1. Introduction and Preliminaries
The Cauchy–Schwarz inequality is a fundamental inequality in mathematics, with applications in many areas such as linear algebra, analysis, and probability theory. In the context of Hilbert spaces, the Cauchy–Schwarz inequality takes on a particularly elegant form, which has led to its extensive use in functional analysis and related fields.
Semi-Hilbert spaces are mathematical spaces that extend the concept of Hilbert spaces, which are commonly studied in analysis and linear algebra. Unlike Hilbert spaces, semi-Hilbert spaces allow for more flexible inner products that do not need to satisfy all the usual requirements. This flexibility makes semi-Hilbert spaces useful for investigating various mathematical and physical problems involving unbounded operators, singularities, or non-local interactions.
In the field of operator theory, semi-Hilbert spaces have attracted significant interest. They can be constructed by starting with a positive semidefinite sesquilinear form and then building a space based on this form. Recent research has made notable contributions to this area. Noteworthy works exploring operator theory in semi-Hilbert spaces include references [
1,
2,
3,
4,
5,
6,
7,
8,
9], as well as the additional references provided therein. These resources provide valuable insights and advancements in understanding semi-Hilbert spaces and their applications in operator theory.
In this paper, we establish new types of Cauchy–Schwarz inequalities in the context of semi-Hilbert spaces and apply them to derive novel A-numerical radius inequalities, where A is a positive semidefinite operator in a complex Hilbert space.
To set the foundation for our findings, we initially present certain symbols and remind readers of some commonly acknowledged facts. Our analysis involves a complex Hilbert space denoted as , which comes equipped with an inner product and a corresponding norm . The -algebra, which comprises all bounded linear operators on along with the identity operator (or simply I), is denoted as throughout this paper. When we refer to an “operator”, we are specifically referring to a bounded linear operator that acts on .
For any operator S, we use and to respectively denote its nullspace and range. The orthogonal projection onto any closed linear subspace of is denoted by . We say that an operator S is positive if for all , and we write to indicate this property. When , we introduce the notation for the unique positive bounded linear operator satisfying .
The absolute value of an operator
is defined as
, which is always non-negative. We define the Moore–Penrose inverse of an operator
S as
, which is the unique linear extension of
to
such that
. Here,
is the isomorphism
. The operator
is obtained as the unique solution to the set of four equations known as the Moore–Penrose equations:
For more details on the Moore–Penrose inverse, interested readers can refer to [
10].
From this point forward, the following assumptions will be made:
and
. We can use
A to define a positive semidefinite sesquilinear form on
, denoted by
, given by
for all
. The seminorm induced by
is
for every
. It is worth mentioning that
serves as a norm on
if and only if the operator
A is injective. Moreover, the pair
forms a semi-Hilbert space that is complete if and only if
is a closed subspace in
. Finally, when
, we get
and
for all
. The set of all unit vectors in
is given by
The numerical radius is a crucial concept in matrix analysis and operator theory. In recent times, there have been various extensions of the numerical radius, one of which is the
A-numerical radius of an operator
. The definition of the
A-numerical radius of
Q was first introduced by Saddi in [
11], and it can be expressed as:
This concept has recently received significant attention, with several papers exploring its properties and applications, including [
1,
2,
3] and the references therein. As established in [
12], it is common knowledge that
can be infinite for certain operators
. Therefore, in order to ensure that
is well defined and finite, we must revisit the notion of
A-adjoint operators presented in [
13].
Definition 1 ([
13])
. An operator is said to be an A-adjoint operator of if the identity holds for every . In other words, Q is the solution of the operator equation . To investigate this equation, we can use a theorem developed by Douglas [
14]. This theorem states that the operator equation
has a solution
if and only if
, which is equivalent to the existence of a positive number
such that
for all
. Furthermore, the same theorem developed by Douglas shows that, if
has more than one solution, then there exists only one solution, denoted by
R, that satisfies
. Such a solution
R is referred to as the reduced solution of the equation
. However, it is not guaranteed that an
A-adjoint operator exists or is unique for a given operator
T.
Let us define the sets
and
as the sets of all operators that have
A-adjoints and
-adjoints, respectively. Based on Douglas’s theorem, we can derive the following equivalences:
and
It is worth noting that
and
are two subalgebras of
, but they are not necessarily closed or dense in
. Furthermore, we can show that
using the reference [
12].
An operator is considered
A-bounded if it is a member of the set
. The set
is equipped with the seminorm
, which is defined as follows:
This seminorm is also given by
for
, as shown in [
15]. This definition is well established in the literature and has been used extensively in the study of bounded linear operators in
A-weighted spaces. It can be shown that
if and only if
for any
. Moreover, for all
and
, we have
, which implies the inequality
for all
.
It is noteworthy that the seminorms
and
are equivalent on
. Specifically, for any
, we have (see [
16])
Moreover, it was demonstrated in [
12] that
for any positive integer
n and
.
Let us remember that an operator
is referred to as being
A-selfadjoint if
is selfadjoint. It is clear that if
Q is
A-selfadjoint, then
. If an operator
Q satisfies
, it is referred to as
A-positive, denoted as
. It is important to highlight that in the context of a complex Hilbert space
, an
A-positive operator is also considered
A-selfadjoint. In [
12], it was demonstrated that, for any
A-selfadjoint operator
Q (especially if
), the following equality holds:
Suppose
. The solution to the equation
can be reduced and denoted as
. This reduced solution can be expressed as
. Moreover, if
, then
and we have the properties
and
. For detailed proofs and additional related results, refer to [
10,
13,
17] and the references therein.
One can verify that the operators
and
. Additionally, for any operator
, the following equalities hold (see Proposition 2.3 in [
10]):
The term “
A-normal” is used to describe an operator
that satisfies
(refer to [
11]). It should be noted that, while all selfadjoint operators are normal, an
A-selfadjoint operator may not necessarily be
A-normal (see Example 4 in [
12]).
Many authors have recently demonstrated various improvements to the inequalities shown in Equation (
1). These can be found in studies such as [
1,
16], as well as other references mentioned therein. Specifically, it has been demonstrated in [
16] that, for
, the following inequality
holds. When
in Equation (
5), the resulting inequalities are the well established ones that were proven by Kittaneh in Theorem 1 in [
18].
Conde et al. in [
19] established important numerical radius upper bounds. Specifically, for operators
and a positive integer
n, the following inequalities hold:
These results can be further improved by replacing
T with
and
S with
, yielding the following inequalities:
In this paper, we present new types of Cauchy–Schwarz inequalities within the framework of semi-Hilbert spaces and employ them to derive innovative
A-numerical radius inequalities. Notably, several of our findings expand upon the established body of knowledge concerning the classical numerical radius inequalities of Hilbert space operators. The inspiration for our investigation comes from recent works in this area [
20,
21], which have highlighted the importance of developing new mathematical tools for studying inner product spaces. To establish our results, we employ techniques rooted in semi-Hilbert space theory, which provides a more general framework for studying inner product spaces. Our findings have important implications in various branches of mathematics, including functional analysis and operator theory. We demonstrate the versatility of the new Cauchy–Schwarz inequalities and show how they can be used to derive innovative
A-numerical radius inequalities, where
A represents a positive semidefinite operator in a complex Hilbert space.
Overall, our research contributes to the ongoing study of inner product spaces and provides new insights and tools for various areas of mathematics.