We give an analogue of the Bessel inequality and we state a simple formulation of the Grüss type inequality in inner product modules, which is a refinement of it. We obtain some further generalization of the Grüss type inequalities in inner product modules over proper algebras and unital Banach algebras for seminorms and positive linear functionals.
1. Introduction
A proper algebra is a complex Banach algebra where the underlying Banach space is a Hilbert space with respect to the inner product satisfying the properties and for all . A algebra is a complex Banach algebra such that for every . If is a proper algebra or a algebra and is such that or , then .
For a proper algebra , the trace class associated with is . For every positive there exists the square root of , that is, a unique positive such that , the square root of is denoted by . There are a positive linear functional on and a norm on , related to the norm of A by the equality for every .
Let be a proper algebra or a algebra. A semiinner product module over is a right module over together with a generalized semiinner product, that is with a mapping on , which is valued if is a proper algebra, or valued if is a algebra, having the following properties:
(i) for all ,
(ii) for , ,
(iii) for all ,
(iv) for .
We will say that is a semiinner product module if is a proper algebra and that is a semiinner product module if is a algebra.
If, in addition,
(v) implies ,
then is called an inner product module over . The absolute value of is defined as the square root of and it is denoted by .
Let be a algebra. A seminorm on is a realvalued function on such that for and : , , . A seminorm on is called a seminorm if it satisfies the condition: . By Sebestyen's theorem [1, Theorem 38.1] every seminorm on a algebra is submultiplicative, that is, , and by [2, Section 39, Lemma 2(i)] . For every , the spectral radius of is defined to be .
The Pták function on algebra is defined to be , where . This function has important roles in Banach algebras, for example, on algebras, is equal to the norm and on Hermitian Banach algebras is the greatest seminorm. By utilizing properties of the spectral radius and the Pták function, Pták [3] showed in 1970 that an elegant theory for Banach algebras arises from the inequality .
This inequality characterizes Hermitian (and symmetric) Banach algebras, and further characterizations of algebras follow as a result of Pták theory.
Let be a algebra. We define by
and call the elements of positive.
The set of positive elements is obviously a convex cone (i.e., it is closed under convex combinations and multiplication by positive constants). Hence we call the positive cone. By definition, zero belongs to . It is also clear that each positive element is Hermitian.
We recall that a Banach algebra is said to be an algebra provided there exists on a second norm , not necessarily complete, which is a norm. The second norm will be called an auxiliary norm.
Definition 1.1.
Let be a algebra. A semiinner product module (or semiinner product module) is a complex vector space which is also a right module with a sesquilinear semiinner product , fulfilling
for , . Furthermore, if satisfies the strict positivity condition
then is called an inner product module (or inner product module).
Let be a seminorm or a positive linear functional on and . If is a seminorm on a semiinner product module , then is said to be a semiHilbert module.
If is a norm on an inner product module , then is said to be a preHilbert module.
A preHilbert module which is complete with respect to its norm is called a Hilbert module.
Since and are self adjoint, therefore we get the following Corollary.
Corollary 1.2.
If is a semiinner product module, then the following symmetry condition holds:
Example 1.3.
(a) Let be a algebra and a positive linear functional or a seminorm on . It is known that is a semiHilbert module over itself with the inner product defined by , in this case .
(b) Let be a Hermitian Banach algebra and be the Pták function on . If is a semiinner product module and , then is a semiHilbert module.
(c) Let be a algebra and be the auxiliary norm on . If is an inner product module and , then is a preHilbert module.
(d) Let be a algebra and (a semiinner product) an inner product module. Since tr is a positive linear functional on and for every we have ; therefore is a (semiHilbert) preHilbert module.
In the present paper, we give an analogue of the Bessel inequality (2.7) and we obtain some further generalization and a simple form for the Grüss type inequalities in inner product modules over algebras, proper algebras, and unital Banach algebras.
2. Schwarz and Bessel Inequality
If is a semiinner product module, then the following Schwarz inequality holds:
(e.g. [4, Lemma 15.1.3]).
If is a semiinner product module, then there are two forms of the Schwarz inequality: for every
First Saworotnow in [5] proved the strong Schwarz inequality, but the direct proof of that for a semiinner product module can be found in [6].
Now let be a algebra, a positive linear functional on and let be a semiinner module. We can define a sesquilinear form on by ; the Schwarz inequality for implies that
In [7, Proposition 1, Remark 1] the authors present two other forms of the Schwarz inequality in semiinner module , one for positive linear functional on :
and another one for seminorm on :
The classical Bessel inequality states that if is a family of orthonormal vectors in a Hilbert space , then
Furthermore, some results concerning upper bounds for the expression
and for the expression related to the Grüsstype inequality
have been proved in [8]. A version of the Bessel inequality for inner product modules and inner product modules can be found in [9], also there is a version of it for Hilbert modules in [10, Theorem 3.1]. We provide here an analogue of the Bessel inequality for inner product modules.
Lemma 2.1.
Let be a algebra, let be an inner product module, and let be a finite set of orthogonal elements in such that are idempotent. Then
Proof.
By [11, Lemma 1] or a straightforward calculation shows that
3. Grüss Type Inequalities
Before stating the main results, let us fix the rest of our notation. We assume, unless stated otherwise, throughout this section that is a unital Banach algebra. Also if is a semiinner product module and is a seminorm on , we put , and if is a positive linear functional on , we put . Let be a finite set of orthogonal elements in such that be idempotent, we set and .
Dragomir in [8, Lemma 4] shows that in a Hilbert space , the condition
is equivalent to the condition
where and , . But for semiinner product modules we have the following lemma, which is a generalization of [7, Lemma 1].
Lemma 3.1.
Let be a semiinner product module and , . Then
if and only if
Proof.
Follows from the equalities:
Remark 3.2.
By making use of the previous Lemma 3.1, we may conclude the following statements.
(i)Let be an inner product module and let be a finite set of orthogonal elements in such that are idempotent, then inequality (3.3) implies that
(ii)Let be an inner product module and be a finite set of orthogonal elements in such that are idempotent. If is a seminorm on then inequality (3.3) implies that
and if is a positive linear functional on from inequality (3.3) and [2, Section 37 Lemma 6(iii)], we get
(iii)Let be a proper algebra, let be an inner product module, and let be a finite set of orthogonal elements in such that are idempotent. Since for every , inequality (3.3) is valid only if
We are able now to state our first main result.
Theorem 3.3.
Let be an inner product module and let be a finite set of orthogonal elements in such that are idempotent. If , , , are real numbers and such that
hold, then one has the inequality
Proof.
By [11, Lemma 2] or, a straightforward calculation shows that for every
Therefore
Analogously, for every , we have
The equalities (3.10), (3.13), and (3.14) imply that
Since
therefore the Schwarz's inequality (2.1) holds, that is,
Finally, using the elementary inequality for real numbers
on
we get
Remark 3.4.
(i) Let be an inner product module and let be a finite set of orthogonal elements in such that are idempotent. If and are such that
and if we put , and , then, by (3.15) and (3.16), we have
These and (3.11) imply that
Therefore, (3.11) is a refinement and a simple formulation of [9, Theorem 4.1.].
(ii) If for , we set
then similarly (3.11) is a refinement and a simple form of [9, Corollary 4.3].
Corollary 3.5.
Let be a Banach algebra, let be an inner product module, and let be a finite set of orthogonal elements in such that are idempotent. If , , are real numbers and such that
hold, then one has the inequality
Proof.
Using the schwarz's inequality (2.6), we have
The assumptions (3.26) and the elementary inequality for real numbers (3.19) will provide the desired result (3.27).
Example 3.6.
Let be a Hermitian Banach algebra and let be the Pták function on . If is a semiinner product module and with the properties that
then we have
That is interesting in its own right.
Corollary 3.7.
Let be a proper algebra, let be an inner product module, and let be a finite set of orthogonal elements in such that are idempotent. If , are real numbers and such that
hold, then one has the inequality
Proof.
Using the strong Schwarz's inequality (2.3), we have
The assumptions (3.31) and the elementary inequality for real numbers (3.19) will provide (3.32).
The following companion of the Grüss inequality for positive linear functionals holds.
Theorem 3.8.
Let be an inner product module, let be a positive linear functional on , and let be a finite set of orthogonal elements in such that are idempotent. If , , , are real numbers and such that
hold, then one has the inequality
Proof.
By taking on both sides of (3.12), we have
Analogously
Now, using Aczl's inequality for real numbers, that is, we recall that
and the Schwarz's inequality for positive linear functionals, that is,
we deduce (3.35).
4. Some Related Results
Theorem 4.1.
Let be an inner product module and let be a finite set of orthogonal elements in such that are idempotent. Let and if we define
then we have
Proof.
For every , , by (3.13) and (3.14), we have
Therefore
Now, using the elementary inequality for real numbers
on
we get
Corollary 4.2.
Let be a Banach algebra, let be an inner product module, and let be a finite set of orthogonal elements in such that are idempotent. Let and put
then
Corollary 4.3.
Let be a proper algebra, let be an inner product module, and let be a finite set of orthogonal elements in such that are idempotent. Let and if we consider
then
From a different perspective, we can state the following result as well.
Theorem 4.4.
Let be an inner product module and let be a finite set of orthogonal elements in such that are idempotent. If , , and such that
then we have the inequality
Proof.
We know that for any and one has
Put , , and since
using (4.14), we have
Now, inequality (4.13) follows from inequalities (3.15) and (4.16).
The following companion of the Grüss inequality for positive linear functionals holds.
Theorem 4.5.
Let be an inner product module, let be a positive linear functional on , and let be a finite set of orthogonal elements in such that are idempotent. If , , and are such that
then we have the inequality
Proof.
The inequality (4.14) for , implies that
By making use of inequality (3.12) for instead of and taking on both sides, we have
From (4.19) and (4.20), we easily deduce (4.18).
Remark 4.6.
(i) The constant 1 coefficient of in (3.11) is sharp, in the sense that it cannot be replaced by a smaller quantity. If the submodule of generated by is not equal to , then there exists such that . We put , then and for any , we have
For every , if we put
then
therefore
Now if is a constant such that , then there is a such that ; therefore
(ii) Similarly, the constant 1 coefficient of in (3.32) is best possible, it is sufficient instead of (4.22) to put
(iii) If there is a nonzero element in such that and (resp. ) then the constant 1 coefficient of in (3.27) (resp. (3.35)) is best possible. Also similarly, the inequalities in Theorem 4.1, Corollaries 4.2 and 4.3, and Theorems 4.4 and 4.5 are sharp. However, the details are omitted.
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