ANTIEIGENVALUE INEQUALITIES IN OPERATOR THEORY

We will prove some inequalities among trigonometric quantities of two and three operators. In particular, we will establish an inequality among joint trigonometric quantities of two operators and trigonometric quantities of each operator. As a corollary, we will find an upper bound and a lower bound for the total joint antieigenvalue of two positive operators in terms of the smallest and largest eigenvalues of these operators.


Introduction.
Recall that an operator T is accretive if the numerical range of T is a subset of the right half-plane. For an accretive operator on a Hilbert space H, the first antieigenvalue of T , µ 1 (T ), is defined by Gustafson to be (see [1,3,4]). The quantity µ 1 (T ) is also denoted by cos T or cos R T and is called the cosine (or real cosine) of T . The first antieigenvalue, µ 1 (T ), has important applications in the study of the numerical range of operators. For instance, it is proved in [1] that for two positive selfadjoint operators A and B, the product AB is accretive if sin A ≤ cos B. It is shown in [2] that if for a positive operator T we define sin T = inf ∈>0 ∈ T − I , (1.2) then sin T = √ 1 − cos 2 T . The quantity µ 1 (T ) has numerous applications in numerical analysis as well as statistics and econometrics (see [5,6,8]). A vector f for which the infimum in (1.1) is attained is called an antieigenvector of T . The first total antieigenvalue of an operator T is defined to be The quantity |µ 1 |(T ) is also denoted by | cos |T . A vector f for which the infimum in (1.3) is attained is called a total antieigenvector of T . For an operator T , we also define the quantity cos I (T ) by and call it the imaginary cosine of T . The angle between a pair of operators A and B can be measured by looking at quantities We call µ 1 (T , S) the joint antieigenvalue for T and S, and |µ 1 |(T , S) is called the total joint antieigenvalue for T and S. The quantities µ 1 (T , S) and |µ 1 |(T , S) are also denoted by cos(T , S) and | cos |(T , S), respectively. A vector f for which the infimum in (1.5) is attained is called a joint antieigenvector for T and S. A vector f for which the infimum in (1.6) is attained is called a joint total antieigenvector for T and S. Likewise, the quantity cos I (T , S) is defined by The author and Gustafson have studied µ 1 (T ) and |µ 1 |(T ) for normal operators on finite-and infinite-dimensional spaces (see [9,10,12,13]). Also, in [11], the author has studied µ 1 (T , S), where S and T are two operators belonging to the same closed normal subalgebra of B(H). Note that µ 1 (T ) = µ 1 (T , I) and |µ 1 |(T ) = |µ 1 |(T , I), where I is the identity operator. Our objective in this paper is to establish some inequalities among these trigonometric functions using the Gram determinant between three vectors.

Operator trigonometry.
Recall that any invertible operator T on a Hilbert space is a product of the form T = UP , where U is a unitary operator and P is a positive operator (polar decomposition). On a finite, dimensional Hilbert space, every operator is the product of a unitary operator U and a positive operator P . For a positive operator P , we know that µ 1 (P ) = 2 √ mM/(m + M), where m and M are the smallest and largest eigenvalues of P , respectively. This was first proved by Gustafson in [2] and later independently by Krein. On the other hand, for a unitary operator U , it is easy to see that µ 1 (U ) = inf{cos θ : e iθ ∈ σ (T )} (see [13]). It is however impossible to express µ 1 (T ) in terms of µ 1 (P ) and µ 1 (U ). In the following, we will establish some inequalities among trigonometric functions of composite operators and their components. One may use Theorem 2.1 with A = U and B = P to establish an inequality between trigonometric functions of T , U, and P if T has a polar decomposition T = UP . Our work here is based on the fact that for any three vectors x 1 , x 2 , and x 3 in a Hilbert space, the Gram determinant is nonnegative.
Proof. Let x be any unit vector and let G be the Gram determinant function defined by ( (2.4) Since the Gram determinant G(Ax/ Ax ,Bx/ Bx ,ABx/ ABx ) is nonnegative, we have If we substitute the values of a, b, c, d, e, and f in (2.6), we have

Im(Bx, x) Bx
Im(ABx, Bx) ABx Bx . (2.7) Now inequality (2.2) follows from inequality (2.7) and the properties of the infimum. and computations similar to those carried out in Theorem 2.1, we have Im(x, Ax) Ax (2.13) Now inequality (2.11) follows from inequality (2.13) and the properties of infimum.
For positive operators A and B, we have cos I A = cos I B = 0. We also have | cos |A = cos A, | cos |B = cos B. Therefore, inequalities (2.2) and (2.8) have simpler forms for positive operators.
(2.17) Inequality (2.16) (or its equivalent (2.17)) is known as the Kantorovich inequality. The last inequality is equivalent to