Bessel and Grüss Type Inequalities in Inner Product Modules over Banach -Algebras

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.

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 semi-inner product module over is a right module over together with a generalized semi-inner 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 semi-inner product -module if is a proper -algebra and that is a semi-inner 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 real-valued 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 semi-inner product -module (or semi-inner product -module) is a complex vector space which is also a right -module with a sesquilinear semi-inner 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 semi-inner product -module , then is said to be a semi-Hilbert -module.

If is a norm on an inner product -module , then is said to be a pre-Hilbert -module.

A pre-Hilbert -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 semi-inner 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 semi-Hilbert -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 semi-inner product -module and , then is a semi-Hilbert -module.

(c) Let be a -algebra and be the auxiliary norm on . If is an inner product -module and , then is a pre-Hilbert -module.

(d) Let be a -algebra and (a semi-inner product) an inner product -module. Since tr is a positive linear functional on and for every we have ; therefore is a (semi-Hilbert) pre-Hilbert -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.

If is a semi-inner product -module, then the following Schwarz inequality holds:

(e.g. [4, Lemma  15.1.3]).

If is a semi-inner 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 semi-inner product -module can be found in [6].

Now let be a -algebra, a positive linear functional on and let be a semi-inner -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 semi-inner -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 Grüss-type 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

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 semi-inner 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 semi-inner product -modules we have the following lemma, which is a generalization of [7, Lemma  1].

Lemma 3.1.

Let be a semi-inner 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 semi-inner 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).

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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