Approximations of Antieigenvalue and Antieigenvalue-Type Quantities

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 𝑇 be a positive operator on a Hilbert space 𝐻 with 𝑚𝐼≤𝑇≤𝐼𝑀, then the Kantorovich inequality asserts that 𝑇𝑓,𝑓𝑇⁻¹≤()𝑓,𝑓𝑚+𝑀²,4𝑚𝑀 (1.1) for every unit vector 𝑓 (see [1]). When 𝜆_𝑚 and 𝜆_𝑀 are the smallest and the largest eigenvalues of 𝑇, respectively, it can be easily verified that 𝜆_𝑚+𝜆_𝑀²4𝜆_𝑚𝜆_𝑀≤(𝑚+𝑀)²,4𝑚𝑀 (1.2) for every pair of nonnegative numbers 𝑚, 𝑀, with 𝑚≤𝜆_𝑚 and 𝑀≥𝜆_𝑀. The expression 𝜆_𝑚+𝜆_𝑀²4𝜆_𝑚𝜆_𝑀 (1.3) is called the Kantorovich constant and is denoted by 𝐾(𝑇).

Given an operator 𝑇 on a Hilbert space 𝐻, the antieigenvalue of 𝑇, denoted by 𝜇(𝑇), is defined by Gustafson (see [2–5]) to be 𝜇(𝑇)=inf_𝑇𝑓≠0Re𝑇𝑓,𝑓𝑓.𝑇𝑓 (1.4) Definition (1.4) is equivalent to 𝜇(𝑇)=inf_{𝑇𝑓≠0𝑓=1}Re𝑇𝑓,𝑓.𝑇𝑓 (1.5) A unit vector 𝑓 for which the inf in (1.5) is attained is called an antieigenvector of 𝑇. For a positive operator 𝑇, we have 𝜇(𝑇)=2𝜆_𝑚𝜆_𝑀𝜆_𝑚+𝜆_𝑀. (1.6) Thus, for a positive operator, both the Kantorovich constant and 𝜇(𝑇) 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 inf_{𝐗′𝐗=𝟏𝑝}1𝑋^′𝑇𝑋𝑋^′𝑇⁻¹𝑋,inf_{𝑋′𝑋=1𝑝}𝑋^′𝑇𝑋²𝑋^′𝑇²𝑋−𝑋^′𝑇𝑋²,inf_{𝑋′𝑋=1𝑝}𝑋^′𝑇⁻¹𝑋⁻¹𝑋^′𝑇𝑋−𝑋^′𝑇⁻¹𝑋⁻¹,inf_{𝑋′𝑋=1𝑝}1𝑋^′𝑇²𝑋−𝑋^′𝑇𝑋²,inf_{𝑋′𝑋=1𝑝}1𝑋^′𝑇𝑋−𝑋^′𝑇⁻¹𝑋⁻¹, (1.7)(1.8)(1.9)(1.10)(1.11) where 𝑇 is a positive definite matrix and 𝐗=[𝐗₁,𝐗₂,…,𝐗_𝑝] with 2𝑝≤𝑛. Each 𝐗_𝑘 is a column vector of size 𝑛, and 𝐗^′𝐗=𝟏_𝑝. Here, 𝟏_𝑝 denotes the 𝑝×𝑝 identity matrix, and |𝑌| stands for the determinant of a matrix 𝑌. 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 𝑋 for quantities involved, whereas in (1.5) the objective is to find optimizing vectors 𝑓 for the quantity involved. Also, please note that for any vector 𝑓, we have (𝑇𝑓,𝑓)=𝑋^′𝑇𝑋, where 𝑋 is the matrix of rank one with 𝑋=[𝑋₁] and 𝑋₁=𝑓. 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 inf in expressions similar to those given previously. The terms “an antieigenvector-type 𝑓” or “an antieigenmatrix-type 𝑋” are used for a vector 𝑓 or a matrix 𝑋 for which the inf 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 𝑇 be a positive operator on a Hilbert space 𝐻 satisfying 𝑀≥𝐴≥𝑚>0. Also, let 𝐹(𝑡) be a real valued convex function on [𝑚,𝑀] and let 𝑞 be a real number, then the inequality, 𝐹(𝑇)≤(𝑀)(𝑚)𝑓,𝑓𝑚𝑓−𝑀𝑓()𝑓(𝑀)(𝑚)𝑞−1𝑀−𝑚𝑞−1−𝑓𝑞𝑚𝑓(𝑀)−𝑀𝑓(𝑚)^𝑞(𝑇𝑓,𝑓)^𝑞, (1.12) which holds for every unit vector 𝑓 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 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 [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 𝑇, the equality in (1.1) holds for unit vectors 𝑓 that satisfy the following properties. Assume that {𝜆_𝑖}^𝑛_𝑖=1 is the set of distinct eigenvalues of 𝑇 such that 𝜆₁<𝜆₂⋯<𝜆_𝑛 and 𝐸_𝑖=𝐸(𝜆_𝑖) is the eigenspace corresponding to eigenvalue 𝜆_𝑖 of 𝑇. If 𝑃_𝑖 is the orthogonal projection on 𝐸_𝑖 and 𝑧_𝑖=𝑃_𝑖𝑓, then 𝑧₁²=𝜆_𝑛𝜆₁+𝜆_𝑛,z_𝑛²=𝜆₁𝜆₁+𝜆_𝑛, (1.13) and ‖𝑧_𝑖‖=0 if 𝑖≠1 and 𝑖≠𝑛. Furthermore, for such a unit vector 𝑓, we have 𝑇𝑓,𝑓𝑇⁻¹=𝜆𝑓,𝑓¹+𝜆_𝑛²4𝜆₁𝜆_𝑛. (1.14) Please note that by a change of variable, (1.14) is equivalent to 𝑇𝑓,𝑓=2𝑇𝑓𝜆₁𝜆_𝑛𝜆₁+𝜆_𝑛. (1.15)

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 𝜇²(𝑇)𝑥=inf²𝑦∶𝑥+𝑖𝑦∈𝑊(𝑆), (1.16) where 𝑆=Re𝑇+𝑖𝑇^∗𝑇 and 𝑊(𝑆) denotes the numerical range of 𝑆. Since 𝑇 is normal so is 𝑆. Also, by the spectral mapping theorem, if 𝜎(𝑆) denotes the spectrum of 𝑆, then 𝜎(𝑆)=𝛽_𝑖+𝑖𝜆_𝑖²∶𝜆_𝑖=𝛽_𝑖+𝛿_𝑖𝑖𝜖𝜎(𝑇). (1.17) Hence, in [17], the problem of computing 𝜇²(𝑇) was reduced to the problem of finding the minimum of the convex function 𝑓(𝑥,𝑦)=𝑥²/𝑦 on the boundary of the convex set 𝑊(𝑆). It turns out that 𝜇(𝑇)=𝛽_𝑝/|𝜆_𝑝| or 𝜇(𝑇)=2𝛽_𝑞−𝛽_𝑝𝛽_𝑝𝜆_𝑞²−𝛽_𝑞𝜆_𝑝²𝜆_𝑞²−𝜆_𝑝², (1.18) where 𝛽_𝑝+𝑖|𝜆_𝑝|² and 𝛽_𝑞+𝑖|𝜆_𝑞|² are easily identifiable eigenvalues of 𝑆. In [17], we called them the first and the second critical eigenvalues of 𝑆, respectively. The corresponding quantities 𝜆_𝑝=𝛽_𝑝+𝛿_𝑝𝑖 and 𝜆_𝑞=𝛽_𝑞+𝛿_𝑞𝑖 are called the first and second critical eigenvalues of 𝑇, respectively. Furthermore, the components of antieigenvectors satisfy 𝑧_𝑝²=𝛽_𝑞𝜆_𝑞²−2𝛽_𝑝𝜆_𝑞²+𝛽_𝑞𝜆_𝑝²𝜆_𝑞²−𝜆_𝑝²𝛽_𝑞−𝛽_𝑝,𝑧_𝑞²=𝛽_𝑝𝜆_𝑝²−2𝛽_𝑞𝜆_𝑝²+𝛽_𝑝𝜆_𝑞²𝜆_𝑞²−𝜆_𝑝²𝛽_𝑞−𝛽_𝑝, (1.19) and ‖𝑧_𝑖‖=0 if 𝑖≠𝑝 and 𝑖≠𝑞. An advantage to this technique is that we were able to inductively define and compute higher antieigenvalues 𝜇_𝑖(𝑇) 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 𝑇 is not a finite dimensional normal matrix, (1.16) is still valid, but 𝑊(𝑆) is not a polygon any more. Thus, we cannot use our methods discussed in previous section for an arbitrary bounded accretive operator 𝑇. In this section, we will develop methods for approximating antieigenvalue and antieigenvalue-type quantities for an arbitrary bounded accretive operator 𝑇 by counterpart quantities for finite dimensional matrices. Computing an antieigenvalue-type quantity for an operator 𝑇 is reduced to computing the minimum of a convex or concave function 𝑓(𝑥,𝑦) on 𝜕𝑊(𝑆), the boundary of the numerical range of another operator 𝑆. To make such approximations, first we will approximate 𝑊(𝑆) with polygons from inside and outside. Then, we use techniques developed in [17] to compute the minimum of the convex or concave functions 𝑓(𝑥,𝑦) on the polygons inside and outside 𝑊(𝑆).

Theorem 2.1. Assume that 𝑓(𝑥,𝑦) is a convex or concave function on 𝑊(𝑆), the numerical range of an operator 𝑆. Then, for each positive integer 𝑘, the real part of the rotations 𝑒^𝑖𝜃𝑆 of 𝑆 induces polygons 𝐺_𝑘 contained in 𝑊(𝑆) and polygons 𝐻_𝑘 which contain 𝑊(𝑆) such that inf_{(𝑆)𝑥,𝑦∈𝜕𝑊}𝑓𝑥,𝑦≥inf_{𝑥,𝑦∈𝜕𝐻𝑘}𝑓𝑥,𝑦,inf_{(𝑆)𝑥,𝑦∈𝜕𝑊}𝑓𝑥,𝑦≤inf_{𝑥,𝑦∈𝜕𝐺𝑘}𝑓𝑥,𝑦,inf_{(𝑆)𝑥,𝑦∈𝜕𝑊}𝑓𝑥,𝑦=lim_𝑘⇒∞inf_{𝑥,𝑦∈𝜕𝐻𝑘}𝑓𝑥,𝑦,inf_{(𝑆)𝑥,𝑦∈𝜕𝑊}𝑓𝑥,𝑦=lim_𝑘⇒∞inf_{𝑥,𝑦∈𝜕𝐺𝑘}𝑓𝑥,𝑦 (2.1)(2.2)(2.3)(2.4)𝜕𝐺_𝑘 and 𝜕𝐻_𝑘 denote the boundaries of 𝐺_𝑘 and 𝐻_𝑘, respectively.

Proof. Following the notations in [18], let 0≤𝜃<2𝜋, and 𝜆_𝜃 is the largest eigenvalue of the positive operator Re(𝑒^𝑖𝜃𝑆). If 𝑓_𝜃 is a unit eigenvector associated with 𝜆_𝜃, then the complex number (𝑆𝑓_𝜃,𝑓_𝜃) which is denoted by 𝑝_𝜃 belongs to 𝜕𝑊(𝑆). Furthermore, the parametric equation of the line of support of 𝑊(𝑆) at (𝑆𝑓_𝜃,𝑓_𝜃) is 𝑒^−𝑖𝜃𝜆_𝜃+𝑡𝑖,0≤𝑡<∞. (2.5) Let Θ denote a set of “mesh” points Θ={𝜃₁,𝜃₂,…,𝜃_𝑘}, where 0≤𝜃₁<𝜃₂<⋯𝜃_𝑘<2𝜋. Let 𝑃₁=𝑃_𝜃1, 𝑃₂=𝑝_𝜃2,…,𝑃_𝑘=𝑝_𝜃𝑘, then the polygon whose vertices are 𝑃₁,𝑃₂,…,𝑃_𝑘 is contained in 𝑊(𝑆). This polygon is denoted by 𝑊_in(𝑆,Θ) in [18], but we denote it by 𝐺_𝑘 in this paper for simplicity in notations. Let 𝑄_𝑖 be the intersection of the lines 𝑒^{−𝑖𝜃𝑖}𝜆_𝜃𝑖+𝑡𝑖,0≤𝑡<∞, (2.6)𝛽 and 𝑒^{−𝑖𝜃𝑖+1}𝜆_𝜃𝑖+1+𝑡𝑖,0≤𝑡<∞, (2.7) which are the lines of support of 𝑊(𝑆) at points 𝑝_𝜃𝑖 and 𝑝_𝜃𝑖+1, respectively, where 𝑘+1 is identified with 1. Then we have 𝑄_𝑖=𝑒^{−𝑖𝜃𝑖}𝜆_𝜃𝑖+𝜆_𝜃𝑖cos𝛿_𝑖−𝜆_𝜃𝑖+1sin𝛿_𝑖𝑖, (2.8) where 𝛿_𝑖=𝜃_𝑖+1−𝜃_𝑖. The polygon whose vertices are 𝑄_𝑖, 1≤𝑖≤𝑘−1 contains 𝑊(𝑆). This polygon is denoted by 𝑊_out(𝑆,Θ) in [18], but we denote it by 𝐻_𝑘 here. Hence, for each 𝑘, we have 𝐺_𝑘⊆𝑊(𝑆)⊆𝐻_𝑘. (2.9) Please see Figure 1.

Figure 1 

Therefore, inf𝑓𝑥,𝑦≤inf_{(𝑆)𝑥,𝑦∈𝜕𝑊}𝑓𝑥,𝑦≤inf_{𝑥,𝑦∈𝜕𝐺𝑘}𝑓𝑥,𝑦. (2.10) As a measure of the approximation given by (2.1) and (2.2) we will adopt a normalized difference between the values: inf_{𝑥,𝑦∈𝜕𝐺𝑘}𝑓𝑥,𝑦,inf_{(𝑥,𝑦)∈𝜕𝐻𝑘}𝑓𝑥,𝑦, (2.11) that is Δ𝑓,𝑘=inf_{(𝑥,𝑦)∈𝜕𝐺𝑘}𝑓𝑥,𝑦−inf_{(𝑥,𝑦)∈𝜕𝐻𝑘}𝑓𝑥,𝑦inf_{(𝑥,𝑦)∈𝜕𝐺𝑘}𝑓𝑥,𝑦. (2.12) Once the vertices of 𝜕𝐺_𝑘 and 𝜕𝐻_𝑘 are determined, inf_{𝑥,𝑦∈𝜕𝐺𝑘}𝑓𝑥,𝑦,inf_{𝑥,𝑦∈𝜕𝐻𝑘}𝑓𝑥,𝑦 (2.13) are computable by methods we used in [17], where 𝑊(𝑆) was a convex polygon. For the convex or concave functions 𝑓(𝑥,𝑦) arising in antieigenvalue-type problems, the minimums on 𝜕𝐺_𝑘 and 𝜕𝐻_𝑘 will occur either at the upper-left or upper-right portion of 𝜕𝐺_𝑘 and 𝜕𝐻_𝑘. As our detailed analysis in [17] shows, the minimum of the convex functions whose level cures appear on the left side of 𝜕𝐻_𝑘 is attained at either the first or the second critical vertex of 𝜕𝐻_𝑘 or on the line segment connecting these two vertices. The same can be said about the minimum of such functions on 𝜕𝐺_𝑘. In Figure 1 above, 𝑄₆ and 𝑄₅ are the first and second critical vertices of 𝜕𝐻_𝑘, respectively. Similarly, 𝑃₇ and 𝑃₆ are the first and the second critical vertices of 𝜕𝐺_𝑘, 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 𝑓(𝑥,𝑦) at all vertices of 𝜕𝐺_𝑘 and 𝐻_𝑘. It also eliminates the need for computing and comparing the minimums of 𝑓(𝑥,𝑦) on all edges of 𝜕𝐺_𝑘 and 𝜕𝐻_𝑘. Thus, to compute the minimum of 𝑓(𝑥,𝑦) on 𝜕𝐻_𝑘, for example, we only need to evaluate 𝑓(𝑥,𝑦) for the components of the first and the second critical vertices and use Lagrange multipliers to compute the minimum of 𝑓(𝑥,𝑦) 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 𝑇𝑓,𝑓^𝑞𝐹(𝑇)≥()𝑓,𝑓𝑞−1𝑀−𝑚(𝑀)(𝑚)𝑞(𝑀)(𝑚)𝑚𝑓−𝑀𝑓𝑚𝑓−𝑀𝑓𝑞−1𝑓(𝑀)−𝑓(𝑚)^𝑞. (2.14) Therefore, for a positive operator, one can define a new antieigenvalue-type quantity by 𝜇_(𝐹,𝑞)(𝑇)=inf_{𝐹(𝑇)𝑓,𝑓≠0}𝑇𝑓,𝑓^𝑞𝐹(𝑇).𝑓,𝑓 (2.15) If we can compute the minimizing unit vectors 𝑓 for (2.15), then obviously for these vectors (1.12) becomes equality. 𝜇_(𝐹,𝑞)(𝑇) 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 𝜇_(𝐹,𝑞)(𝑇). It is easily seen that the standard antieigenvalue 𝜇(𝑇) 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 𝜇_(𝐹,𝑞)(𝑇) to an arbitrary operator where 𝐹 is an analytic function. One way is to extend the definition of 𝜇_(𝐹,𝑞)(𝑇) by 𝜇_(𝐹,𝑞)(𝑇)=inf_{|𝐹(𝑇)|𝑓,𝑓≠0}Re(𝑇𝑓,𝑓)^𝑞|𝐹(𝑇,)|𝑓,𝑓 (2.16) where 𝐹 is a complex-valued analytic function defined on the spectrum of 𝑇. The problem then becomes 𝜇_(𝐹,𝑞)(𝑇)𝑥=inf^𝑞𝑦∶𝑥+𝑖𝑦∈𝑊(𝑆), (2.17) where 𝑆=(Re𝑇)^𝑞+|𝐹(𝑇)|𝑖. For an arbitrary operator 𝑇, the set 𝑊(𝑆) is not in general a polygon but a bounded convex subset of the complex plane. Nevertheless, we can approximate 𝑊(𝑆) with polygons from inside and outside and thus obtain an approximation for 𝜇_(𝐹,𝑞)(𝑇) by looking at the minimum of the function 𝑓(𝑥,𝑦)=𝑥^𝑞/𝑦 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 𝛿(𝑇)=inf_{𝐗′𝐗=𝟏𝑃}1𝑋^′𝑇𝑋𝑋^′𝑇⁻¹𝑋=^𝑝_𝑖=14𝜆_𝑖𝜆_{𝑛−𝑖+1}𝜆_𝑖+𝜆_{𝑛−𝑖+1}². (2.18) To compute the first of these simpler antieigenvalue-type quantities, one has 𝛿₁(𝑇)=4𝜆₁𝜆_𝑛𝜆₁+𝜆_𝑛². (2.19) To find this quantity, we converted the problem to finding the minimum of the function 𝑓(𝑥,𝑦)=1/𝑥𝑦 on the convex set 𝑊(𝑆), where 𝑆=𝑇+𝑖𝑇⁻¹. If 𝑇 is not a positive 𝛽 operator on a finite dimensional space, 𝑊(𝑆) is not a polygon. We can, however, approximate 𝑊(𝑆) with polygons from inside and outside and thus obtain an approximation for 𝛿₁(𝑇)=inf_{𝑇𝑓≠0𝑓=1}1(𝑇𝑥,𝑥)𝑇⁻¹,𝑥,𝑥 (2.20) an antieigenvalue-type quantity, by looking at the minimum of the function 𝑓(𝑥,𝑦)=1/𝑥𝑦 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 𝑇, there is a sequence of finite-dimensional normal matrices {𝑇_𝑘} such that 𝜇(𝑇)≤𝜇𝑇_𝑘𝜇(𝑇),𝑘=1,2,3,…=lim_𝑘⇒∞𝜇𝑇_𝑘. (2.21)

Proof. Recall that for any operator 𝑇, we have 𝜇²(𝑇)𝑥=inf²𝑦∶𝑥+𝑖𝑦∈𝑊(𝑆), (2.22) where 𝑆=Re𝑇+𝑖𝑇^∗𝑇. (2.23) Using the notations in Theorem 2.1, for each 𝑘, there is an accretive normal operator 𝑆_𝑘 with 𝑊(𝑆_𝑘)=𝐺_𝑘. We can define 𝑆_𝑘 to be the diagonal matrix whose eigenvalues are 𝑃₁,𝑃₂,…,𝑃_𝑘. By the spectral mapping theorem, there exists an accretive normal matrix 𝑇_𝑘 such that 𝑆_𝑘=Re𝑇_𝑘+𝑖𝑇^∗_𝑘𝑇_𝑘. (2.24) To see this, let the complex representation of the vertex 𝑃_𝑗, 1≤𝑗≤𝑘, of 𝐺_𝑘 be 𝑃_𝑗=𝑥_𝑗+𝑖𝑦_𝑗, (2.25) then 𝑇_𝑘 can be taken to be the diagonal matrix whose eigenvalues are 𝑥_𝑗+𝑖𝑦_𝑗−𝑥²_𝑗. (2.26) Note that since 𝑃_𝑗 is on the boundary of 𝑊(𝑆), we have 𝑦_𝑗≥𝑥²_𝑗. Since we have 𝜇²(𝑇)=inf_{(𝑆)𝑥,𝑦∈𝜕𝑊}𝑥²𝑦=lim_𝑘⇒∞inf_{𝑥,𝑦∈𝜕𝐺𝑘}𝑥²𝑦, (2.27) we have 𝜇²(𝑇)=lim_𝑘⇒∞inf_{𝑆𝑥,𝑦∈𝜕𝑊𝑘}𝑥²𝑦. (2.28) However, lim_𝑘⇒∞inf_{(𝑥,𝑦)∈𝜕𝑊(𝑆𝑘)}𝑥²𝑦=lim_𝑘⇒∞𝜇²𝑇_𝑘. (2.29) This implies, 𝜇²(𝑇)=lim_𝑘⇒∞𝜇²𝑇_𝑘. (2.30) Since 𝜇(𝑇) is positive a 𝜇(𝑇_𝑘) is positive for each 𝑘, we have 𝜇(𝑇)=lim_𝑘⇒∞𝜇𝑇_𝑘. (2.31)

Theorem 2.6. For any bounded accretive operator 𝑇 there is a sequence of normal matrices {𝑇_𝑘} such that 𝜇_(𝐹,𝑞)(𝑇)≤𝜇_(𝐹,𝑞)𝑇_𝑘,𝑘=1,2,3,… (2.32) for each 𝑘 and 𝜇_(𝐹,𝑞)(𝑇)=lim_𝑘⇒∞𝜇_(𝐹,𝑞)𝑇_𝑘. (2.33)

Proof. Recall that for any operator 𝑇 we have 𝜇²_𝐹,𝑞(𝑇)𝑥=inf^𝑞𝑦∶𝑥+𝑖𝑦∈𝑊(𝑆), (2.34) where 𝑆=(Re𝑇)^𝑞+|𝐹(𝑇)|𝑖. (2.35) Using the notations in Theorem 2.1, for each 𝑘 there is an accretive normal matrix 𝑆_𝑘 with 𝑊(𝑆_𝑘)=𝐺_𝑘. We can define 𝑆_𝑘 to be the diagonal matrix whose eigenvalues are 𝑃₁,𝑃₂,…,𝑃_𝑘. By the spectral mapping theorem, there exists a normal matrix 𝑇_𝑘 such that 𝑆_𝑘=Re𝑇_𝑘^𝑞+𝐹𝑇_𝑘𝑖. (2.36) To see this, let the complex representation of the vertex 𝑃_𝑗, 1≤𝑗≤𝑘, of 𝐺_𝑘 be 𝑃_𝑗=𝑥_𝑗+𝑖𝑦_𝑗, (2.37) then by the spectral mapping theorem 𝑇_𝑘 can be taken to be any diagonal matrix with eigenvalues 𝑢_𝑗+𝑖𝑣_𝑗 where (𝑢_𝑗𝑣_𝑗) are any solution to the system 𝑢_𝑗=𝑥_𝑗^1/𝑞,Re𝐹𝑢_𝑗+𝑖𝑣_𝑗²+Im𝐹𝑢_𝑗+𝑖𝑣_𝑗²=𝑦²_𝑗. (2.38) Since we have 𝜇²_(𝐹,𝑞)=inf_{(𝑥,𝑦)∈𝜕𝑊(𝑆)}𝑥^𝑞𝑦=lim_𝑘⇒∞inf_{(𝑥,𝑦)∈𝜕𝐺𝑘}𝑥^𝑞𝑦, (2.39)

we have 𝜇²_(𝐹,𝑞)=lim_𝑘⇒∞inf_{(𝑥,𝑦)∈𝜕𝑊(𝑆𝑘)}𝑥^𝑞𝑦. (2.40) However, lim_𝑘⇒∞inf_{(𝑥,𝑦)∈𝜕𝑊(𝑆𝑘)}𝑥^𝑞𝑦=lim_𝑘⇒∞𝜇²_𝐹,𝑞𝑇_𝑘. (2.41) This implies that 𝜇²_(𝐹,𝑞)(𝑇)=lim_𝑘⇒∞𝜇²_(𝐹,𝑞)𝑇_𝑘. (2.42) Since 𝜇_(𝐹,𝑞) is positive a 𝜇_(𝐹,𝑞)(𝑇_𝑘) is positive for each 𝑘, we have 𝜇_(𝐹,𝑞)(𝑇)=lim_𝑘⇒∞𝜇_(𝐹,𝑞)𝑇_𝑘. (2.43)

Theorem 2.7. For any bounded accretive operator 𝑇 there is a sequence of finitedimensional normal matrices {𝑇_𝑘} such that 𝛿₁(𝑇)≤𝛿₁𝑇_𝑘𝛿𝑘=1,2,3,…,₁(𝑇)=lim_𝑘⇒∞𝛿₁𝑇_𝑘. (2.44)

Proof. Recall that for any operator 𝑇 we have 𝛿²₁(𝑇)𝑓=1=inf𝑥,𝑦(𝑆,𝑥𝑦∶𝑥+𝑖𝑦∈𝑊) (2.45) where 𝑆=𝑇+𝑖𝑇⁻¹. (2.46) Using the notations in Theorem 2.1, for each 𝑘 there is a normal operator 𝑆_𝑘 with 𝑊(𝑆_𝑘)=𝐺_𝑘. We can define 𝑆_𝑘 to be the diagonal matrix whose eigenvalues are 𝑃₁,𝑃₂,…,𝑃_𝑘. By the spectral mapping theorem there exist a normal matrix 𝑇_𝑘 such that 𝑆_𝑘=𝑇_𝑘+𝑖𝑇_𝑘⁻¹. (2.47) To see this, let the complex representation of the vertex 𝑃_𝑗, 1≤𝑗≤𝑘, of 𝐺_𝑘 be 𝑃_𝑗=𝑥_𝑗+𝑖𝑦_𝑗=𝑧_𝑗. (2.48) Take 𝑇_𝑘 to be any diagonal matrix whose eigenvalues 𝜆_𝑗 satisfy 1𝜆_𝑗+1/𝜆_𝑗=𝑧_𝑗. (2.49) To fund the eigenvalues of such a finite-dimensional diagonal matrix 𝑇_𝑘 explicitly, we solve the previous equation for 𝜆_𝑗. The solutions are 𝜆_𝑗1=−2𝑧_𝑗−4𝑧²_𝑗+1−1, (2.50) or 𝜆_𝑗=12𝑧_𝑗−4𝑧²_𝑗+1+1. (2.51)
Since we have 𝛿²₁(𝑇)=inf_{(𝑆)𝑥,𝑦∈𝜕𝑊}1𝑥𝑦=lim_𝑘⇒∞inf_{𝑥,𝑦∈𝜕𝐻𝑘}1,,𝑥𝑦 (2.52) we have 𝛿²₁(𝑇)=lim_𝑘⇒∞inf_{(𝑥,𝑦)∈𝜕𝑊(𝑆𝑘)}1𝑥𝑦. (2.53) However, lim_𝑘⇒∞inf_{(𝑥,𝑦)∈𝜕𝑊(𝑆𝑘)}𝑥^𝑞𝑦=lim_𝑘⇒∞𝛿²₁𝑇_𝑘. (2.54) This implies that 𝛿²₁(𝑇)=lim_𝑘⇒∞𝛿²₁𝑇_𝑘. (2.55) Since 𝜇_(𝐹,𝑞) is positive a 𝜇_(𝐹,𝑞)(𝑇_𝑘) is positive for each 𝑘, we have 𝛿₁(𝑇)=lim_𝑘⇒∞𝛿₁𝑇_𝑘. (2.56)

In the proofs of Theorems 2.5 through 2.7 previous, we considered accretive normal matrices 𝑆_𝑘 whose spectrum are vertices of 𝐺_𝑘, 𝑘=1,2,3,…. We could also consider matrices whose spectrums are 𝐻_𝑘, 𝑘=1,2,3,…. However, notice that those matrices may not be accretive for small values of 𝑘.

The term antieigenvalue was initially defined by Gustafson for accretive operators. For an accretive operator 𝑇 the quantity 𝜇(𝑇) is nonnegative. However, in some of our previous work we have computed 𝜇(𝑇) 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 𝑇 is bounded accretive to ensure that 𝑊(𝑆) in these theorems is a subset of the first quadrant. Thus, for each 𝑘, 𝑊(𝑆_𝑘) in Theorems 2.5 through 2.7 is a finite polygon in the first quadrant, making it possible to compute 𝜇(𝑇_𝑘) in terms of the first and the second critical eigenvalues (see (1.18)). If 𝑇 is not accretive in Theorems 2.5 through 2.7, then we can only say that 𝑊(𝑆) is a subset of the upper-half plane which implies for each 𝑘, we can only say 𝑊(𝑆_𝑘) 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 𝑇 in these theorems is not accretive. The only challenge in this case is computing 𝜇(𝑇_𝑘), for each 𝑘, in these theorems. What we know from our previous work in [16] is that 𝜇(𝑇_𝑘) can be expressed in terms of one or a pair of eigenvalues of 𝑇_𝑘. However, we do not know which eigenvalue or which pair of eigenvalues of 𝑇_𝑘 expresses 𝜇(𝑇_𝑘). Of course, one can use Theorem 2.2 of [16] to compute 𝜇(𝑇_𝑘), however, this requires a lot of computations, particularly for large values of 𝑘.

Example 2.8. Consider a normal matrix 𝑇 whose eigenvalues are 1+√6𝑖, −2+√7𝑖, 3+4𝑖,   −4+√34𝑖, 1+√15𝑖, and −5+2𝑖. For this matrix, we have 𝜇(𝑇)<0. If 𝑆=Re𝑇+𝑖𝑇^∗𝑇, then 𝑊(𝑆) is the polygon whose vertices are 1+7𝑖, −2+11𝑖, 3+25𝑖, −4+50𝑖, 1+16𝑖, and −5+29𝑖. This polygon is not a subset of the first quadrant. Therefore, 𝜇(𝑇), not readily found using the first and second critical eigenvalues. Using Theorem 2.2 of [16], we need to perform a set of 6𝐶2+6=21 computations and compare the values obtained to find 𝜇(𝑇).
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 𝜕𝑊(𝑆) is simple enough, we can use more elementary methods to find the minimum of 𝑓(𝑥,𝑦) on 𝜕𝑊(𝑆) without approximating it with polygons. For example, in [21], we computed 𝜇(𝑇) by direct applications of the Lagrange multipliers when 𝜕𝑊(𝑆) is just an ellipse. Also, in [23] we used Lagrange multipliers directly to compute 𝜇(𝑇), when 𝑇 is a matrix of low dimension on the real field.