Advances in Linear Algebra & Matrix Theory
Vol.4 No.2(2014), Article ID:46838,8 pages DOI:10.4236/alamt.2014.42010

Proving and Extending Greub-Reinboldt Inequality Using the Two Nonzero Component Lemma

Morteza Seddighin

Indiana University East, Richmond, IN 47374, USA

Email: mseddigh@iue.edu

Received 12 May 2014; revised 2 June 2014; accepted 9 June 2014

ABSTRACT

We will use the author’s Two Nonzero Component Lemma to give a new proof for the GreubReinboldt Inequality. This method has the advantage of showing exactly when the inequality becomes equality. It also provides information about vectors for which the inequality becomes equality. Furthermore, using the Two Nonzero Component Lemma, we will generalize GreubReinboldt Inequality to operators on infinite dimensional separable Hilbert spaces.

Keywords:Greub-Reinboldt Inequality, Two Nonzero Component Lemma

1. Introduction

Many authors have established Kantorovich inequality and its generalizations such as Greub-Reinboldt Inequality by variational methods. In a variational approach, one differentiates the functional involved to arrive at an “Euler Equation” and then solves the Euler Equating to obtain the minimizing or maximizing vectors of the functional involved. Solving these Euler Equations is tedious and generally provides little information (see [1] , subsection 4.4 for an example of this method). Others have established Kantorovich-type inequalities for positive operators by going through a two-step process which consists of first computing upper bounds for suitable functions on intervals containing the spectrum of suitable matrix and then applying the standard operational calculus to that matrix (see [2] ) for an example of this method. This method, which we refer to as “the operational calculus method”, has the following two limitations:

First, it does not provide any information about vectors for which the established inequalities become equalities. Second, the operational calculus method is futile in extending Kantorovich-type inequalities to operators on an infinite dimensional Hilbert space. A number of Kantorovich-type inequalities are discussed in [3] .

In this paper we use the author’s Two Nonzero Component Lemma to prove, improve and extend matrix form of Greub-Reinboldt Inequality.

2. The Two Nonzero Component Lemma

In his investigation on problems of antieigenvalue theory the author has discovered a useful lemma which he calls it the Two Nonzero Component Lemma (see [4] -[6] ). Although this Lemma is implicitly used in all of the papers just cited, it was not until 2008 that he stated a formal description of the Lemma in his paper titled, “Antieigenvalue Techniques in Statistics”. Below is the statement of the lemma. For the proof of the lemma please see the author’s work cited above.

Lemma 1 (The Two Nonzero Component Lemma) Let be the set of all sequences with nonnegative terms in the Banach Space. That is, let

(1)

Let

(2)

be a function from to. Assume for, , and. Then the minimizing vectors for the function

(3)

on the convex set have at most two nonzero components.

What make the proof of the Lemma possible are the following two facts: First, the convexity of the set

(4)

Second, a special property that the functions

(5)

involved possess. If we set

(6)

then all restrictions of the form

(7)

of

(8)

have the same algebraic form as itself. For example if

(9)

then we have

(10)

which has the same algebraic form as

(11)

Indeed, for any,; all restrictions of the function

(12)

obtained by setting an arbitrary set of components of equal to zeros have the same algebraic form as. Obviously, not all functions have this property. For instance, for the function, , which does not have the same algebraic form as.

3. Greub-Reinboldt Inequality

Let and be two real n-tuples. Suppose that, , , are constants such that and. Then, for we have

(13)

A slightly different form of the above inequality was proved by J. W. S. Cassels in 1951 (see Appendix 1 of [7] ). In the following section we provide a proof for the matrix form of Greub-Reinboldt Inequality based on the Two Nonzero Component Lemma. The proof is completely different than the proofs given by others, including Greub and Reinboldt themselves (see [8] ). This proof has the advantage of providing information about when the inequality becomes equality and gives information about vectors which make the inequality equality. Furthermore, as we will discuss in the Section 5, our method will indeed extend the Greub-Reinboldt Inequality to operators on an infinite dimensional Hilbert space.

4. The Matrix Form of Greub-Reinboldt Inequality

Theorem 2 Let and be two commuting positive operators with eigenvalues and respectively. Assume

(14)

and

Also assume and are diagonalized with diagonal elements and respectively , then

(15)

for every vector. In this, case if is any unit vector which makes the inequality (15) an equality then we have

(16)

and

(17)

and

(18)

where is the projection of on the eigenspace corresponding to eigenvalue.

Proof. Without loss of generality we can assume. Consider the functional

(19)

(19) can be written as

(20)

The reciprocal of (20) is

(21)

The square root of (21) is

(22)

To prove (15) we first find

(23)

Since is invertible, by a change of variable we have

(24)

By the spectral mapping theorem the inf on the right hand side of (24) can be represented as

(25)

over the set

(26)

where is the set of eigenvalues of The fact that and commute implies that

(27)

for and

(28)

where

(29)

and

(30)

If we set the problem is reduced to finding

(31)

over

(32)

By the Two Nonzero Component Lemma we need to look at

(33)

over the convex set

(34)

for pairs of and Notice that since the expression in (33) is positive, for simplicity, we can first compute the infimum of the square of that expression on the convex set (34) and then take square root of the result Therefore, the problem is now reduced to finding

(35)

on (34). By substituting in (35) the problem is now reduced to finding

(36)

for. To find (36), simply differentiate the expression in (36) and set its derivative with respect to equal to zero (we omit the straight forward computations). The expression in (36) is minimized when

Substituting this value of in (34) and the expression in (36) gives us

(37)

and

(38)

Hence

(39)

and the inf in (39) is attained at

(40)

and

(41)

Assume that. Now we show that we must have and. To prove this we must show

(42)

for and Squaring both sides of (42) gives us

(43)

Thus instead of proving inequality (42) we can prove inequality (43). Let and It is obvious that

(44)

If we substitute and in (43) we get

(45)

which is equivalent to

(46)

Hence proving inequality (42) is reduced to proving inequality (46). To prove (46), note that based on (44)

(47)

Therefore, we must have

(48)

The inequality

(49)

can be proved the same way we just proved (42). Hence we have

(50)

The right side of 50 is simplified to

(51)

Thus (50) becomes

(52)

Finally (50) is equivalent to

(53)

5. Generalizing Greub-Reinboldt Inequality to Operators on a Separable Hilbert Space

There are many proofs for Greub-Reinboldt Inequality in the literature. A significant advantage of proving Greub-Reinboldt Inequality by The Two Nonzero Component Lemma is that we can generalize this inequality to the case of positive operators and on an separable infinite dimensional separable Hilbert space. This is because, as the statement of the Two Nonzero Component Lemma shows, this lemma is also when the functions infinite linear combinations of Thus we can replace finite summations in (25), (26), (31), (32) with infinite sums and the arguments made in this paper remain valid. However, in this case it seems difficult to the pinpoint the exact pair of and for which the projections and of minimizing unit vectors are nonzero.

Theorem 3 Let and be two commuting positive operators on a separable Hilbert space such that

and where and represent the spectrums of and respectively, then

(54)

for any vector. In this, case if is any unit vector which makes the inequality (54) an equality then there exist a pair of and such that

(55)

and

(56)

and

(57)

where is the projection of on the eigenspace corresponding to eigenvalue.

There are other generalizations of Greub-Reinboldt Inequality. For example in ([9] ) Gustafson extends this inequality to pair noncommuting positive matrices and. However, he replaces the standard norm of the Hilbert space with the norm relative to.

Conclusion 4 The Two Nonzero Component Lemma provides an effective way of proving the GreubReinboldt Inequality and extending it to positive operators on separable infinite dimensional Hilbert spaces. The author has also utilized this lemma to prove other Kantorovich-type inequalities. Please see ([4] -[6] [10] [11] ).

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