#### Abstract

We will extend the definition of antieigenvalue of an operator to antieigenvalue-type quantities, in the first section of this paper, in such a way that the relations between antieigenvalue-type quantities and their corresponding Kantorovich-type inequalities are analogous to those of antieigenvalue and Kantorovich inequality. In the second section, we approximate several antieigenvalue-type quantities for arbitrary accretive operators. Each antieigenvalue-type quantity is approximated in terms of the same quantity for normal matrices. In particular, we show that for an arbitrary accretive operator, each antieigenvalue-type quantity is the limit of the same quantity for a sequence of finite-dimensional normal matrices.

#### 1. Introduction

Since 1948, the Kantorovich and Kantorovich-type inequalities for positive bounded operators have had many applications in operator theory and other areas of mathematical sciences such as statistics. Let be a positive operator on a Hilbert space with , then the Kantorovich inequality asserts that for every unit vector (see [1]). When and are the smallest and the largest eigenvalues of , respectively, it can be easily verified that for every pair of nonnegative numbers , , with and . The expression is called the Kantorovich constant and is denoted by .

Given an operator on a Hilbert space , the antieigenvalue of , denoted by , is defined by Gustafson (see [2–5]) to be Definition (1.4) is equivalent to A unit vector for which the in (1.5) is attained is called an antieigenvector of . For a positive operator , we have Thus, for a positive operator, both the Kantorovich constant and are expressed in terms of the smallest and the largest eigenvalues. It turns out that the former can be obtained from the latter.

Matrix optimization problems analogues to (1.4), where the quantity to be optimized involves inner products and norms, frequently occur in statistics. For example, in the analysis of statistical efficiency one has to compute quantities such as where is a positive definite matrix and with . Each is a column vector of size , and . Here, denotes the identity matrix, and stands for the determinant of a matrix . Please see [6–12]. Notice that in the references just cited, the sup’s of the reciprocal of expressions involved in (1.8), (1.9), (1.10), and (1.11) are sought. Nevertheless, since the quantities involved are always positive, those sup’s are obtained by finding the reciprocals of the inf's found in (1.8), (1.9), (1.10) and (1.11), while the optimizing vectors remain the same. Note that in (1.7) through (1.11). one wishes to compute optimizing matrices for quantities involved, whereas in (1.5) the objective is to find optimizing vectors for the quantity involved. Also, please note that for any vector , we have , where is the matrix of rank one with and . Hence the optimizing vectors in (1.5) can also be considered as optimizing matrices for respective quantity. When we use the term “an antieigenvalue-type quantity” throughout this paper, we mean a real number obtained by computing the in expressions similar to those given previously. The terms “an antieigenvector-type ” or “an antieigenmatrix-type ” are used for a vector or a matrix for which the in the associated expression is attained. A large number of well-known operator inequalities for positive operators are indeed generalizations of Kantorovich inequality. The following inequality, called the Holder-McCarthy inequality is an example. Let be a positive operator on a Hilbert space satisfying . Also, let be a real valued convex function on and let be a real number, then the inequality, which holds for every unit vector under certain conditions, is called the Holder-McCarthy inequality (see [13, 14]). Many authors have established Kantorovich-type inequalities, such as (1.12), for a positive operator by going through a two-step process which consists of computing upper bounds for suitable functions on intervals containing the spectrum of and then applying the standard operational calculus to (see [14]). These methods have limitations as they do not shed light on vectors or matrices for which inequalities become equalities. Also, they cannot be used to extend these inequalities from positive matrices to normal matrices. To extend these kinds of inequalities from positive operators to other types of operators in our previous papers, we have developed a number of effective techniques which have been useful in discovering new results.

In particular, a technique which we have frequently used is the conversion of a matrix/operator optimization problem to a convex programming problem. In this approach the problem is reduced to finding the minimum of a convex or concave function on the numerical range of a matrix/operator. This technique is not only straight forward but also sheds light on the question of when Kantorovich-type inequalities become equalities. For example, the proof given in [1] for inequality (1.1) does not shed light on vectors for which the inequality becomes equality. Likewise, in [14], the methods used to prove a number of Kantorovich-type inequalities do not provide information about vectors for which the respective inequalities become equalities. From the results we obtained later (see [15–17]) it is now evident that, for a positive definite matrix , the equality in (1.1) holds for unit vectors that satisfy the following properties. Assume that is the set of distinct eigenvalues of such that and is the eigenspace corresponding to eigenvalue of . If is the orthogonal projection on and , then and if and . Furthermore, for such a unit vector , we have Please note that by a change of variable, (1.14) is equivalent to

Furthermore, in [15–17], we have applied convex optimization methods to extend Kantorovich-type inequalities and antieigenvalue-type quantities to other classes of operators. For instance, in [16], we proved that for an accretive normal matrix, antieigenvalue is expressed in terms of two identifiable eigenvalues.

This result was obtained by noticing the fact that where and denotes the numerical range of . Since is normal so is . Also, by the spectral mapping theorem, if denotes the spectrum of , then Hence, in [17], the problem of computing was reduced to the problem of finding the minimum of the convex function on the boundary of the convex set . It turns out that or where and are easily identifiable eigenvalues of . In [17], we called them the first and the second critical eigenvalues of , respectively. The corresponding quantities and are called the first and second critical eigenvalues of , respectively. Furthermore, the components of antieigenvectors satisfy and if and . An advantage to this technique is that we were able to inductively define and compute higher antieigenvalues and their corresponding higher antieigenvectors for accretive normal matrices (see [17]). This technique can also be used to approximate antieigenvalue-type quantities for bounded arbitrary bounded accretive operators as we will show in next section.

#### 2. Approximations of Antieigenvalue-Type Quantities

If is not a finite dimensional normal matrix, (1.16) is still valid, but is not a polygon any more. Thus, we cannot use our methods discussed in previous section for an arbitrary bounded accretive operator . In this section, we will develop methods for approximating antieigenvalue and antieigenvalue-type quantities for an arbitrary bounded accretive operator by counterpart quantities for finite dimensional matrices. Computing an antieigenvalue-type quantity for an operator is reduced to computing the minimum of a convex or concave function on , the boundary of the numerical range of another operator . To make such approximations, first we will approximate with polygons from inside and outside. Then, we use techniques developed in [17] to compute the minimum of the convex or concave functions on the polygons inside and outside .

Theorem 2.1. *Assume that is a convex or concave function on , the numerical range of an operator . Then, for each positive integer , the real part of the rotations of induces polygons contained in and polygons which contain such that
** and denote the boundaries of and , respectively. *

*Proof. *Following the notations in [18], let , and is the largest eigenvalue of the positive operator . If is a unit eigenvector associated with , then the complex number which is denoted by belongs to . Furthermore, the parametric equation of the line of support of at is
Let denote a set of “mesh” points , where . Let , , then the polygon whose vertices are is contained in . This polygon is denoted by in [18], but we denote it by in this paper for simplicity in notations. Let be the intersection of the lines
and
which are the lines of support of at points and , respectively, where is identified with . Then we have
where . The polygon whose vertices are , contains . This polygon is denoted by in [18], but we denote it by here. Hence, for each , we have
Please see Figure 1.

Therefore, As a measure of the approximation given by (2.1) and (2.2) we will adopt a normalized difference between the values: that is Once the vertices of and are determined, are computable by methods we used in [17], where was a convex polygon. For the convex or concave functions arising in antieigenvalue-type problems, the minimums on and will occur either at the upper-left or upper-right portion of and . As our detailed analysis in [17] shows, the minimum of the convex functions whose level cures appear on the left side of is attained at either the first or the second critical vertex of or on the line segment connecting these two vertices. The same can be said about the minimum of such functions on . In Figure 1 above, and are the first and second critical vertices of , respectively. Similarly, and are the first and the second critical vertices of , respectively. In [17], an algebraic algorithm for determining the first and the second critical vertices of a polygon is developed based on the slopes of lines connecting vertices of a polygon. This eliminates the need for computing the values of the function at all vertices of and . It also eliminates the need for computing and comparing the minimums of on all edges of and . Thus, to compute the minimum of on , for example, we only need to evaluate for the components of the first and the second critical vertices and use Lagrange multipliers to compute the minimum of on the line segment connecting these two vertices.

*Example 2.2. *The Holder-McCarthy inequality for positive operators given by (1.12) can be also written as
Therefore, for a positive operator, one can define a new antieigenvalue-type quantity by
If we can compute the minimizing unit vectors for (2.15), then obviously for these vectors (1.12) becomes equality. is the antieigenvalue-type quantity associated with the Holder-McCarthy inequality, which is a Kantorovich-type inequality. The minimizing unit vectors for (2.15) are antieigenvector-type vectors associated with . It is easily seen that the standard antieigenvalue is a special case of this antieigenvalue-type quantity.

*Example 2.3. *There are a number of ways that we can extend the definition of to an arbitrary operator where is an analytic function. One way is to extend the definition of by
where is a complex-valued analytic function defined on the spectrum of . The problem then becomes
where . For an arbitrary operator , the set is not in general a polygon but a bounded convex subset of the complex plane. Nevertheless, we can approximate with polygons from inside and outside and thus obtain an approximation for by looking at the minimum of the function on those inside and outside polygons.

*Example 2.4. *In [19], in the study of statistical efficiency, we computed the value of a number of antieigenvalue-type quantities. Each antieigenvalue-type quantity there is itself the product of several simpler antieigenvalue-type quantities [23, Theorem 3]. One example is
To compute the first of these simpler antieigenvalue-type quantities, one has
To find this quantity, we converted the problem to finding the minimum of the function on the convex set , where . If is not a positive operator on a finite dimensional space, is not a polygon. We can, however, approximate with polygons from inside and outside and thus obtain an approximation for
an antieigenvalue-type quantity, by looking at the minimum of the function on those inside and outside polygons. We can compute other simpler antieigenvalue-type quantities involved in the previous product by same way.

Theorem 2.5. *For any bounded accretive operator , there is a sequence of finite-dimensional normal matrices such that
*

*Proof. *Recall that for any operator , we have
where
Using the notations in Theorem 2.1, for each , there is an accretive normal operator with . We can define to be the diagonal matrix whose eigenvalues are . By the spectral mapping theorem, there exists an accretive normal matrix such that
To see this, let the complex representation of the vertex , , of be
then can be taken to be the diagonal matrix whose eigenvalues are
Note that since is on the boundary of , we have . Since we have
we have
However,
This implies,
Since is positive a is positive for each , we have

Theorem 2.6. * For any bounded accretive operator there is a sequence of normal matrices such that
**
for each and
*

*Proof. *Recall that for any operator we have
where
Using the notations in Theorem 2.1, for each there is an accretive normal matrix with . We can define to be the diagonal matrix whose eigenvalues are . By the spectral mapping theorem, there exists a normal matrix such that
To see this, let the complex representation of the vertex , , of be
then by the spectral mapping theorem can be taken to be any diagonal matrix with eigenvalues where are any solution to the system
Since we have

we have However, This implies that Since is positive a is positive for each , we have

Theorem 2.7. *For any bounded accretive operator there is a sequence of finitedimensional normal matrices such that
*

*Proof. *Recall that for any operator we have
where
Using the notations in Theorem 2.1, for each there is a normal operator with . We can define to be the diagonal matrix whose eigenvalues are . By the spectral mapping theorem there exist a normal matrix such that
To see this, let the complex representation of the vertex , , of be
Take to be any diagonal matrix whose eigenvalues satisfy
To fund the eigenvalues of such a finite-dimensional diagonal matrix explicitly, we solve the previous equation for . The solutions are
or

Since we have
we have
However,
This implies that
Since is positive a is positive for each , we have

In the proofs of Theorems 2.5 through 2.7 previous, we considered accretive normal matrices whose spectrum are vertices of , . We could also consider matrices whose spectrums are , . However, notice that those matrices may not be accretive for small values of .

The term antieigenvalue was initially defined by Gustafson for accretive operators. For an accretive operator the quantity is nonnegative. However, in some of our previous work we have computed for normal operators or matrices which are not necessarily accretive (see [15, 16, 20, 21]). In Theorems 2.5 through 2.7 above we assumed is bounded accretive to ensure that in these theorems is a subset of the first quadrant. Thus, for each , in Theorems 2.5 through 2.7 is a finite polygon in the first quadrant, making it possible to compute in terms of the first and the second critical eigenvalues (see (1.18)). If is not accretive in Theorems 2.5 through 2.7, then we can only say that is a subset of the upper-half plane which implies for each , we can only say is a bounded polygon in the upper-half plane. This is despite the fact that (2.22), (2.34), and (2.45) in the proofs of Theorems 2.5, 2.6, and 2.7, respectively, are still valid. Therefore, Theorems 2.5 through 2.7 are valid if the operator in these theorems is not accretive. The only challenge in this case is computing , for each , in these theorems. What we know from our previous work in [16] is that can be expressed in terms of one or a pair of eigenvalues of . However, we do not know which eigenvalue or which pair of eigenvalues of expresses . Of course, one can use Theorem 2.2 of [16] to compute , however, this requires a lot of computations, particularly for large values of .

*Example 2.8. *Consider a normal matrix whose eigenvalues are , , , , , and . For this matrix, we have . If , then is the polygon whose vertices are , , , , , and . This polygon is not a subset of the first quadrant. Therefore, , not readily found using the first and second critical eigenvalues. Using Theorem 2.2 of [16], we need to perform a set of computations and compare the values obtained to find .

In [20, 22] the concepts of slant antieigenvalues and symmetric antieigenvalues were introduced. These antieigenvalue-type quantities can also be approximated by their counterparts for normal matrices. However, since slant antieigenvalues or symmetric antieigenvalues are reduced to regular antieigenvalue of another operator (see [20]), we do not need to develop separate approximations for these two antieigenvalue-type quantities. If is simple enough, we can use more elementary methods to find the minimum of on without approximating it with polygons. For example, in [21], we computed by direct applications of the Lagrange multipliers when ) is just an ellipse. Also, in [23] we used Lagrange multipliers directly to compute , when is a matrix of low dimension on the real field.