Computation of antieigenvalues

We will use a theorem by M. G. Kreĭn and properties of the numerical range of an operator to develop a method of computation for the
first antieigenvalue of a strictly accretive operator.


Introduction
An operator on a Hilbert space is called accretive if Re(T f , f ) ≥ 0 and strictly accretive if Re(T f , f ) > 0, for every vector f .For an accretive operator or matrix T on a Hilbert space, the first antieigenvalue of T, denoted by µ 1 (T), is defined by Gustafson to be (see [3,4,5]).The quantity µ 1 (T) is also denoted by cosT and is called the cosine of T. Definition (1.1) is equivalent to µ 1 (T) measures the maximum turning capability of T. A vector f for which the inf in (1.1) is attained is called an antieigenvector of T. Cosine of T has important applications in pure operator theory as well as numerical analysis (see [3,4,6]).In [7,8,11], we have computed the first antieigenvalue of normal accretive operators in terms of eigenvalues of these operators.In [12], we computed the first antieigenvalue for certain classes of matrices on the real field.In this note, we will focus on computation and approximation of the first antieigenvalue of general accretive operators.Our work in this note is a generalization of our work in [7,8,11] to certain classes of nonnormal operators.We will start with the case of a general two-by-two matrix.

Computing µ 1 (T) for an accertive two-by-two matrix
Theorem 2.1.Let T be a strictly accretive two-by-two matrix and let S = Re T + iT * T.
There exist the following two cases.If S has two distinct eigenvalues α = α 1 + iα 2 , β = β 1 + iβ 2 and the corresponding eigenvectors, so normalized that f = g = 1, then µ 1 (T) = x 2 0 /y 0 , where (x 0 , y 0 ) is the point where a member of the family of convex functions y = x 2 /k touches just one point of the ellipse whose equation is ) x 2 0 /y 0 , where (x 0 , y 0 ) is the point where a member of the family of convex functions y = x 2 /k touches just one point of the circle whose equation is Proof.Let Ω(S) be the numerical range of S. By Kreȋn [10], we have On the other hand, by Halmos [9], if α and β are distinct, then Ω(S) is the closed elliptical disk with foci at α and β whose major axis is |α − β|/δ and minor axis is γ(|α − β|/δ).If S has only one eigenvalue α, then Ω(S) is the circular disk with center at α and radius S − α /2.Using the convexity property of the curves f (x, y) = x 2 /y = k, we conclude that the minimum of the function f (x, y) = x 2 /y on Ω(S) is exactly at the point where a member of the family of convex functions y = x 2 /k touches one point of Ω(S).This point of contact must be on the boundary of Ω(S), which is an ellipse or a circle.We denote the boundary of Ω(S) by ∂Ω(S).First, assume that S has two distinct eigenvalues α = α 1 + iα 2 and β = β 1 + β 2 .Let d = |α − β|.If we select the system of coordinate axis x -y so that x -axis coincides with the major axis of Ω(S), then the equation of ∂Ω(S) with respect to x -y system is (2.4) To obtain the equation of this ellipse with respect to x-y axis, we need to perform a rotation and a translation of axis.This will give us (2.1) which represents the equation of ∂Ω(S) with respect to x-y axis.Next, assume that α = α 1 + α 2 is the only eigenvalue of S, then the equation of ∂Ω(S) is (2.2).
In [7,8,11], we used (1.2) to first compute antieigenvectors, and then from there compute antieigenvalues for normal matrices and normal operators.Also in [12], we used (1.2) to first compute antieigenvectors, and then from there compute antieigenvalues for arbitrary matrices on the field of real numbers.However, it is very difficult to use (1.2) to compute antieigenvectors in the case of an arbitrary matrix or operator on the field of complex numbers.To see this, consider the accertive matrix T = [ 1 0 1 1 ] and let f = z1 z2 represent a unit vector.Assume z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 .Direct computations show that the problem of finding inf T f =0 (Re(T f , f )/ T f ), subject to the constraint f = 1, is equivalent to the problem of finding the infimum of the function (2.9) subject to the constraint (2.10) If one wishes to solve this optimization problem using Lagrange multiplier method, then the following nonlinear system consisting of five equations and five unknowns will result (2.11) Notice that three equations and three unknowns resulted when Lagrange multiplier method was applied in Example 2.2 above to compute the antieigenvalue for the same matrix.
In computing µ 1 (T) for a two-by-two matrix, we took advantage of the fact that the minimum of the convex function f (x, y) = x 2 /y on the convex set Ω(S) is attained on the boundary of Ω(S), where Ω(S) is the numerical range of the operator S = Re T + iT * T. The same argument can also be made in the case of a general matrix or operator.However, If T is not a two-by-two matrix, then the boundary of Ω(S) is no longer an ellipse.Indeed, the boundary of the numerical range of a finite-dimensional matrix can be very complicated.We can generalize the technique of computing µ 1 (T) for two-dimensional matrices to matrices which are direct sums of two-by-two matrices.For instance, suppose that T = T 1 ⊕ T 2 , where T 1 and T 2 are accretive two-by-two matrices.If as before we define S = Re T + iT * T, then S = S 1 ⊕ S 2 where S 1 = Re T 1 + iT *   1 T 1 and S 2 = Re T 2 + iT * 2 T 2 .Hence, by Halmos [9], we have Ω(S) = co(Ω(S 1 ),Ω(S 2 )), where co(Ω(S 1 ),Ω(S 2 )) denotes the convex hull of the numerical ranges of S 1 and S 2 .Hence, the boundary of Ω(S) consists of a finite number of possible elliptic, circular, and flat (straight) arcs.The minimum of f (x, y) = x 2 /y on each arc can be obtained using Lagrange multiplier method as demonstrated in Example 2.2 above.Therefore, µ 2 1 (T) is the smallest of minimum values of f (x, y) = x 2 /y on those arcs.This can be further generalized to the case when T is a direct sum of a finite number of two-by-two accretive matrices T 1 ,T 2 ,T 3 ,...,T n .

The general case
To apply these techniques to a general operator T, we need to understand the boundary of the numerical range of the operator S where S = Re T + iT * T. If Ω(S) is closed, then its boundary ∂Ω(S) consists of a countable set of differentiable arcs plus a countable set of points at which the boundary is not differentiable.Furthermore, the points at which the boundary is not differentiable are eigenvalues of S (see [2]).This leads us to the following theorem.
Theorem 3.1.Let T be a strictly accretive operator and S = Re T + iT * T. Assume that Ω(S) is closed and ∂Ω(S) = M ∪ N, where and N = ∪ i γ i , where each γ i is a differentiable arc represented by the differentiable function g i (x, y) = 0. Let N be the set of all points (u i ,v i ) such that (u i ,v i ) is the solution of the Lagrange multiplier equation at which the function f (x, y) = x 2 /y attains its minimum on γ i .If ) Under certain conditions, the differentiable arcs γ i are simple arcs such as conic sections and straight lines.In those situations, equations (3.2) can be solved explicitly.One of those special cases is discussed in the next theorem.Recall that for two operators U and V , we have U ≥ V if (U f , f ) ≥ (V f , f ) for every vector f .An operator A is hyponormal if A * A ≥ AA * .An operator A is seminormal if either A or A * is hyponormal.In the following theorem, σ(T) denotes the spectrum of an operator T, and σ p (T) denotes the point spectrum of T (i.e., σ p (T) denotes the set of all eigenvalues of T).Theorem 3.2.Let T be a bounded strictly accretive operator such that the numerical range of T * T Re T is a subset of upper half-plane or lower half-plane, that is, Im(T * T Re T f , f ) ≥ 0 for all vectors f or Im(T * T Re T f , f ) ≤ 0 for all vectors f .Let S = Re T + iT * T, and C = σ(S) ∩ ∂co(σ(S)), where ∂co(σ(S)) denotes the boundary of the convex hull of σ(S).Assume that σ p (S) ∩ C is dense in C. Let R + denote the set of all positive numbers and define the set M by M = λ i = α i + β i : λ i ∈ σ p (S), ∂Ω(S) is not differentiable at λ i .(3.4) ) This implies that S is a seminormal operator.By [1], the denseness of σ p (S) ∩ C in C implies that Ω(S) is closed and Ω(S) = co(σ p (S)).Therefore, in this case the differentiable curves γ i are simple line segments whose endpoints are pairs of points from set M. First assume that the infimum of the function f (x, y) = x 2 /y is attained at an interior point of one of these line segments.We denote the endpoints of this line segment by (α i ,β i ) and (α j ,β j ).The slope of this line segment is and its equation is To find the infimum of the function f (x, y) = x 2 /y on this line segment, we use the Lagrange multiplier method.This will give Hence from the fact that µ 2 1 (T) = inf{x 2 /y : x + iy ∈ Ω(S)}, we conclude that µ 1 (T) = 2 (α i β j − α jβi )(α j −α i )/(β j −β i ).Next, assume that the infimum of the function f (x, y)= x 2 /y on ∂Ω(S) occurs at a nondifferentiable point of ∂Ω(S).Since a nondifferential point of ∂Ω(S) is an eigenvalue of S, then µ 1 (T) is equal to an element from set E.
Note that every finite-dimensional hyponormal matrix is normal (see [9]).The author and Karl Gustafson have treated the cases of normal matrices and normal operators in [7,8,11].