$ and $\sigma _{a}(\mathcal{U})$ denote the point spectrum and approximate point spectrum of $\mathcal{U}$. If $\mu \in \sigma _{p}(\mathcal{U})$ and $\overline{\mu} \in \sigma _{p}(\mathcal{U}^{*})$, then μ is in the joint point spectrum, $\sigma _{jp}(\mathcal{U})$. If $\mu \in \sigma _{a}(\mathcal{U})$ and $\overline{\mu} \in \sigma _{a}(\mathcal{U}^{*})$, then we say that μ is in the joint approximate point spectrum, $\sigma _{ja}(\mathcal{U})$.

Definition 2.1

([15]) Let ${\mathbf{U}}=(\mathcal{U}_{1},\ldots ,\mathcal{U}_{m})$ on $\mathcal{H}$.

$(1)$ A point $\gamma =(\gamma _{1},\ldots ,\gamma _{m})\in \mathbb{C}^{m}$ is called a joint point eigenvalue of U if there exists a non-zero vector $\omega \in \mathcal{H}$ such that

$$ \big(\mathcal{U}_{k}-\gamma _{k}I_{\mathcal {H}}) \big)\omega =0\;\; \text{for}\;\;k=1,\ldots ,m, $$

or equivalently if

$$ \displaystyle \bigcap _{1\leq k\leq d}\ker (\mathcal{U}_{k}-\gamma _{k}) \neq\{0\}. $$

The joint point spectrum, denoted by $\widetilde{{\sigma _{jp}}}({\mathbf{U}})$ is the set of all joint eigenvalues of U.

$(2)$ $\gamma =(\gamma _{1},\ldots ,\gamma _{d})\in \mathbb{C}^{m}$ is in the joint approximate point spectrum ${\widetilde{\sigma _{jp}}}({\mathbf{U}})$ if and only if there exists a sequence of unit vector $(\omega _{n})_{n}$ such that

$$ \lim _{n\to \infty}\|\big(\mathcal{U}_{k} -\gamma _{k} ) \omega _{n} \| =0,\quad \;\;k=1,\ldots ,m. $$

It was proved that if $\mathcal{U}$ is M-hyponormal operator then for all $\gamma \in \sigma _{p}(\mathcal{U})$.

$$ \ker (\mathcal{U}-\gamma )\subset \ker (\mathcal{U}-\gamma )^{*}. $$

In the following proposition, we extend this result to M-hyponormal tuple of operators.

Proposition 2.2

Let ${\mathbf{U}}=\left ( \mathcal{U}_{1},\ldots ,\mathcal{U}_{m} \right ) \in {\mathcal {B}}\left ( {\mathcal {H}}\right )^{m}$ be an M-hyponormal tuple. Then

$$ \bigcap _{k=1}^{m}\ker \big(\mathcal{U}_{k}-\gamma _{k} \big)\subset \ker \bigg( \sum _{1\leq k \leq m}\big(\mathcal{U}_{k}-\gamma _{k} \big)^{*}\bigg) $$

for $(\gamma _{1},\ldots ,\gamma _{m})\in \mathbb{C}^{m}$.

Proof

Let $\omega \in \displaystyle \bigcap _{k=1}^{m}\ker \big(\mathcal{U}_{k}- \gamma _{k} \big)$ and it follows that $\big(\mathcal{U}_{k}-\gamma _{k} \big)\omega =0$ for $k=1,\ldots ,m$. According to (2.1) we get $- \bigg\|\displaystyle \sum _{1\leq k\leq m} \big(\mathcal{U}_{k}- \gamma _{k})^{*}\omega \bigg\|^{2}\geq 0$ and hence

$$ \sum _{1\leq k\leq m} \big(\mathcal{U}_{k}-\gamma _{k})^{*}\omega =0. $$

 □

Corollary 2.1

Let ${\mathbf{U}}=\left ( \mathcal{U}_{1},\ldots ,\mathcal{U}_{m} \right ) \in {\mathcal {B}}\left ( {\mathcal {H}}\right ) ^{m}$ be an M-hyponormal tuple. If $\gamma =(\gamma _{1},\ldots , \gamma _{m})\in \widetilde{\sigma _{jp}}({\mathbf{U}})$, then $\overline{\displaystyle \sum _{1\leq k \leq m}\gamma _{k}} \in \sigma _{p}\bigg( \displaystyle \sum _{1\leq k \leq m}\mathcal{U}_{k}^{*} \bigg)$.

Proof

Since U is M-hyponormal tuple and $(\gamma _{1},\ldots ,\gamma _{m})\in \widetilde{\sigma _{jp}}( \mathcal{U})$, it follows from Corollary 2.1 that

$$ \bigcap _{k=1}^{m}\ker \big(\mathcal{U}_{k}-\gamma _{k} \big)\subset \ker \bigg( \sum _{1\leq k \leq m}\big(\mathcal{U}_{k}-\gamma _{k} \big)^{*}\bigg). $$

 □

Corollary 2.2

Let ${\mathbf{U}}=\left ( \mathcal{U}_{1},\ldots ,\mathcal{U}_{m} \right ) \in {\mathcal {B}}\left ( {\mathcal {H}}\right )^{m}$ be an M-hyponormal tuple. If $\gamma =(\gamma _{1},\ldots , \gamma _{m})\in \widetilde{\sigma _{jp}}({\mathbf{U}})$ and $\gamma ^{\prime }=(\gamma _{1}^{\prime },\ldots ,\gamma _{m}^{ \prime })\in \widetilde{ \sigma _{jp}}(\mathbf{{U})}$ such that $\displaystyle \sum _{1\leq k\leq m}\big(\gamma _{k}-\gamma _{k}^{ \prime })\neq0$. Then

$$ \bigcap _{k=1}^{m}\ker \big(\mathcal{U}_{k}-\gamma _{k} \big)\bot \bigcap _{k=1}^{m}\ker \big(\mathcal{U}_{k}-\gamma _{k}^{\prime } \big). $$

Proof

Since $\gamma =(\gamma _{1},\ldots ,\gamma _{m})\in \widetilde{\sigma _{jp}}({\mathbf{U}})$ there exists $\omega _{1} \in \mathcal{K}$ such that

$$ \big(\mathcal{U}_{k}-\gamma _{k})\omega _{1}=0 \quad k=1,\ldots ,m. $$

Similarly, $\gamma ^{\prime }=(\gamma _{1}^{\prime },\ldots ,\gamma _{m}^{ \prime })\in \widetilde{\sigma _{jp}}({\mathbf{U}})$ there exists $\omega _{2} \in \mathcal{K}$ such that

$$ \big(\mathcal{U}_{k}-\gamma _{k}^{\prime})\omega _{2}=0 \quad k=1, \ldots ,m. $$

Since U is M-hyponormal tuple, we get from Corollary 2.1 that

$$\begin{aligned} \displaystyle \sum _{1\leq k\leq m}\gamma _{k}\left \langle \omega _{1} \mid \omega _{2} \right \rangle =& \left \langle \displaystyle \sum _{1 \leq k\leq m}\gamma _{k} \omega _{1}\mid \omega _{2} \right \rangle \\ =& \left \langle \displaystyle \sum _{1\leq k\leq m}\mathcal{U}_{k} \omega _{1}\mid \omega _{2} \right \rangle \\ =& \left \langle \omega _{1}\mid \displaystyle \sum _{1\leq k\leq m} \mathcal{U}_{k}^{*} \omega _{2} \right \rangle \\ =& \left \langle \omega _{1}\mid \overline{\displaystyle \sum _{1\leq k\leq m}\gamma _{k}^{\prime}} \omega _{2} \right \rangle \\ =&{\displaystyle \sum _{1\leq k\leq m}\gamma _{k}^{\prime}} \left \langle \omega _{1}\mid \omega _{2} \right \rangle . \end{aligned}$$

It will be $\displaystyle \sum _{1\leq k\leq m}\big(\gamma _{k}-\gamma _{k}^{ \prime}\big)\left \langle \omega _{1}\mid \omega _{2} \right \rangle =0$ and so it shall be $\left \langle \omega _{1}\mid \omega _{2} \right \rangle =0$. □

Proposition 2.3

Let ${\mathbf{U}}=\left ( \mathcal{U}_{1},\ldots ,\mathcal{U}_{m} \right ) \in {\mathcal {B}}\left ( {\mathcal {H}}\right )^{m}$ be an M-hyponormal tuple. If $\gamma =(\gamma _{1},\ldots ,\gamma _{m})\in \widetilde{\sigma _{ja}}({\mathbf{U}})$ and $\gamma ^{\prime }=(\gamma _{1}^{\prime },\ldots ,\gamma _{m}^{ \prime })\in \widetilde{ \sigma _{ja}}(\mathbf{{U})}$ such that $\displaystyle \sum _{1\leq k\leq m}\big(\gamma _{k}-\gamma _{k}^{ \prime })\neq0$. The following hold,

$(1)$ $\overline{\displaystyle \sum _{1\leq k \leq m}\gamma _{k}} \in \sigma _{ap}\bigg( \displaystyle \sum _{1\leq k \leq m}\mathcal{U}_{k}^{*} \bigg)$.

$(2)$ If $(\omega _{n})_{n} \subset \mathcal{K}$ and $(\omega _{n}^{\prime})_{n} \subset \mathcal{K}$ such that $\|\omega _{n}\|=\|\omega _{n}^{\prime}\|=1$ and

$$ \|\big(\mathcal{U}_{k}-\gamma _{k})\omega _{n}\big)\|\longrightarrow 0 \quad \|\big(\mathcal{U}_{k}-\gamma _{k}^{\prime})\omega _{n}^{\prime} \big)\|\longrightarrow 0 \quad k=1,\ldots ,m, $$

then

$$ \left \langle \omega _{n}\mid \omega _{n}^{\prime }\right \rangle \longrightarrow 0 \quad n\longrightarrow \infty . $$

Proof

The steps of the proof are similar to what was done in the Corolaries 2.1 and 2.2, so we left it. □

Theorem 2.4

([8]) Let $\mathcal{H}$ be a complex Hilbert space. Then there exists a Hilbert space $\mathcal{K} \supset \mathcal{H}$ and $\psi : \mathcal{B}(\mathcal{H}) \longrightarrow \mathcal{B}( \mathcal{K})$ satisfying the following properties for every $T,S \in \mathcal{B}(\mathcal{H})$ and $\varrho ,\mu \in \mathbb{C}$.

(1) $\psi (T^{*})=\psi (T)^{*}$, $\psi (I_{\mathcal{H}})=I_{\mathcal{K}}$, $\psi (\varrho T+\mu S)=\varrho \psi (T)+\mu \psi (S)$.

(2) $\psi (TS)=\psi (T)\psi (S)$, $\|\psi (T)\|= \|T\|$, $\psi (T)\geq \psi (S)$, for $T\geq S$,

(3) $\psi (T) \geq 0$ if $T\geq 0$,

(4) $\sigma _{a}(T)=\sigma _{a}(\psi (T))=\sigma _{p}(\psi (T))$,

(5) $\sigma _{ja}(T)=\sigma _{jp}(\psi (T))$.

Theorem 2.5

Let $\mathcal{U} \in \mathcal{B}(\mathcal{H})$ be an M-hyponormal. Then $\sigma _{a}({\mathcal{U}})=\sigma _{ja}({ \mathcal{U}})$.

Proof

Since $\mathcal{U}$ is M-hyponormal, we have

$$ \bigg(M^{2}(\mathcal{U}-\gamma )^{*}(\mathcal{U}-\gamma )- ( \mathcal{U}-\gamma )(\mathcal{U}-\gamma )^{*}\bigg)\geq 0 \quad \forall \; \gamma \in {\mathbb{C}}. $$

(2.10)

In view of Theorem 2.4, we have

$$\begin{aligned} &\bigg(M^{2}(\psi (\mathcal{U})-\gamma )^{*}(\psi (\mathcal{U})- \gamma )- (\psi (\mathcal{U})-\gamma )(\psi (\mathcal{U})-\gamma )^{*} \bigg) \\ =& \bigg(M^{2}(\psi (\mathcal{U}-\gamma )^{*}(\psi (\mathcal{U}- \gamma )- (\psi (\mathcal{U}-\gamma )(\psi (\mathcal{U}-\gamma )^{*} \bigg) \\ =& \psi \bigg(\bigg(M^{2}(\mathcal{U}-\gamma )^{*}(\mathcal{U}- \gamma )- (\mathcal{U}-\gamma )(\mathcal{U}-\gamma )^{*}\bigg)\bigg) \\ \geq &0 \big(\text{by Theorem 2.4 and} \;\;\;( 2.10) \big). \end{aligned}$$

Hence $\psi (\mathcal{U})$ is an M-hyponormal.

From Theorem 2.4, we have $\sigma _{a}(\mathcal{U})=\sigma _{p}(\psi (\mathcal{U}))$. Since $\psi (\mathcal{U})$ is an M-hyponormal, we have $\ker \big(\psi (\mathcal{U})-\mu \big)\subset \ker \big(\psi ( \mathcal{U})-\mu \big)^{*}$ (from [22, Proposition 2]). Hence $\sigma _{p}(\psi (\mathcal{U}))=\sigma _{jp}(\psi (\mathcal{U}))$. According to Theorem 2.4, we have $\sigma _{jp}(\psi (\mathcal{U}))=\sigma _{ja}(\mathcal{U})$. Hence $\sigma _{a}(\mathcal{U})=\sigma _{ja}(\mathcal{U})$. □