1. Introduction
In functional analyses, many authors have studied the tuples of operators. For example, we refer to [
1,
2,
3,
4,
5] and the references therein.
Consider a complex Hilbert space , where the norm induced by is denoted by . The set denotes the -algebra of all bounded linear operators acting on with identity (or shortly I). If is n-dimensional, we identify with the space of all matrices with entries in the complex field and denote its identity by . In what follows, by an operator, we mean a bounded linear operator. We will mention some specific notions of an operator, i.e., the null space of every operator T is denoted by , its range by , and is the adjoint of T. An operator is said to be positive if for all . We write if T is positive. If , then means the square root of T. The commutator of two operators is defined as . It is easy to see that , for every and . Recall that is called normal (respectively hyponormal) if (respectively, ).
Next, we present some inequalities related to operators that we need in the future. First, we give the classical Schwarz inequality for a positive operator
:
where
.
In [
6], Halmos obtains a result similar to the inequality above
for every
and for any
. In [
7], Kato proves a Schwarz-type inequality (
1), which generalizes the inequality of Halmos:
for all operators
, for every vector
, and
. McCarthy [
8] gives an important inequality in the theory of operators as follows:
Lemma 1 (Theorem 1.4 in [
8])
. Let be a positive operator and satisfy . Then, for ,For , the above inequality is reversed. In what follows, we assume that
A is a positive nonzero operator that defines the following positive semi-definite sesquilinear form:
The seminorm induced by
is given by
. It can be seen that
is a norm on
if and only if
A is injective, and the semi-Hilbert space
is complete if and only if
is closed in
.
Definition 1 ([
9])
. An operator is called an A-adjoint of , if we have () for every . The existence of an
A-adjoint operator is not guaranteed. Thus, we denote by
the set of all operators that admit
A-adjoints. Using Douglas’ theorem [
10], we obtain the following:
and
When
, we say that
T is
A-bounded. The sets
and
are subalgebras of
, which are neither closed nor dense in
. Moreover, the inclusions
hold. We have equality if
A is injective and
, where
means the closure of
in the norm topology of
(see [
11]). Further,
gives the following seminorm on
(see [
12] and the references therein). It is useful to note that if
, then
. Further,
for any
. Let
denote the Moore–Penrose pseudo-inverse of an operator
X (for more details concerning this operator, see [
11]). Following [
11], we have:
implies that
and
In 2012, Saddi [
13] introduced the
A-numerical radius of an operator
by
In 2020, the concept of the
A-spectral radius of
A-bounded operators was defined in [
14] as follows:
Note that
and
may equal
for some
(see [
14]). However, the following relation shows that
and
are equivalent seminorms on
:
For the proof of (
6), we refer to the following references [
12,
14]. If
, then the classical definitions of the operator norm, numerical radius, and spectral radius for Hilbert space operators are obtained and are simply denoted by
,
and
.
If
, then by Douglas’s theorem [
10] there exists a unique solution, given by
, of the following problem
Note that
, where
is the Moore–Penrose pseudo-inverse of
A (see [
11]). If
, then
for every
and we have
and
. Moreover, if
denotes the orthogonal projection onto
, then for a given
, we have
,
and
. For more details about the operator
, one can see [
9,
11,
15]. Furthermore, we recall that an operator
T is said to be
A-positive if
is a positive operator and we write
. It can be observed that
A-positive operators are in
. For
, the notation
means
. When
, then
will simply be denoted by
.
The structure of this paper is organized as follows: in
Section 2, we give some notions that characterize a
d-tuple of operators
. In
Section 3, we introduce a new joint norm of tuples of operators that generalizes the joint norm given in (
12) and define the class of doubly-commuting tuples of hyponormal operators acting on an
A-weighted Hilbert space, where
A is a positive operator that is not assumed to be invertible. We proved a generalization of the well-known result due to G. Popescu [
16]. We also present an inequality that characterizes the Euclidean norm of an operator tuple
. In
Section 4, we give several characterizations related to the operators from
and the operators from
. For an
A-doubly-commuting
d-tuple of hyponormal operators, we prove the equalities
and
The motivation for our investigation comes from a recent paper [
17].
2. Preliminaries
To prepare the framework in which we will work, we present in this section some notions and notations that will be useful in this paper.
Let and denote the set of nonnegative and positive integers, respectively. Let and be a d-tuple of operators. If for all , then is said to be a commuting tuple. Moreover, if is a commuting d-tuple of operators and for every , then it is called a doubly-commuting operator tuple.
In the next definition, we recall two important classes of operators in semi-Hilbert spaces.
Definition 2 ([
14])
. An operator is called- (i)
A-normal if ;
- (ii)
A-hyponormal if .
For some results concerning the above two classes of operators, see [
14] and the references therein. For
, the equalities
hold for the class of
A-normal,
A-hyponormal, and
A-positive operators (see [
14]). Since
and
, then an application of the second equality in (
7) together with the last equality in (
3) shows that
Now, associated with a
d-tuple of operators,
(not necessarily commuting), the following quantities
and
are defined in [
12]. If
for all
, then one can verify that
and
two seminorms on
. Notice that
and
are called the
A-joint numerical radius and the
A-joint operator seminorm of
, respectively.
In [
18], H. Baklouti et al. introduced the concept of the
A-joint spectral radius associated with a
d-tuple of commuting operators
as follows
where
. Moreover, for the multi-index
, we will use the following notations:
We mention here that the second equality in (
9) has also been proved by Baklouti et al. in [
18]. Notice that for every commuting operator tuple
, we have
(see Theorem 2.4 in [
12] and Theorem 2.2 in [
19]). In [
19], it is stated that if
is any
d-tuple of commuting
A-normal operators, then
One of the main targets of this work is to establish the equalities in (
11) for a new class of multivariable operators.
Next, for
, we define
,
, and
which will simply be denoted by
,
and
, respectively. Thus, we obtain
and
The last equality is given in [
20] by M. Chō and M. Takaguchi and is the Euclidean operator radius of an operator tuple
, see also [
16].
For
, G. Popescu defined in [
16] the following quantity
where
denotes the open unit ball of
with respect to the Euclidean norm, i.e.,
It is clear that we can change
with its closure in (
12) without changing the value of
. Note that
defines a norm on
. Moreover, in [
17], the following equality is established:
for every doubly-commuting
d-tuple of hyponormal operators
. It is important to mention that G. Popescu proved in [
16] that the following inequalities hold
for any
d-tuple
. Furthermore, it has been shown in [
16] that the constants
and 1 are the best choices possible.
3. New Joint Seminorm for Operator Tuples
In this section, we aim to introduce and investigate a new joint seminorm for
d-tuples of
A-bounded operators. An alternative and easy proof of a well-known result due to G. Popescu [
16] is established.
First, we introduce the following definition, which is a natural generalization of (
12).
Definition 3. Let . The A-Euclidean seminorm of the d-tuple of A-bounded operators is given by In the next proposition, we state some connections between the seminorms and .
Proposition 1. Let be the d-tuple of the operators. Then, the following assertions hold:
- (1)
;
- (2)
If for all , then andwhere .
Proof. (1) Let
and
. Then, by applying the Cauchy–Schwarz inequality (in short (C–S)) and making several calculations, we deduce that
By applying the inequality (C–S) again, we obtain the following inequality
Then, by taking the supremum over all
with
, we find
So, the desired inequality is proved by taking the supremum over all
.
(2) The fact that
follows trivially since
for all
. Now, in order to prove (
15), we need to recall from [
16] the following facts: if we denote by
the unit sphere of
and
the rotation-invariant positive Borel measure on
for which
, then for all
, we have
Now, let
denote the closed unit ball of
. It is clear that
Further, by using (
8), we see that
On the other hand, since
, then it follows that
for all
. This implies that, through (
16),
This proves that
By replacing
by
in (
17) and then using the fact that
, we have
Combining (
17) together with (
18) yields (
15) as desired. □
Remark 1. (1) If , then clearly . Hence, a direct application of (7) shows thatIt should be mentioned here that the equality may not be correct even if is a commuting operator tuple. Indeed, let us consider the following matrices in : , and . We remark that . Furthermore, by using the fact that with , it can be seen that and . Further, by applying (19) and (4), we can show thatand(2) In virtue of Proposition 1, we infer that and are equivalent seminorms on . The following corollary provides a generalization and improvement of the well-known result due to G. Popescu [
16].
Corollary 1. Let be a d-tuple of operators. Then, the inequalityholds, where and Proof. By applying Proposition 1 together with (
19), we deduce that
By replacing
by
in (
21), we can see that
from which we have
A combination of (
21) together with (
22) yields (
20) as desired. □
Remark 2. Note that the following equalitymay not be correct for some d-tuple of operators even if . Indeed, we consider the following matrices in : , and . It is not difficult to check that In the next theorem, we give a new formula of
for
, which allows us to prove that
and
are two equivalent seminorms on
. Notice that our new techniques provide an alternative and easy proof of the inequalities (
14), which were first proved in [
16].
Theorem 1. Let . Then, the equalityholds. Proof. By using (
3), we see that
Moreover, recall from [
19] that for complex numbers
, we have
Now, let
. By using (
25), we have
Hence, by taking the supremum over all
with
in the last equality we have
This yields that
On the other hand, it is not difficult to check that
Thus, by using (
26) and (
27), we obtain
Combining (
28) together with (
24) yields (
23) as required, and, hence, the proof is complete. □
Remark 3. By letting in (23), we obtain a well-known result established by Dragomir in Theorem 9 in [21], and when the 2-tuple is , where , we obtain a recent result in [22]. The following corollary is an application of Theorem 1 and provides an improvement of the results given in Proposition 1 since
. Moreover, the new Formula (
23) enables us to derive an alternative and easy proof of the inequalities (
14).
Corollary 2. Let be a d-tuple of operators. Then, Proof. By using (
23) and then applying the inequality (C–S), we easily prove the second inequality in (
29). Now, let
be such that
. Assume that
for all
and let
(If
for some
, we choose
). We clearly have
Thus, by applying (
23), we have
Similarly, we prove that
for all
. This yields
Therefore, by taking the supremum over all
with
in the last inequality, we have
Hence, the proof is complete. □
Remark 4. By letting in (29) and then replacing with for all we easily obtain the inequalities (14) that have been already established by G. Popescu in [16] by using a different argument. To establish our next result, we require the following lemma.
Lemma 2. For any vectors in and for arbitrary complex numbers , with , , we havefor any . Proof. We use, as in [
23] or [
24], the technique of the monotony of a sequence. Consider the sequence
By studying the monotony of sequence
, we have
For two vectors
and for complex numbers
, the following equality holds:
Since the term on the right side of equality (
30) is positive, then we have
Now, using the inequality from (
31), we have:
It is easy to see that
, that is, the sequence
is increasing. Therefore, we deduce that
However, by applying relation (
30) for
,
,
and
, we obtain
Taking into account that we can rearrange the terms of the two sequences, we obtain the inequality:
Consequently, we deduce the inequality of the statement. □
We are now able to establish the following result.
Theorem 2. Let be a d-tuple of operators. Then, the inequalityholds for any . Proof. In Lemma 2, set
for all
, then
First, we take the supremum over all
with
in relation (
33), we deduce
Therefore, it we take the supremum over all
in relation (
34), then we find the inequality of the statement. □
Remark 5. By letting in (32), we obtainfor any d-tuple of operators and . Next, we will present a result that characterizes the Euclidean norm of an operator tuple .
Proposition 2. Let be a d-tuple of operators. The following inequality holds:where Here . Proof. First, we will prove a radon-type inequality,
for every
and
with
. If we apply the Jensen inequality for the function
, which is concave for
, we deduce
which is equivalent to inequality (
36). In [
25], Dragomir applied Hölder’s inequality for this. For
, the function
is convex and the inequality sign in (
36) is flipped, obtaining the classical Radon inequality
Thus, we have
Therefore, we deduce
Consequently, by taking the supremum over
with
in inequality (
37) and taking into account the equality from Theorem 1 for
, we obtain the desired result. □
Remark 6. If we take in relation (35), then and for in the same relation, we obtain . Using the Kittaneh–Manasrah inequality [26] and inequality (35), we found the following inequality:for all a d-tuple of positive operators. 4. -Doubly-Commuting Tuples of -Hyponormal Operators
In this section, we give several characterizations related to the operators from and the operators from . For an A-doubly-commuting d-tuple of hyponormal operators, we proved the equalities and
Let us introduce the following definition.
Definition 4. Let . The d-tuple is said to be A-doubly commuting if:
- (i)
it is commuting, i.e., for all ,
- (ii)
for all .
In this section, we will study the connection between
and
, when
is a
d-tuple of
A-doubly-commuting tuples of
A-hyponormal operators. For this purpose, we need to recall some aspects: the semi-inner product
induces an inner product on the quotient space
is given by
for any
. We remark that
is not complete unless
is closed in
. However, L. de Branges and J. Rovnyak [
27] proved that the completion of
is isometrically isomorphic to the Hilbert space
, where
is given by
It is obvious that
stands for the norm induced by the inner product
. We mention here that, in view of Proposition 2.1. in [
15], we have
is dense in
. Further, since
, then we observe that
which implies that
In the next proposition, we give an interesting connection between operators in and operators in . Furthermore, we summarize in the same proposition some useful properties.
Proposition 3 ([
14,
15,
19,
28])
. Let . Then there exists such that if and only if . In such cases is unique. Here is defined by . Furthermore, for every and , we have the following properties:- (1)
and .
- (2)
If , then and
- (3)
If and for all , then we have
- (i)
and ;
- (ii)
and .
- (4)
If such that for all , then , where .
The following lemma is also useful in proving our results in this section.
Lemma 3. Let . Then, the following assertions hold:
- (i)
If , then ;
- (ii)
If is A-hyponormal, then is hyponormal on .
Proof. (1) Let
. By applying (
38) together with Proposition 3 (3), we have
where the last inequality follows by using the fact that
. Now, by using the fact that
is dense in
, we can check that
Hence,
is a positive operator on the Hilbert space
. Therefore,
as required.
(2) Since T is A-hyponormal, then . Hence, by the first assertion of this lemma, we deduce that . By Proposition 3, this implies that . Therefore, is an hyponormal operator on . □
Now, we are able to prove the following result.
Theorem 3. Let be an A-doubly-commuting d-tuple of A-hyponormal operators. Then, the following equalityholds. Proof. Since
, then by applying Proposition 3 we deduce that for each
there exists
such that
. Another application of Proposition 3 shows that
and hence,
where
. On the other hand, we observe that the
A-joint seminorm of
can be written as:
where
If
, we simply denote
by
. In particular, we observe that
Now, by using the decomposition
together with (
39), we obtain
from which
This immediately implies that
Furthermore, it can be seen that
Now, let
. Then, there exists
satisfying
Since the subspace
is dense in
, then there exists a sequence
, which may be assumed to be in
(because of the fact that
) such that
. Thus, by taking (
44) into consideration, it holds that
for all
. Now, set
. Clearly, we have
. Further, by using (
45), it can be checked that
This implies that, through (
42),
and, therefore,
From (
43) and (
46), we deduce that
. Therefore, we infer that
On the other hand, since
is an
A-doubly-commuting
d-tuple of
A-hyponormal operators, then
Therefore, by applying Proposition 3 together with Lemma 3 (ii), we have
Hence,
is a
d-tuple of doubly-commuting hyponormal operators on the Hilbert space
. Therefore, by (
13), we have
This completes the proof by taking (
41) and (
47) into consideration. □
Remark 7. Note that the converse of Theorem 3 need not be correct as shown in the next example.
Example 1. Let us consider the same matrices in given in Remark 1, i.e., , and . By Remark 1, we have . Further, we see thatwhere means the closed unit ball of . So, by using (4) and making direct calculations, we show thatBy making use of the Cauchy–Schwarz inequality, it can be easily checked that . On the other hand, by using (48) and then (4), we see thatfrom which . So,However, it can be verified that and this is not an A-doubly-commuting 2-tuple of A-hyponormal operators. Remark 8. According to our proof in Theorem 3, we remark that the equalityholds for every d-tuple of operators . Here, and verify for all . Note that (49) provides an improvement of a result of the second author in [19] since is in general a proper subspace of . In order to derive an important consequence from Theorem 3, we first introduce the following definition that is inspired by the work of G. Popescu [
16].
Definition 5. For , we define a new A-joint numerical radius and a new A-joint spectral radius of by settingand Remark 9. If , then will simply be denoted by . Further, it is worth mentioning that the equalityholds for every commuting operator tuple (see Theorem 2.1 in [29] or [2]). Now, as an application of Theorem 3, we state the following result.
Theorem 4. Let be an A-doubly-commuting d-tuple of A-hyponormal operators. Then Proof. Since
is a
d-tuple of
A-doubly-commuting
A-hyponormal operators, then in particular we have
for every
Now, for
, we let
. Clearly,
. By making simple calculations and using (
52), we see that
Hence,
is an
A-hyponormal operator. This implies that, by (
7),
for all
. If we take the supremum over all
in the last equalities, then we obtain
Taking into account relation (
25), we have
This implies that
This proves
In view of Theorem 3, we have
. So, all that remains to be proven is that
. Since we have
, then by applying Proposition 3 we deduce that for each
there exists
such that
. Hence, another application of Proposition 3 shows that
and so,
Since
is an
A-doubly-commuting operator tuple, then it is commuting. So, similar to the proof of Theorem 3, we find that
is a commuting
d-tuple of operators in the Hilbert space
. Therefore, by (
51), we conclude that
. Further, by Proposition 3 (4), we have
. Hence, by taking (
55) into account, we deduce that
as desired. Thus, combining (
53) with (
54), (
56) and (
40) yields the desired result and the proof is complete. □
5. Conclusions
In this paper, we introduced a definition that is a generalization of (
12). For
, the
A-Euclidean seminorm of
is given by
Consequently, our objective was to study a new joint norm of tuples of operators which generalizes the joint norm given in (
12) and define the class of doubly-commuting tuples of hyponormal operators acting on an
A-weighted Hilbert space, where
A is a positive operator that is not assumed to be invertible. The motivation for our investigation comes from the recent paper [
17].
This article was structured as follows: In
Section 3, we investigated a new joint seminorm for
d-tuples of
A-bounded operators. An alternative and easy proof of a well-known result due to G. Popescu [
16] was established. In
Section 4, we give several characterizations related to the operators from
and the operators from
. For the
A-doubly-commuting
d-tuple of hyponormal operators, we proved the equalities
and
In this paper, the ideas and methodologies used may serve as a starting point for future studies in this field. We will look for other connections of these seminorms for d-tuples of A-bounded operators by studying other possible characterizations. In future work, we will generalize the results given a countable collection of operators.