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We will extend some of the Kantorovich-Type inequalities for positive finite dimensional matrices to infinite dimensional normal operators by applying The Two-Nonzero Component Lemma and converting them to an An-tieigenvalue-Type problem.

Kantorovich-Type inequalities for positive matrices on finite dimensional Hilbert Spaces are extensions of the original Kantorovich inequality. Let T be a positive matrix with the smallest eigenvalue m and the largest eigenvalue M, then the original Kantorovich inequality states that

( T − l x , x ) ( T x , x ) ≤ ( m + M ) 2 4 m M for all x ∈ H , ‖ x ‖ = 1. (1)

In fact

sup ( T x , x ) ( T − 1 x , x ) = ( M + m ) 2 4 m M for all x ∈ H , ‖ x ‖ = 1. (2)

Please see [

inf ‖ x ‖ = 1 ( T x , x ) ‖ T x ‖ = 2 m M m + M . (3)

Please see [

inf ‖ x ‖ = 1 ( T x , x ) ‖ T x ‖ (4)

the antieigenvalue of T. Gustafson’s proof of (3) was based on his min-max theorem which states that for such a positive operator we have

inf ∈ > 0 ‖ ∈ T − I ‖ = M − m M + m . (5)

At the same time, Gustafson defined the antieigenvalue of an arbitrary operator T to be

inf ‖ x ‖ = 1 Re ( T x , x ) ‖ T x ‖ . (6)

Thus computing (6) for an arbitrary operator T is equivalent to extend the original Kantorovich inequality to an arbitrary operators T. The first attempt to compute a value for (6) was made by Davis in [

Lemma 1 (The Two Nonzero Component Lemma) Let l 1 + be the set of all sequences with nonnegative terms in the Banach Space l 1 , i.e.,

l 1 + = { t = ( t i ) ∈ l 1 , t i ≥ 0 } . (7)

Let F ( x 1 , x 2 , ⋯ , x m ) be a function from R^{m} to R. Assume

g k ( t ) = ∑ c i k t i for ( c i k ) ∈ l 1 + , t ∈ l 1 + ,1 ≤ k ≤ m . (8)

Then the minimizing vectors for the functional

F ( g 1 ( t ) , g 2 ( t ) , ⋯ , g m ( t ) ) (9)

on the convex set

C = { ( t i ) ∈ l 1 + : ∑ t i = 1 } (10)

have at most two nonzero components.

A geometric proof for this lemma in the finite dimensional case is implicit in the proof of Theorem 5.1 in [

F ( g 1 , g 2 ) = g 1 g 2 , (11)

g 1 ( t ) = β 1 t 1 + β 2 t 2 + β 3 t 3 , (12)

and

g 2 ( t ) = | λ 1 | 2 t 1 + | λ 2 | 2 t 2 + | λ 3 | 2 t 3 . (13)

Here we have replaced v 1 , v 2 , v 3 in Theorem 5.1 of [

F ( x 1 , x 2 ) = x 1 x 2 , (14)

with

g 1 ( t ) = ∑ β i t , (15)

and

g 2 ( t ) = ∑ | λ i | 2 t i . (16)

Please note there is a harmless error in expression 2.18 in [

C = { ( t i ) ∈ l 1 + : ∑ t i = 1 } . (17)

Second, a special property that the functions

F ( g 1 ( t ) , g 2 ( t ) ) , (18)

involved possess. If we set

D ( t 1 , t 2 , t 3 , ⋯ ) = F ( g 1 ( t ) , g 2 ( t ) ) , (19)

then all restrictions of the form

D ( t 1 , t 2 , ⋯ , t i − 1 ,0, t i + 1 , ⋯ ) (20)

of

D ( t 1 , t 2 , t 3 , ⋯ ) , (21)

have the same algebraic form as

D ( t 1 , t 2 , t 3 , ⋯ ) (22)

itself. For example, if

D ( t 1 , t 2 , ⋯ , t n ) = β 1 t 1 + β 2 t 2 + ⋯ + β n t n | λ 1 | 2 t 1 + | λ 2 | 2 t 2 + ⋯ + | λ 3 | 2 t 3 , (23)

(this is the function appearing in the proof of Theorem 2.2 in [

D ( 0, t 2 , ⋯ , t n ) = β 2 t 2 + ⋯ + β n t n | λ 2 | 2 t 2 + ⋯ + | λ 3 | 2 t 3 , (24)

which has the same algebraic form as

D ( t 1 , t 2 , ⋯ , t n ) = β 1 t 1 + β 2 t 2 + ⋯ + β n t n | λ 1 | 2 t 1 + | λ 2 | 2 t 2 + ⋯ + | λ 3 | 2 t 3 . (25)

Indeed, for any j, 1 ≤ j < n ; all restrictions of the function

D ( t 1 , t 2 , ⋯ , t n ) = β 1 t 1 + β 2 t 2 + ⋯ + β n t n | λ 1 | 2 t 1 + | λ 2 | 2 t 2 + ⋯ + | λ 3 | 2 t 3 , (26)

obtained by setting an arbitrary set of j components of

D ( t 1 , t 2 , ⋯ , t n ) (27)

equal to zeros have the same algebraic form as

D ( t 1 , t 2 , ⋯ , t n ) . (28)

Obviously, not all functions have this property. For instance, for the function

G ( t 1 , t 2 ) = 2 t 1 + t 1 t 2 , (29)

we have

G ( t 1 ,0 ) = t 1 , (30)

which does not have the same algebraic form as

G ( t 1 , t 2 ) . (31)

To avoid repetitions in our papers, we will not present a separate proof for Lemma 1 here. Instead, we note that the proof of this Lemma is embedded in the proof of Theorem 2.2 of [

The Two Nonzero Component Lemma was formulated as above by Seddighin in [

Let T be a positive matrix on a finite dimensional space satisfying M ≥ A ≥ m > 0 . Also let F ( t ) be a real valued convex function on [ m , M ] and q be a real number, then the inequality

( F ( T ) x , x ) ≤ ( m f ( M ) − M f ( m ) ) ( q − 1 ) ( M − m ) ( ( q − 1 ) ( f ( M ) − f ( m ) ) q ( m f ( M ) − M f ( m ) ) ) q ( T x , x ) q (32)

holds for every unit vector x under one of the following conditions

F ( M ) > F ( m ) , F ( M ) M > F ( m ) m , F ( m ) m q ≤ F ( M ) − F ( m ) M − m ≤ F ( M ) M q , (33)

or

F ( M ) < F ( m ) , F ( M ) M < F ( m ) m , F ( m ) m q ≤ F ( M ) − F ( m ) M − m ≤ F ( M ) M q . (34)

The Inequality (32) is a nontrivial Kantorovich-Type inequality which is a generalization of the original Kantorovich inequality. In this paper we call (32) the (q,F) Kantorovich-Type inequality. Please see [

( T x , x ) q ( F ( T ) x , x ) ≥ ( q − 1 ) ( M − m ) ( m f ( M ) − M f ( m ) ) ( q ( m f ( M ) − M f ( m ) ) ( q − 1 ) ( f ( M ) − f ( m ) ) ) q , (35)

under the conditions stated above. Therefore, the inequality is established if we show

inf ‖ x ‖ = 1 ( T x , x ) q ( F ( T ) x , x ) = ( q − 1 ) ( M − m ) ( m f ( M ) − M f ( m ) ) ( q ( m f ( M ) − M f ( m ) ) ( q − 1 ) ( f ( M ) − f ( m ) ) ) q . (36)

The quantity inf ‖ x ‖ = 1 ( T x , x ) q ( F ( T ) x , x ) resembles inf ‖ x ‖ = 1 ( T x , x ) ‖ T x ‖ and in accordance to

Antieigenvalue Theory we call it ( q , F ) antieigenvalue of T and denote it by μ ( q , F ) . In general in accordance to Antieigenvalue Theory, if T is a normal

operator, we call inf ‖ x ‖ = 1 Re ( T x , x ) q ( F ( T ) x , x ) the ( q , F ) Antieigenvalue of T and denote it by μ ( q , F ) .

The following is a generalization of ( q , F ) Kantorovich-Type inequality to normal Hilbert space operators.

Theorem 2 Let T be a normal operator on a separable Hilbert space. Suppose λ i = β i + δ i i , 1 ≤ i < ∞ , are the eigenvalues of T. Let E ( λ i ) be the eigenspace corresponding to λ i and let P ( λ i ) be the orthogonal projection on E ( λ i ) . Assume F is an analytic function defined on σ ( T ) . For each vector x let

z i = P ( λ i ) x . If x is a minimizing vector with ‖ x ‖ = 1 for Re ( T x , x ) q ( F ( T ) x , x ) , then

we have one of the following cases: 1) Only one of the vectors z i is nonzero, i.e., ‖ z i ‖ = 1 , for some i, and ‖ z j ‖ = 0 for j ≠ i . In this case we have

μ ( q , F ) = β i q | F ( λ i ) | . (37)

2) Only two of the vectors z i and z j are nonzero and the rest of the components of f are zero. i.e., ‖ z i ‖ ≠ 0 , ‖ z j ‖ ≠ 0 and ‖ z k ‖ = 0 if k ≠ i and k ≠ j . In this case we have

‖ z i ‖ 2 = β j ( | F ( λ i ) | − | F ( λ j ) | ) + q | F ( λ i ) | ( β j − β i ) ( q − 1 ) ( | F ( λ i ) | − | F ( λ j ) | ) ( β i − β j ) , (38)

and

‖ z j ‖ 2 = β i ( | F ( λ j ) | − | F ( λ i ) | ) + q | F ( λ j ) | ( β i − β j ) ( q − 1 ) ( | F ( λ i ) | − | F ( λ j ) | ) ( β i − β j ) . (39)

Furthermore,

μ ( q , F ) = ( q − 1 ) ( β j − β i ) ( β i | F ( λ j ) | − β j | F ( λ i ) | ) ( q ( β i | F ( λ j ) | − β j | F ( λ i ) | ) ( q − 1 ) ( | F ( λ j ) | − | F ( λ i ) | ) ) q . (40)

Proof. Direct computations show that

μ ( q , F ) = i n f ( ∑ i = 1 ∞ β i ‖ z i ‖ 2 ) q ( ∑ i = 1 ∞ | F ( λ i ) | ‖ z i ‖ 2 ) . (41)

Let t i = ‖ z i ‖ 2 . Then the problem is reduced to finding

μ ( q , F ) = i n f ( ∑ i = 1 ∞ β i t i ) q ( ∑ i = 1 ∞ | F ( λ i ) | t i ) , (42)

on the convex set

C = { ( t i ) ∈ l 1 + : ∑ t i = 1 } . (43)

Now by the Two Nonzero Component Lemma, a minimizing vector t for

( ∑ i = 1 ∞ β i t i ) q ( ∑ i = 1 ∞ | F ( λ i ) | t i ) (44)

has either one or two nonzero components. First, if for a minimizing vector t we have t i ≠ 0 and t j = 0 , i ≠ j then

μ ( q , F ) = β i q | F ( λ i ) | . (45)

Second, if a minimizing vector t for

( ∑ i = 1 ∞ β i t i ) q ( ∑ i = 1 ∞ | F ( λ i ) | t i ) (46)

has two nonzero components t i and t j then the problem is reduced to finding the minimum of the function

( β i t i + β j t j ) q | F ( λ i ) | t i + | F ( λ j ) | t j , (47)

on the line segment

t i + t j = 1 , 0 ≤ t i ≤ 1 , 0 ≤ t i ≤ 1. (48)

An application of Lagrange Multipliers shows that we must have

t i = β j ( | F ( λ i ) | − | F ( λ j ) | ) + q | F ( λ i ) | ( β j − β i ) ( q − 1 ) ( | F ( λ i ) | − | F ( λ j ) | ) ( β i − β j ) , (49)

and

t j = β i ( | F ( λ j ) | − | F ( λ i ) | ) + q | F ( λ j ) | ( β i − β j ) ( q − 1 ) ( | F ( λ i ) | − | F ( λ j ) | ) ( β i − β j ) . (50)

If we substitute (49) and (50) in (47) and simplify, we obtain

( q − 1 ) ( β j − β i ) ( β i | F ( λ j ) | − β j | F ( λ i ) | ) ( q ( β i | F ( λ j ) | − β j | F ( λ i ) | ) ( q − 1 ) ( | F ( λ j ) | − | F ( λ i ) | ) ) q . (51)

The following corollary states the (q,F) Kantorovich-Type inequality for normal operators on a separable Hilbert space, without mentioning the

minimizing vectors for Re ( T x , x ) q ( F ( T ) x , x ) (which as we saw make the inequality an

equality). Traditionally, some inequalities are written without stating when the inequality becomes equality. The reason is that, as we explain later in this paper, they were driven by other methods without computing the vectors that make the inequality an equality. However, as we remark at the end of this paper, vectors which make an inequality equality have applications of their own.

Corollary 3 Let T be a normal operator on a separable Hilbert space. Suppose λ i = β i + δ i i , 1 ≤ i < ∞ , are the eigenvalues of T, F is an analytic function defined on σ ( T ) , and q is a real number. Then one of the following inequalities is satisfied: 1) There exist an eigenvalue λ i = β i + δ i i such that

( F ( T ) x , x ) ≤ | F ( λ i ) | β i q ( Re ( T x , x ) ) q , (52)

for all unit vectors x. 2) There exist a pair of eigenvalues λ i = β i + δ i i and λ j = β j + δ j i such that

( F ( T ) x , x ) ≤ ( β i | F ( λ j ) | − β j | F ( λ i ) | ) ( q − 1 ) ( β j − β i ) ) ( ( q − 1 ) ( | F ( λ j ) | − | F ( λ i ) | ) q ( β i | F ( λ j ) | − β j | F ( λ i ) | ) ) q ( Re ( T x , x ) ) q , (53)

for all unit vectors x.

In [

μ w ( T ) = inf ‖ x ‖ = 1 a Re ( T x , x ) + b Im ( T x , x ) a 2 + b 2 ‖ T x ‖ ( ‖ x ‖ ) , (54)

where a and b are a pair of real numbers, at least one of them nonzero. In particular when a = b = 1 we have

μ s = inf ‖ x ‖ = 1 Re ( T x , x ) + Im ( T x , x ) 2 ‖ T x ‖ ( ‖ x ‖ ) , (55)

which is called the symmetric antieigenvalue of T and is denoted by μ s . The symmetric antieigenvalue is a balanced definition of antieigenvalue because it depends on both Re T and Im T by the same factor. Note that, as we proved in [

μ w ( T ) = μ ( A ) , (56)

where

A = ( a − i b ) T . (57)

If we define weighted ( q , F ) antieigenvalue of T by

μ ( q , F ) w ( T ) = a Re ( T x , x ) + b Im ( T x , x ) a 2 + b 2 ( F ( T ) x , x ) , (58)

then we have,

Theorem 4 For any normal operator T we have

μ ( q , F ) w ( T ) = μ ( q , F ) ( A ) , (59)

where

A = ( a − i b ) T . (60)

Proof. Using spectral mapping theorem we have

Re A = Re ( ( a − i b ) T ) = Re ( ( a − i b ) T ) = a Re ( T x , x ) + b Im ( T x , x ) , (61)

and

F ( A ) = F ( ( a − i b ) T ) = | a − i b | F ( T ) = a 2 + b 2 F ( T ) . (62)

Theorem 5 Let T be a normal operator on a separable Hilbert space. Suppose λ i = β i + δ i i , 1 ≤ i < ∞ , are the eigenvalues of T. Let E ( λ i ) be the eigenspace corresponding to λ i and let P ( λ i ) be the orthogonal projection on E ( λ i ) . Assume F is an analytic function defined on σ ( T ) and q is a real number. Furthermore Let a and b be real numbers, at least one of them nonzero. For each vector x let z i = P ( λ i ) x . If x is a minimizing vector with ‖ x ‖ = 1 for

μ ( q , F ) w ( T ) = a Re ( T x , x ) + b Im ( T x , x ) a 2 + b 2 ( F ( T ) x , x ) ,

then we have one of the following cases: 1) Only one of the vectors z i is nonzero. i.e., ‖ z i ‖ = 1 , for some i, and ‖ z j ‖ = 0 for j ≠ i . In this case we have:

μ ( q , F ) w = ( a β i + b δ i ) q a 2 + b 2 | F ( λ i ) | . (63)

2) Only two of the vectors z i and z j are nonzero and the rest of the components of f are zero. i.e., ‖ z i ‖ ≠ 0 , ‖ z j ‖ ≠ 0 and ‖ z k ‖ = 0 if k ≠ i and k ≠ j . In this case we have

‖ z i ‖ 2 = ( a β i + b δ i ) ( | F ( λ i ) | − | F ( λ j ) | ) + q | F ( λ i ) | ( a β i + b δ i − a β j − b δ j ) ( q − 1 ) ( | F ( λ i ) | − | F ( λ j ) | ) ( a β i + b δ i − a β j + b δ j ) , (64)

and

‖ z j ‖ 2 = β i ( | F ( λ j ) | − | F ( λ i ) | ) + q | F ( λ j ) | ( β i − β j ) ( q − 1 ) ( | F ( λ i ) | − | F ( λ j ) | ) ( a β i + b δ i − a β j + b δ j ) . (65)

Furthermore

μ ( q , F ) w = ( ( a β i + b δ i ) | F ( λ i ) | − ( a β j + b δ j ) | F ( λ i ) | ) ( q − 1 ) ( a β i + b δ i − a β j − b δ j ) ( ( q − 1 ) ( | F ( λ j ) | − | F ( λ i ) | ) q ( ϵ i | F ( λ j ) | − ϵ j | F ( λ i ) | ) ) q . (66)

Proof. Let

= ( a − i b ) T , (67)

and let γ i = ϵ i + θ i , 1 ≤ i < ∞ , be the set of antieigenvalues of A. By the spectral mapping theorem, we have

( a − i b ) ( β i + i δ i ) = ( a β i + b δ i ) + i ( a δ i − b β i ) . (68)

By Theorem 2 we have one of the following two cases. 1) Only one of the vectors z i is nonzero. i.e., ‖ z i ‖ = 1 , for some i, and ‖ z j ‖ = 0 for j ≠ i . In this case we have:

μ ( q , F ) ( A ) = ϵ i i q | F ( γ i ) | . (69)

2) Only two of the vectors z i and z j are nonzero and the rest of the components of f are zero. i.e., ‖ z i ‖ ≠ 0 , ‖ z j ‖ ≠ 0 and ‖ z k ‖ = 0 if k ≠ i and k ≠ j . In this case we have

‖ z i ‖ 2 = ϵ j ( | F ( γ i ) | − | F ( γ j ) | ) + q | F ( γ j ) | ( ϵ j − ϵ i ) ( q − 1 ) ( | F ( γ i ) | − | F ( γ j ) | ) ( ϵ i − ϵ j ) , (70)

and

‖ z j ‖ 2 = β i ( | F ( γ j ) | − | F ( γ i ) | ) + q | F ( γ j ) | ( ϵ i − ϵ j ) ( q − 1 ) ( | F ( γ i ) | − | F ( γ j ) | ) ( ϵ i − ϵ j ) . (71)

Furthermore,

μ ( q , F ) ( A ) = ( q − 1 ) ( ϵ j − ϵ i ) ( ϵ i | F ( γ j ) | − ϵ j | F ( γ i ) | ) ( q ( ϵ i | F ( γ j ) | − ϵ j | F ( γ i ) | ) ( q − 1 ) ( | F ( γ j ) | − | F ( γ i ) | ) ) q . (72)

By Theorem 4 we have

μ ( q , F ) w ( T ) = μ ( q , F ) ( A ) . (73)

The proof is completed by substituting A in terms of T and eigenvalues of A in terms of eigenvalues of T in (69), (70), (71), and (72).

Corollary 6 Let T be a normal operator on a separable Hilbert space. Suppose λ i = β i + δ i i , 1 ≤ i < ∞ , are the eigenvalues of T, F is an analytic function defined on σ ( T ) , and q is a real number. Also assume a and b are real numbers, at least one of them nonzero. Then one of the following inequalities is satisfied, 1) There exist an eigenvalue λ i = β i + δ i i such that

( F ( T ) x , x ) ≤ a 2 + b 2 | F ( λ i ) | ( a β i + b δ i ) q ( a Re ( T x , x ) + b Im ( T x , x ) ) q , (74)

for all unit vectors x. 2) There exist a pair of eigenvalues λ i = β i + δ i i and λ j = β j + δ j i such that for all unit vectors x,

( F ( T ) x , x ) ≤ C D ( a Re ( T x , x ) + b Im ( T x , x ) ) , (75)

where C and D are defined by

C = ( ( a β i + b δ i ) | F ( λ i ) | − ( a β j + b δ j ) | F ( λ i ) | ) ( q − 1 ) ( a β i + b δ i − a β j − b δ j ) (76)

and

D = ( ( q − 1 ) ( | F ( λ j ) | − | F ( λ i ) | ) q ( ϵ i | F ( λ j ) | − ϵ j | F ( λ i ) | ) ) q . (77)

Remark 7 In this paper and in some of our other papers we have proved Kantorovich-Type inequalities by converting them to an Antieigenvalue-Type problem and then finding the minimizing vectors for the Antieigenvalue-Type problems. These vectors are the vectors that Kantorovich-Type inequalities become equalities. Traditionally authors have established Kantorovich-Type inequalities for a positive operator T by going through a two-step process which consists of computing upper bounds for suitable functions on intervals containing the spectrum of T and then applying the standard operational calculus to T (see [

Remark 8 While we have used TNCL to show that one or two eigenvalues are involved in expressing the (q,F) Kantorovich Type inequality for normal operators acting on a separable Hilbert space, TNCL does not enable us to pinpoint one or two eigenvalues involved. However, if T is a normal matrix it is possible to pinpoint exactly which eigenvalues express Antieigenvalue-Type quantities. We have shown this in [

Remark 9 If T is an arbitrary operator on an infinite dimensional Hilbert space we can numerically approximate its Antieigenvalue-Type quantities for T and hence establish approximate Kantorovich-Type inequalities for T (please see [

Conclusion 10 The results in this paper are milestones in the evolution of Kantorovich-Type inequalities. The simplest Kantorovich-Type inequality is the Kantorovich inequality for real numbers which states

( ∑ i = 1 n λ i x i ) ( ∑ i = 1 n λ i x i − 1 ) ≤ ( x 1 + x n ) 2 4 x 1 x n , (78)

where

0 < x 1 < x 2 < ⋯ < x n , (79)

and λ 1 , λ 2 , ⋯ , λ n are non-negative numbers with

∑ i = 1 n λ i = 1. (80)

The Inequality (1) was the first generalization of (78) to positive matrices by Kantorovich himself. In 1980, C. Davis extended the Kantorovich inequality from positive matrices to normal matrices. However, Davis assumed that the numerical range of the normal matrix is contained in the right half plain (i.e., the matrix is accretive). Furthermore, Davis was not able to identify the vectors for which the inequality becomes equality. Our results here are major steps in generalizing Kantorovich-Type inequalities for the following reasons: 1) We generalized Kantorovich-Type inequalities from positive matrices to infinite dimensional normal operators. 2) There are no conditions on the numerical range of normal operators. 3) We shed light on the vectors for which Kantorovich-Type inequalities become equalities.

Seddighin, M. (2018) Extending Kantorovich-Type Inequalities to Normal Operators. Advances in Linear Algebra & Matrix Theory, 8, 41-52. https://doi.org/10.4236/alamt.2018.81005