1. Introduction and Preliminaries
In functional analysis, particularly in the context of operator theory, the Euclidean operator radius refers to the supremum (least upper bound) of the norms of operators acting on a Hilbert space. If we have a bounded linear operator
T on a Hilbert space
, then its norm is defined as
, where
denotes the norm of the vector
x in the Hilbert space. The Euclidean operator radius extends this concept to a tuple of operators or matrices (see [
1,
2,
3,
4]).
This study delves into Euclidean operator spaces and their associated inequalities, likely in a finite-dimensional context. It explores symmetries and operator norms and derives new inequalities governing various operator properties. While not directly related to the existence of Hilbert spaces, this research contributes to our understanding of functional analysis, which is closely intertwined with Hilbert space theory, demonstrating its significance within the broader mathematical landscape.
Numerous scholars have extensively studied the theory of Hilbert modules over non-commutative
-algebras, as demonstrated by various works, such as [
5,
6,
7,
8,
9,
10,
11,
12]. The notion of symmetry plays a pivotal role in exploring non-commutative topology, extensions of
-algebras, and the field of
K-theory. Symmetry is closely linked with the attributes of Hilbert modules, and grasping their symmetrical aspects offers valuable insights into how they differ from Hilbert spaces.
In non-commutative topology, symmetry shows up in how different algebraic structures relate and change. Hilbert modules, which do not follow the usual commutative rules, have their own kind of symmetry that is different from what we see in Hilbert spaces. They do not have this special “self-dual” quality that Hilbert spaces often do, which marks a big difference in their symmetry.
Another way that Hilbert modules stand apart from Hilbert spaces is that they do not have properties like being orthogonally complementary. In Hilbert spaces, being orthogonal and complementary are important for symmetry, especially when it comes to self-adjoint and unitary operators. Since Hilbert modules lack these properties, they have a kind of imbalance that shapes their unique mathematical features.
Studying symmetry here helps us see the subtle contrasts between Hilbert modules and Hilbert spaces more clearly. It shows us how the absence of specific symmetrical aspects in Hilbert modules affects how they work in non-commutative topology, extensions of -algebras, and K-theory. By looking into these symmetrical details, mathematicians obtain a better grasp of the patterns and connections in these areas of math, making functional analysis and operator theory more interesting and comprehensive.
Hilbert
-modules extend the concept of Hilbert spaces by allowing scalars to belong to a
-algebra instead of being restricted to real or complex numbers. In these modules, an operator matrix refers to a matrix with entries being operators acting within the module, serving as linear mappings from one part of the module to another. Exploring inequalities related to the Euclidean operator radius of
n-tuple operators and operator matrices within Hilbert
-modules typically involves an examination of norm behaviors and inequalities amidst various operations, such as addition, scalar multiplication, and operator composition. This exploration also entails delving into the characteristics of
-algebras and their associated modules, along with methods for analyzing operator and operator matrix norms within these frameworks. For more detail, the reader should refer to [
8,
9,
10,
13].
We begin with the definition of a -algebra.
Definition 1 ([
14])
. A -algebra is a (non-empty) set with the following algebraic operations:- (i)
Addition, which is commutative and associative;
- (ii)
Multiplication, which is associative;
- (iii)
Multiplication by complex scalars;
- (iv)
An involution (that is, for all ).
Both types of multiplication distribute over addition. For , we have . The involution is conjugate linear; that is, for and , we have . For and , we have and . In addition, has a norm in which it is a Banach algebra; that is,
- (N1)
;
- (N2)
;
- (N3)
for all and , and is complete in the metric . Moreover, for all , we have .
Let us revisit how a Hilbert module over a
-algebra
is defined, as described in [
6].
Definition 2 ([
6])
. Consider a linear space over a complex field endowed with the structure of a right -module. We assume that , where ξ belongs to , a belongs to , and λ is a complex number. The space is referred to as a pre-Hilbert -module if it possesses an inner product that adheres to the following conditions:- (i)
and if and only if ;
- (ii)
;
- (iii)
;
- (iv)
,
where , , and . Set . This is a norm on . If is complete, is called a Hilbert module over .
Let us consider Hilbert -modules and . We define as the set containing all mappings , where there exists another mapping satisfying for every and . It is widely known that t must meet the criteria of being a bounded -linear mapping, indicating that t is a linear mapping that satisfies certain boundedness conditions. Additionally, it satisfies for all and .
In the event that , then forms a -algebra, complete with the operator norm.
Definition 3 ([
11])
. Consider a -algebra denoted by . A linear functional ψ acting on is deemed positive if it satisfies the condition for all a within . In the scenario where possesses a unit element (i.e., it is unital), the linear functional ψ is referred to as a state if it is positive and, furthermore, . It is worth noting that when ψ is a linear functional operating on a unital -algebra and , then ψ automatically qualifies as a state. We represent the collection of all states of as . 2. Euclidean Operator Radius of n-Tuple Operators
In this section, we explore the fundamental characteristics of the joint numerical radius and various other operator radii related to the Euclidean operator radius of an n-tuple of operators denoted by . A general guideline emerges: for , it holds that . However, these two concepts coincide with the classical numerical radius when . We present precise inequalities and demonstrate that the operator radius within forms a norm that is equivalent to both the operator norm and the joint numerical radius.
Definition 4 ([
8])
. Let and . Then,where . It is known from [
8] that
is a norm on
. And if
is a Hilbert space, then
.
The following result was investigated in [
8].
Definition 5. Let and . The joint numerical radius for is defined bywhere the supremum is taken over all families of vectors with , where is the free semigroup with the generators and the neutral element . Definition 6. Let be an n-tuple operator. Then, Ƒ is said to be commuting if for all .
For n-tuple operators , we write , and for any scalar .
Definition 7. Let be an n-tuple of bounded linear operators on . The Euclidean operator norm is defined by Lemma 2. defines a norm on .
Theorem 1 ([
9])
. Let . If is a Hilbert -Modules, then Recall that an operator is said to be self-adjoint if .
Theorem 2 ([
9])
. If is self-adjoint, then Definition 8. Let be an n-tuple of bounded linear operators on . The Euclidean operator radius is defined byThe joint (spatial) numerical range of , defined by In general, when , .
Theorem 3. The joint numerical radius and Euclidean operator radius coincide with the classical numerical radius of an operator if , i.e., Proof. If
and
,
,
Taking
, we obtain
. On the other hand, let
,
and
and define
,
, where
. It is easy to see that
and
for any
. Hence, we deduce the inequality
, which proves our assertion, i.e.,
. □
The result outcomes are highly relevant for the subsequent discussions, as detailed in [
9].
Lemma 3. Let and . Then, the following are equivalent:
- (a)
for every with ;
- (b)
for every .
Lemma 4. Let . Then, if and only if for every and .
Lemma 5. Let . Then, for every and , Theorem 4. If , thenOr, equivalently,Here, the constants and 1 are the best possible. Theorem 5. defines a norm that is equivalent to the norm on .
Consider an n-dimensional complex Hilbert
-module denoted by
, equipped with an orthonormal basis
, where
n is a natural number or
. We introduce the full Fock space of
, defined as follows:
Here,
, and
represents the Hilbert tensor product of
k copies of
. We define the left creation operators
, where
, as follows:
Now, let
denote the unital free semigroup on
n generators
, along with the identity element
. We define the length of
as
, where
for
, and
for
. We also introduce
and set
. It is evident that
forms an orthonormal basis for
.
The joint spectral radius linked with an
n-tuple of operators
is defined as follows:
where
denotes the norm.
Theorem 6. The joint numerical radius for n-tuples of operators satisfies the following properties:
- (a)
for any operator in ;
- (b)
for any unitary operator in ;
- (c)
;
- (d)
.
Proof. (a) Let
be an arbitrary sequence of vectors in
such that
. Fix an operator
and define the vectors
,
, where
. Observe that
and
. On the other hand, we have
Taking the supremum over all sequences
with
, we deduce inequality (a).
(b) A closer look reveals that, when is a unitary operator, we have equality in the above inequality. Therefore, relation (b) holds true.
(c) Any vector
with
,
, has the form
, where the sequence
is such that
. If
are the left creation operators on the full Fock space
, note that for
,
Hence, we infer that
Conversely, it is a widely acknowledged fact that the classical numerical radius of an operator
y adheres to the inequalities
(as established in ([
9], Theorem 2.13)). Utilizing relation (
4) and considering the orthogonal ranges of the left creation operators, we can derive
Similarly, one can prove the first inequality in (c).
(d) To prove (vii), notice that, since
,
, we have
Consequently, we deduce that
Therefore, inequality (d) is established. □
In this section, we outline the fundamental characteristics of the Euclidean operator radius for an
n-tuple of operators. We introduce a novel norm and a concept akin to the “spectral radius” on
by defining them as follows:
where
, i.e., the unit ball of
.
And
where
denotes the usual spectral radius of an operator
. Notice that
is a norm
,
In what follows, we show that
is equivalent to the operator norm on
.
Theorem 7. If , thenwhere the constants 1 and are the best possible, and Proof. Let
be the rotation-invariant normalized positive Borel measure on the unit sphere
. Using the relations
we have
On the other hand, we have
Now, notice that if
are the left creation operators, then
This shows that the inequalities in (
7) are the best possible. In order to prove (
8), note that
□
The following outcome summarizes several key properties of the Euclidean operator radius for an n-tuple of operators.
Theorem 8. The Euclidean operator radius for n-tuples of operators satisfies the following properties:
- (i)
;
- (ii)
;
- (iii)
for any separable Hilbert -Modules;
- (iv)
is a continuous map in the norm topology.
Proof. Observe that, for every
and
,
Consequently, we obtain
It is known from [
9], Theorem 2.13, that
for every
. Applying these inequalities to the operator
for
and using relation (
10), we deduce (i) and (ii).
To prove (iii), we use relation (
10) and the fact that the classical numerical radius satisfies the equation
. Indeed, we have
According to (i) and Theorem (7), we obtain
Therefore, we deduce that
is continuous in the norm topology. The proof is complete. □
We will now establish one of our central results, which provides an inequality involving the Euclidean operator radius.
Theorem 9. Let . Then, for every and , the following inequality holds: Proof. Let
and
(
) be real numbers with
. Then, we have
Hence,
Since
for all
and
,
. Consequently,
Suppose
for all
, and we choose
in such a way that
and
for all
. Then, we have
Therefore, the Cauchy–Schwartz inequality implies that
Also, this inequality holds when
for all or some
. This completes the proof. □
Utilizing Theorem 9, we obtain the following corollary.
Corollary 1. If , then Proof. From Theorem 9, together with
we have
Taking
, we obtain
This implies
Therefore,
Taking the supremum over all
with
and
, we obtain the desired inequality. □
Definition 9. Let . Then, we define by setting Next, we obtain a refinement of the first inequality in (
3).
Theorem 10. If and , then Proof. Taking
in (
11), we obtain
Hence,
Taking the supremum over all
with
and
, we obtain
and so the result. □
We will now demonstrate the following inequalities for the joint operator norm of n-tuple normal operators. To this end, we will leverage the well-known characterization of normal operators. An operator t in is considered normal if and only if holds for all in .
Theorem 11. Let be an n-tuple normal operator. Then, Proof. Let
with
and
. Then, we have
Now,
Also, we have
Also,
Therefore, the proof is complete. □
Remark 1. It is worth noting that if we choose () as an matrix where only the diagonal entries at position are 1, and all other entries are zero, then the first inequality in Theorem 11 becomes an equality. Similarly, if we set (where I is the identity matrix) for , then the second inequality in Theorem 11 also becomes an equality. Therefore, it can be concluded that the inequalities presented in Theorem 11 are indeed sharp.
In the subsequent theorem, we establish an inequality involving the joint numerical radius of n-tuple operators in terms of powers.
Theorem 12. If , then Proof. Suppose we have
such that
and
. The inequality
implies
for each
. Thus, if
, then
for each
. The power inequality [
15] implies that
for each
, whenever
. Therefore, if
, then
Now, if we take
for all
, then
, where
, and so
. Thus,
, and this gives
. □
The example below illustrates Theorem 12.
Example 1. Let be the vector space of all continuous real-valued functions on with the norm defined byand the inner product defined byNow, let with , . Then,Also,Now,Thus, the validity of the outcome stated in Theorem 12 is confirmed. Applying Theorem 12, we derive the following inequality.
Corollary 2. Let . If , then Proof. It follows from inequality (
3), together with Theorem 12, that
□
Now, we will establish the inequalities for the Euclidean operator radius when multiplying n-tuple operators. To show the Euclidean operator norm’s submultiplicative property and the Euclidean operator radius’s subadditive property, we require the following lemma.
Lemma 6. Let , . Then, the following inequalities hold:
- (a)
.
- (b)
.
Proof. (a) Let
with
and
. Then, we have
(b) See the proof of Theorem 5. □
Theorem 13. Let , . Then, Proof. We have
, where the second inequality is derived from Lemma 6(a), and the third inequality is derived from (
3). □
Moreover, we derive an inequality concerning the collective numerical radius for the multiplication of two n-tuple operators and , given the condition .
Theorem 14. Let , . If , i.e., ( for all ), then Proof. Assume that
. Then, we have
This proof is complete. □
The next bound for the product of two n-tuple normal operators reads as follows.
Theorem 15. Let , . If Ƒ and are normal, then Proof. We have
, where the last equality follows from
and
, as
are both normal (see [
16]). □
We wrap up this section with the subsequent theorem concerning the joint spectral radius and joint numerical radius. We begin by introducing the notion of the joint approximate point spectrum for an
n-tuple operator
, represented by
, which is defined as follows:
This definition can be equivalently expressed as the existence of a sequence
with
such that
for all
. For a commuting
n-tuple operator
,
represents the joint spectrum of
, as defined in [
16]. Notably, it follows that
.
For an n-tuple commuting operator , we introduce the non-negative number , defined as , which is referred to as the joint spectral radius of . For an n-tuple commuting operator , the inequality is valid.
Theorem 16. Let be commuting. Then, the following statements are equivalent.
- (i)
.
- (ii)
.
Proof. (i)⟹ (ii). Assume that . It follows from that .
(ii) ⟹ (i). Let
. Then, there exists a sequence
with
such that
Without loss of generality, assume that
converges to
, and the sequence
converges to
b. Then,
. Now,
Hence,
, and so
. This implies
. Hence,
. □
3. Euclidean Operator Radius of Matrices
For
,
, the
operator matrix, whose entries are
n-tuple operators
, is defined as
Note that
is a Hilbert
-Module with the inner product defined as
for all
and
. We prove the next lemma to start this section.
Lemma 7. Let , . Then, the following assertions hold:
- (i)
.
- (ii)
. In particular, .
- (iii)
.
- (iv)
for all .
- (v)
. In particular, - (vi)
. In particular, .
Proof. (i) Let
and
such that
, i.e.,
. Then,
Taking the supremum over all
with
and
, we obtain
Suppose
with
and
. Then, we have
Taking the supremum over all
with
, we obtain
This implies that
. Similarly,
. Hence,
.
(ii) Let
and
such that
, i.e.,
. Then,
Taking the supremum over all
with
and
, we obtain
Suppose
with
and
. Then, we have
Taking the supremum over all
with
, we obtain
This implies that
. Similarly,
. Therefore,
In particular, if
, then
(because
).
(iii) It follows from Theorem 11(b) that
for every unitary operator
. If we let
, then
(iv) The proof follows from (
13) by taking
.
(v) Let
and
. Then,
. Using (i) and (
13), we obtain
. In particular, if we take
, then
.
(vi) Let
and
such that
, i.e.,
. Then,
Taking the supremum over all
with
and
, we obtain
Suppose
with
and
. Then, we have
Taking the supremum over all
with
, we obtain
This implies that
. Similarly,
. Therefore,
□
Hereafter, we set an upper limit for the Euclidean operator radius of operator matrices, where the elements comprise n-tuple operators.
Theorem 17. Let , . Then, Proof. Let
and
such that
, i.e.,
. Then, by Minkowski’s inequality, we have
Thus,
□
As a consequence of Theorem
3, we can conclude that
and
. Additionally, it is evident that
for all
. Hence, the subsequent corollary directly stems from Theorem 17.
Corollary 3. Let , . Then, It is important to highlight that Theorem 17 provides a stronger bound compared to Corollary 3. To proceed with the next conclusion, we must refer to the following lemma.
Lemma 8 ([
15])
. Consider an matrix , where for all . In this case, , where represents the spectral radius. The subsequent series of corollaries are derived by utilizing Lemma 8, Theorem 17, and Corollary 3.
Corollary 4. Let , . Then,where and . Corollary 5. Let , . Then,where and . Next, we establish a lower limit for the Euclidean operator radius of operator matrices composed of n-tuple operators, utilizing the power inequality derived in Theorem 12.
Theorem 18. Let . Then, Proof. Let
. Then,
for all
. Using Lemma 7 and Theorem 12, we obtain
So, the proof is complete. □
Next, we will establish the subsequent lower and upper bounds.
Theorem 19. Let . Then, Proof. It follows from Lemma 7(v) that
By replacing
with
, we have
Consequently,
To prove the second inequality, consider a unitary operator. Let
. Then, we have
The proof of the theorem is now concluded. □
The following example illustrates Theorem 19.
Example 2. Let , where , , , and . First, let us compute :Next, let us compute : Now, consider and . We will evaluate the expression With these values, we haveTherefore, the inequality becomes We derive the following inequalities by employing Theorem 19.
Corollary 6. Let for all . Then, Proof. Replacing
with
in Theorem 19 and then using Lemma 7, we have
This implies that
But we know that
and so the result. □