-Normal Operators in Several Variables
We consider an extension of the concept of -normal operators in single variable operator to tuples of operators, similar to those extensions of the concepts of normality to joint normality, hyponormality to joint hyponormality, and quasi-hyponormality to joint quasi-hyponormality.
Through this paper, we denote Hilbert space over the field of complex numbers by . We write for the set of all bounded linear operators on . For an operator , the adjoint, the kernel, and the range of are denoted by , , and , respectively. For is said to be positive if for all and if and only if . We let .
The class of normal operators on Hilbert spaces plays a critical role in operator theory. This class of operators has been generalized by many authors to non-normal operators.
Let such that . An operator is called(1)Normal if .(2)Hyponormal if [2, 3].(3)-normal  ifor equivalentlyfor all . It should be noted that for , is a normal operator. For , we observe that is hyponormal and for , we obtain that is hyponormal. Interested readers can find more details on -normal operators in [4–9]. The concept of --normal operators was introduced by Senthilkumar and Shanthi . An operator is said to be --normal operator for if
When , this coincides with -normal operators.
The study of tuples of commuting operators has received great attention in recent years by many authors, and the interested reader can refer to [11–20] for complete details.
Let be a -tuple of operators. is said to be jointly normal if and every is a normal operator for all (see ).
Given an -tuple , let denote the self-commutator of , defined by for all . In , the author has introduced the concept of joint hyponormality of operators. An -tuple of operators is called joint hyponormal, if the operator matrixis positive on the direct sum of copies of (i.e., ) or equivalently if , for each finite collection of (see ).
Recently, the first named author in  has introduced the concept of joint -quasi-hyponormal tuple of operators as follows. An -tuple of operators is said to be joint -quasi-hyponormal if and only if satisfiesfor . Or equivalently, is joint -quasi-hyponormal tuple if the operator matrixis positive on the space .
For , the numerical radius of an operator is defined by
These concepts were later generalized by Dekker  to joint numerical radius of as follows.
The joint numerical radius of is defined  by
Now, for a given -tuple of operators , we consider the joint operator norm of defined by
Note that was introduced by Cho and Takaguchi in .
It was proved in  that if is an -normal operator and , thenand in ,
The study of classes of operators and related topics is one of the hottest areas in operator theory in Hilbert spaces. In this work, we are going to consider an extension of the concept of -normal operators in single variable to tuples of operators, similar to those extensions of the concepts of normality to joint normality, hyponormality to joint hyponormality, and quasi-hyponormality to joint quasi-hyponormality (for more details, see [21–23]).
The outline of the paper is as follows. Section 2 is devoted to the study of our new class of multioperators. We give several remarks and examples, which try to clarify the context of the concept of joint -normal tuples. We show that the product of a joint -normal tuple by an -tuple of operators satisfying suitable conditions is joint -normal tuple. In Section 3, some inequalities involving joint operator norms and joint numerical radius for joint -normal tuples are proved under suitable conditions.
2. Joint -Normal Operators
This section is devoted to the study of the class of jointly -normal operators acting on complex Hilbert spaces. We show some basic results related to this class of operators.
Let and set
Definition 1. Let be an -tuple of operators and let . We say that is a joint -normal tuple of operators if the operator matrix satisfieson the direct sum of -copies of or equivalently
Remark 1. If , (14) coincides with (1).
Remark 2. Let be a joint normal tuple of operators; then, is a joint -normal tuple.
Proposition 1. Let such that for . Then, is joint -normal if and only iffor in .
Proof. Under the assumptions that for and , it follows that for each finite collections ,andFor , let Re(z) denote the real part of .
Theorem 1. Let be an -tuple of operators. If is a joint -normal operator, thenfor .
Proof. Since is a joint -normal operator, it follows from (14)thatWe deduce from the above inequalities thatfrom which (18) follows.
Question. Assume that satisfies (18); is it then true that is a joint -normal operator?
Remark 3. If is joint -normal, then is joint hyponormal. In fact, taking and in (14), we getfor , or equivalentlywhich yields on . Therefore, is joint hyponormal.
Remark 4. Let be a commuting tuple of operators. If is joint -normal, then is a joint normal operator.
In fact, since is joint -normal, it follows from (14) thatandWe deduce thatThis implies thatfor each . In particular,Consequently, for . So, is joint normal.
Example 1. Consider where and . A simple calculation shows that is joint -normal for all .
The following example shows that there exists a -tuple of operators such that each is an -normal operator for all , but is not joint -normal.
Example 2. Let and be the standard orthonormal basis of . Consider where and . It was observed that and are -normal with and (see [4, 8]). However, a short calculation shows thatConsequently, is not joint -normal.
Example 3. Let be -normal. Then, is joint -normal. Indeed, Since is -normal, it follows thatLet . Using these inequalities for , we can writeandMoreover, one hasandfor . This means that is joint -normal.
Proposition 2. Let and consider . Then, is joint -normal if and only if is -normal.
Proof. Assume that is -normal. In view of Example 3, we know that is joint -normal. Conversely, assume that is joint -normal. From Definition 1, it follows thatandfor each collection . In particular, for , we getandor equivalently,for all . So, is -normal.
In the general case, we have the following theorem.
Theorem 2. Let be -tuple of operators on and let . Assume that
Then, is joint normal tuple if and only if each is normal for .
Proof. We have that is joint normal tuple if and only ifHowever, by using (37), we get
Proposition 3. Let and let such that . If for all , then is joint -normal if and only if is joint -normal.
Proof. Assume that is joint -normal and prove that is joint -normal. To do so, we have the following in view of (15):from which it follows thatTherefore, is joint -normal tuple. Conversely, if is joint -normal, we getThis means thatHence, is a joint -normal tuple by (13).
The following corollary is an immediate consequence of Proposition 3.
Corollary 1. Let . Let such that and . If for , then is joint -normal if and only if is joint -normal.
Proposition 4. Let be joint -normal. The following statements hold.(1) is joint -normal for all .(2)If is an isometry, then is joint -normal.
Proof. (1) Since is joint -normal, it follows for each collection thatOn the other hand, for same reason, we haveConsequently,Thus, is joint -normal as required. (2) Under the assumption that is an isometry, we have for each collection ,On the other hand,More precisely, we have shown that satisfiesIt is obvious that if is -normal, then . The following lemma extended this result to the multivariable case.
Lemma 1. Let be joint -normal. Then,
Proof. Since is joint -normal, it follows thatandfor . In particular, for , we getandIf we take one can see thatThis implies that , and we deduce that for all . So, . Conversely, let . From the above inequalities, we infer immediately thatHence, we always have , and we deduce that for all . This yields .
Proposition 5. Let and let such that for . If and are joint -normal, then so is .
Proof. Let . Under the assertions that for , we getandSince and are -normal, it follows thatandFrom the previous calculations, we see that is a joint -normal operator.
In [26, Theorem 4], the authors have proved that if is an -normal operator and is an -normal operator such that , then and are -normal operators. The following example proves that even if and are joint -normal operators, their product is not in general a joint -normal operator.
Example 4. (1)Consider and which are -normal and their product is -normal. In view of Proposition 2, it follows that , , and are joint -normal operators.(2)Consider and which are -normal, whereas their product is not -normal. Hence, by applying Proposition 2, we observe that and are joint -normal operators. However, is not a joint -normal operator.In the following theorems, we show that the product of a joint -normal tuple by an -tuple of operators satisfying suitable conditions is joint -normal tuple.
Theorem 3. Let be joint -normal and let be joint normal operators such that
Then, is a joint -normal operator.
Proof. Since is joint normal and for all , it follows from Fuglede theorem  that