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Bessel and Grüss Type Inequalities in Inner Product Modules over Banach
-Algebras
Journal of Inequalities and Applications volume 2011, Article number: 562923 (2011)
Abstract
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 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.
2. Schwarz and Bessel Inequality
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

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 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
(3.22)
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
(3.25)
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).
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
(4.21)
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
(4.26)
-
(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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Ghazanfari, A.G., Dragomir, S.S. Bessel and Grüss Type Inequalities in Inner Product Modules over Banach -Algebras.
J Inequal Appl 2011, 562923 (2011). https://doi.org/10.1155/2011/562923
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DOI: https://doi.org/10.1155/2011/562923
Keywords
- Hilbert Space
- Real Number
- Straightforward Calculation
- Symmetry Condition
- Previous Lemma