Power vector inequalities for operator pairs in Hilbert spaces and their applications

1 Introduction

Let H be a complex Hilbert space equipped with an inner product ⟨ ⋅ , ⋅ ⟩ and a corresponding norm ‖ ⋅ ‖ . We denote the C * -algebra of bounded linear operators on H as B ( H ) . For any operator A ∈ B ( H ) , the adjoint of A is denoted as A * , and ∣ A ∣ = ( A * A ) 1 2 represents the positive square root of A . The real and imaginary parts of A are defined as Re ( A ) = 1 2 ( A + A * ) and Im ( A ) = 1 2 i ( A − A * ) , respectively. We define the numerical range of A , denoted by W ( A ) , as the set of values { ⟨ A x , x ⟩ : x ∈ H , ‖ x ‖ = 1 } .

Let ‖ A ‖ and w ( A ) denote the operator norm and the numerical radius of A , respectively. The operator norm is given by

while the numerical radius is defined as

It is known that the numerical radius w ( ⋅ ) defines a norm on B ( H ) and is equivalent to the operator norm ‖ ⋅ ‖ . In fact, the following double inequality holds:

These inequalities are sharp. The first inequality becomes an equality if A 2 = 0 , while the second inequality becomes an equality if A is a normal operator. An improvement to these inequalities was established by Kittaneh [1], who showed that

For further advancements in (1.1) and (1.2), interested readers can refer to [2–9].

For an operator T ∈ B ( H ) , let r ( T ) denote the spectral radius of T . It is well known that for every T ∈ B ( H ) , we have the fundamental inequality

and that equality holds in inequality (1.3) if T is normal.

In addition to inequality (1.3), the most important properties of the spectral radius are the spectral radius formula

a special case of the spectral mapping theorem, which asserts that

for every natural number m , and the commutativity property, which asserts that

It follows from the spectral radius formula (1.4) that if A , B ∈ B ( H ) are commutative, then the following subadditivity

and submultiplicativity

properties hold. We also observe that any upper bound for the numerical radius will provide an upper bound for the spectral radius, which may motivate finding upper bounds for the latter in order to estimate the former. For additional properties of the spectral radius, the reader is referred to the classical book [10].

Let B and C be operators in B ( H ) . The Euclidean operator radius between B and C , denoted as w e ( B , C ) , is defined as

More information can be found in [11–13].

According to [13], the function w e ( ⋅ , ⋅ ) : B ( H ) × B ( H ) → [ 0 , ∞ ) is a norm that satisfies the inequality

The constants 2 4 and 1 are the optimal choices in (1.9). If B and C are self-adjoint operators, then (1.9) simplifies to

Note that for self-adjoint operators B and C , w e ( B , C ) = w ( B + i C ) . The proof of this follows easily from the definition of w e ( B , C ) .

In a previous study [14], the second author derived a lower bound as follows:

It was shown that the constant 2 2 in (1.10) cannot be replaced by a larger constant. The same study also obtained the following results:

where the constant 2 2 is sharp in both inequalities. Furthermore, the study presented the following inequality:

which is also sharp. Another sharp inequality derived in the study is as follows:

By considering ( B , C ) = ( A , A ∗ ) or ( B , C ) = ( Re ( A ) , Im ( A ) ) for an operator A ∈ B ( H ) , the second author derived several norm and numerical radius inequalities for a single operator A in [14].

For additional refinements of the aforementioned results, readers are advised to refer to the recent study by Jana et al. [15]. In the same study, they also presented the following interesting results as well:

and

for all α ∈ [ 0 , 1 ] .

In [16], for r ≥ 1 , the generalization of the numerical radius for a pair of operators B , C ∈ B ( H ) , referred to as the generalized s - r -numerical radius, is defined by

and the generalized s - r -norm is defined by

If we choose r = 2 in the previous definitions, then we obtain the Euclidean numerical radius

and the Euclidean norm

studied in [13].

For r = 1 , we denote

and

The above notations can also be extended for r ∈ ( 0 , 1 ) ; however in this case, they are no longer norms.

In 2017, Moslehian et al. [11] obtained the following fundamental inequalities for the s-r-numerical radius of the pair of operators B , C ∈ B ( H ) :

and

to mention a few. Moslehian et al. [11] also applied their results for the Cartesian decomposition of an operator A .

We conclude this introductory section by recalling a recent generalization of the Boas-Bellman result (see [17,18] or [19, p. 392]) given by Mitrinović et al. [19, p. 392]. They proved that if x , y 1 , … , y n are elements of H (which need only to be an inner product space, not necessarily complete), the following inequality holds:

It should be noted that if we choose c i = ⟨ x , y i ⟩ ¯ in (1.11), we obtain the well-known inequality in [19]. For further results regarding the Boas-Bellman-type inequality, Mitrinović-Pečarić-Fink-type inequality, and Bessel inequality, we invite readers to consult [19–22] and references therein.

Motivated by the results in [15], in this work, we first employ the Mitrinović-Pečarić-Fink-type inequality (1.11) to derive several power vector inequalities for a pair of operators ( B , C ) in Section 2.

The main result in this section that plays a crucial role in the other sections is the following power inequality for B , C ∈ B ( H ) and α , β ∈ C :

for all x , y ∈ H and p ≥ 1 .

In particular, for p = 1 , we establish that for B , C ∈ B ( H ) , α , β ∈ C and x , y ∈ H , we have the following sharp inequality

which, for an appropriate choice of the operators, reduces to Bessel’s inequality. By considering ( B , C ) = ( A , A ∗ ) or ( B , C ) = ( Re ( A ) , Im ( A ) ) for A ∈ B ( H ) , we obtain vector, norm, and numerical radius inequalities for a single operator.

In Section 3, motivated by the results of Moslehian et al. in [11], we explore several power inequalities for the s - r -norm and s - r -numerical radius of the pair of operators ( B , C ) ∈ B ( H ) , which generalize the Euclidean norm and Euclidean numerical radius.

Two of the main results we want to emphasize here are

and

for B , C ∈ B ( H ) , p ≥ 1 , and r > 1 . They provide different generalizations for powers and more diverse upper bounds than those from [11,14,15] mentioned above. Due to their quite different analytic expressions, they cannot be compared in general with the bounds listed above. Finally, we apply these results to derive the some power inequalities for a single operator A ∈ B ( H ) .

2 Vector inequalities

The main objective of this section is to derive power vector inequalities for a pair of operators ( B , C ) . To establish our first result in this direction, we will utilize the Mitrinović-Pečarić-Fink-type inequality (1.11) for n = 2 . Specifically, for n = 2 in (1.11), we obtain

for complex numbers c 1 , c 2 and vectors x , y 1 , y 2 ∈ H .

We are now prepared to state the following result.

The next corollary presents a specific instance of the aforementioned outcome.

Theorem 2.1 leads to several important results and practical uses. One significant outcome is described in the next corollary.

Some other notable consequences of Theorem 2.1 are summarized in the following remark.