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A Grüss type inequality for vector-valued functions in Hilbert -modules
Journal of Inequalities and Applications volume 2014, Article number: 16 (2014)
In this paper we prove a version of Grüss’ integral inequality for mappings with values in Hilbert -modules. Some applications for such functions are also given.
MSC:46L08, 46H25, 26D15.
In 1934, Grüss  showed that for two Lebesgue integrable functions ,
provided m, M, n, N are real numbers with the property and a.e. on . The constant is best possible in the sense that it cannot be replaced by a smaller constant.
The following inequality of Grüss type in real or complex inner product spaces is well known .
Theorem 1 Let be an inner product space over () and , . If and are such that the conditions
hold, or, equivalently, if
hold, then the following inequality holds:
The constant is the best possible in equation (1.1).
Let be a real or complex Hilbert space, be a Lebesgue measurable set and a Lebesgue measurable function with . We denote by the set of all strongly measurable functions f on Ω such that .
A further extension of the Grüss-type inequality for Bochner integrals of vector-valued functions in real or complex Hilbert spaces is given in .
Theorem 2 Let be a real or complex Hilbert space, a Lebesgue measurable set and a Lebesgue measurable function with . If f, g belong to and there exist vectors such that
then the following inequalities hold:
The constant is sharp in the sense mentioned above.
Also Jocić et al. in  presented the following Grüss-type inequality:
for all bounded self-adjoint fields satisfying and for all and some bounded self-adjoint operators C, D, E, and F, and for all .
The main aim of this paper is to obtain a generalization of Theorem 2 for vector-valued functions in Hilbert -modules. Some applications for such functions are also given.
Hilbert -modules are used as the framework for Kasparov’s bivariant K-theory and form the technical underpinning for the -algebraic approach to quantum groups. Hilbert -modules are very useful in the following research areas: operator K-theory, index theory for operator-valued conditional expectations, group representation theory, the theory of -algebras, noncommutative geometry, and others. Hilbert -modules form a category in between Banach spaces and Hilbert spaces and obey the same axioms as a Hilbert space except that the inner product takes values in a general -algebra rather than in the complex numbers ℂ. This simple generalization gives a lot of trouble. Fundamental and familiar Hilbert space properties like Pythagoras’ equality, self-duality and decomposition into orthogonal complements must be given up. Moreover, a bounded module map between Hilbert -modules does not need to have an adjoint; not every adjoinable operator needs to have a polar decomposition. Hence to get its applications, we have to use it with great care.
Let be a -algebra. A semi-inner product module over is a right module X over together with a generalized semi-inner product, that is, with a mapping on , which is -valued and has the following properties:
for all ,
for , ,
for all ,
We will say that X is a semi-inner product -module. The absolute value of is defined as the square root of , and it is denoted by . If, in addition,
then is called a generalized inner product and X is called an inner product module over or an inner product -module. An inner product -module which is complete with respect to the norm () is called a Hilbert -module.
As we can see, an inner product module obeys the same axioms as an ordinary inner product space, except that the inner product takes values in a more general structure than in the field of complex numbers.
If is a -algebra and X is a semi-inner product -module, then the following Schwarz inequality holds:
(e.g. [, Proposition 1.1]).
It follows from the Schwarz inequality equation (2.1) that is a semi-norm on X.
Now let be a ∗-algebra, φ a positive linear functional on , and let X be a semi-inner -module. We can define a sesquilinear form on by ; the Schwarz inequality for σ implies that
In [, Proposition 1, Remark 1] the authors present two other forms of the Schwarz inequality in semi-inner -module X, one for a positive linear functional φ on :
where r is the spectral radius, and another one for a -seminorm γ on :
3 The main results
Let be a -algebra; first we state some basic properties of integrals of -value functions with respect to a positive measure for Bochner integrability of functions which we need to use in our discussion. For basic properties of the integrals of vector-valued functions with respect to scalar measures and the integrals of scalar-valued functions with respect to vector measures, see Chapter II in .
Lemma 1 Let be a -algebra, X a Hilbert -module and be a measure space. If is Bochner integrable and then
fa is Bochner integrable, where (),
the function defined by is Bochner integrable and .
if and f is positive, i.e., for all , then .
Proof (a) Suppose is a simple function with finite support i.e., with for each non-zero (), then for every the function defined by () is a simple function and . Since is Bochner integrable and , fa is strongly measurable and , therefore fa is Bochner integrable.
For every simple function we have and consequently . The result therefore follows.
Suppose that and f is a Bochner integrable function and positive, i.e., for all . Since f is Bochner integrable, by Theorem 2 in [, Chapter II, Section 2]. Using the Holder inequality for Lebesgue integrable functions we get
So is Bochner integrable; thus there is a sequence of simple functions such that for almost all and in the norm topology in .
This implies that , i.e., for every positive Bochner integrable function f there is a sequence of positive simple functions such that for almost all and in the norm topology in . By proposition (1.6.1) in  the set of positive elements in a -algebra is a closed convex cone, therefore since . □
If μ is a probability measure on Ω, we denote by the set of all strongly measurable functions f on Ω such that .
For every , we define the constant function by (). In the following lemma we show that a special kind of invariant property holds, which we will use in the sequel.
Lemma 2 If , , and , are measurable then
Proof We must state that the functions under the integrals of equation (3.1) are Bochner integrable on Ω, since they are strongly measurable and we can state the following obvious results.
For every we have . Therefore
Also for almost all we have
A simple calculation shows that
and for and we deduce equation (3.2). □
The following result concerning a generalized semi-inner product on may be stated.
Lemma 3 If , , then
the following inequalities, equations (3.3) and (3.4), are equivalent:(3.3)(3.4)
the map ,(3.5)
is a generalized semi-inner product on .
Proof (i) If , since for any
showing that, indeed, the inequalities of equations (3.3) and (3.4) are equivalent.
(ii)v We note that the first integral in equation (3.5) belongs to and the later integrals are in X, and the following Korkine-type identity for Bochner integrals holds:
By an application of the identity equation (3.7),
It is easy to show that is a generalized semi-inner product on . □
The following theorem is a generalization of Theorem 2 for Hilbert -modules.
Theorem 3 Let X be a Hilbert -module, μ a probability measure on Ω. If f, g belong to and there exist vectors such that
Then the following inequalities hold:
The coefficient 1 in the second inequality and the constant in the last inequality are sharp in the sense that they cannot be replaced by a smaller quantity.
Proof Since equation (3.5) is a generalized semi-inner product on , the Schwarz inequality holds, i.e.,
Using equation (3.2) with and equation (3.6) we get
By the Schwarz inequality (3.12) and the inequalities (3.13) and (3.14) we deduce equation (3.11).
Now, suppose that equation (3.11) holds with the constants in the third and fourth inequalities. That is,
Every Hilbert space H can be regarded as a Hilbert ℂ-module. If we choose , , ,
then for , and μ a Lebesgue measure on Ω, the conditions (3.10) hold. By equation (3.15) we deduce
giving and , and the theorem is proved. □
1. Let X be a Hilbert -module and the set of all adjoinable operators on X. We recall that if then its operator norm is defined by
with this norm is a -algebra.
Let and for , where A is an invertible element in . Since for each one has
an application of the first inequality in equation (3.13) for , gives
This implies that
2. For square integrable functions f and g on and
Landau proved 
Jocić et al. in  have proved for a probability measure μ and for square integrable fields and () of commuting normal operators that the following Landau-type inequality holds:
for all and for all unitarily invariant norms .
Every -algebra can be regarded as a Hilbert -module over itself with the inner product defined by . If we apply the first inequality in equation (3.11) of Theorem 3, we obtain the following result.
Corollary 1 Let be a -algebra,μ a probability measure on Ω. If f, g belong to , then the following inequality holds:
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The author would like to thank the referee for some useful comments and suggestions.
The author declares that he has no competing interests.
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Ghazanfari, A.G. A Grüss type inequality for vector-valued functions in Hilbert -modules. J Inequal Appl 2014, 16 (2014). https://doi.org/10.1186/1029-242X-2014-16
- Hilbert -modules
- Grüss inequality
- Landau-type inequality
- Bochner integral