Shapes of Numerical Ranges of Operators on a 3-dimensional Krein

International Journal of Algebra, Vol. 1, 2007, no. 4, 185 - 204

Shapes of Numerical Ranges of Operators on a 3-dimensional Krein Space Hiroshi Nakazato Hirosaki University Department of Mathematical System Science 036-8561 Hirosaki, Japan Natalia Bebiano University of Coimbra Mathematics Department P 3001-454 Coimbra, Portugal Jo˜ao da Providˆ encia University of Coimbra Physics Department P 3004-516 Coimbra, Portugal providencia@teor.fis.uc.pt Abstract Shapes of numerical ranges of operators on a 3-dimensional Krein space are investigated. Namely, a criterion for convexity is stated, an inclusion domain is obtained and properties like closedness are studied.

Mathematics Subject Classification: 46C20; 47A12; 15A60 Keywords: Krein space

1. Introduction and Results Indefinite inner product spaces are a classical subject of functional analysis and geometry. The matrix J = Ir ⊕ −In−r , r ≥ n − r > 0 endows Cn with an

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indefinite inner product x, yJ := y ∗ Jx, for x, y ∈ Cn . A matrix U ∈ Mn (C), the algebra of n × n complex matrices, is said to be J-unitary if UJU ∗ J = JU ∗ JU = In . For a Hermitian matrix J of signature (r, s), the J-unitary matrices with determinant 1 form a semi-simple Lie group G = SU(r, s). For X ∈ Mn (C), we have X = 1/2(X + JXJ) + 1/2(X − JXJ). The Lie algebra G is decomposed as G = {X ∈ Mn (C) : JX ∗ J = −X, Tr(X) = 0} = K + P, where K = {X ∈ Mn (C) : X ∗ = −X, JX = XJ, Tr(X) = 0}, P = {X ∈ Mn (C) : X ∗ = X, JXJ = −X, Tr(X) = 0}. The Lie algebra G has a root space decomposition with respect to a maximal Abelian subspace of P . The corresponding extended integral root system is of type BCs ([H] p.347-349, [V] p. 380). The matrices A ∈ SU(r, s), or in its Lie algebra, are J-normal, that is, AJA∗ J = JA∗ JA. The J-Hermitian matrices, defined by JA∗ J = A, constitute a special class of J-normal matrices. Although J-Hermitian matrices share many properties in common with Hermitian matrices, those have some singular properties. For instance, the spectrum of J-Hermitian matrices is symmetric with respect to the real axis. J-Hermitian matrices are not necessarily semi-simple. In the case r = 2, s = 1, the following J-Hermitian matrix A is nilpotent and A2 = 0: ⎛

⎞

0 1 0 ⎜ ⎟ A = ⎝1 0 1⎠. 0 −1 0 More generally, the associative algebra Nilr = {T = (tij ) ∈ Mr (IR) : tij = 0 for 1 ≤ j ≤ i ≤ r} is isomorphically imbedded in the associative algebra of J-Hermitian matrices: {

 1≤i 0, (a2 ) > 0. Now, we prove the assertion (1) of Theorem 1.1. Having in mind Remark 3.1 and (2.7), a general point of WCJ (A) may be represented in the form z = a11 + a2 a12 + c2 a21 + c2 a2 a22 , where the entries of A = (aij ) ∈ OJ (2, 1) satisfy

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the equation F (a11 , a12 , a21 , a22 ) = 0. Solving this equation with respect to a11 , we obtain (a22 − 1)2 a11 = a12 a21 a22 + (a12 + a22 − 1)(a21 + a22 − 1) √ +2 a12 a21 a22 (a12 + a22 − 1)(a21 + a22 − 1),  = {−1, 1}

(3.3)

Consider the domain of A4 {(a12 , a21 , a22 ) : a12 , a21 , a22 ≥ 0, a22 = 1, a12 + a22 − 1 ≥ 0, a21 + a22 − 1 ≥ 0}. (3.4) We claim that the restriction of the 2-valued function a11 to (3.4) satisfies the linear inequalities (3.5), (3.6), (3.7) and (3.8). Indeed, from (3.3) it follows that a11 ≥ 0.

(3.5)

Moreover, from (3.3) we get

(a22 − 1)2 (a11 + a12 − 1) = a21 (a12 + a22 − 1) + a12 a22 (a21 + a22 − 1)

+2 a12 a21 a22 (a12 + a22 − 1)(a21 + a22 − 1), and we clearly have a11 + a12 − 1 ≥ 0.

(3.6)

Analogously, it can also be seen that a11 + a21 − 1 ≥ 0,

(3.7)

a11 + a12 + a21 + a22 − 1 ≥ 0,

(3.8)

and

and so the claim follows. Suppose that z1 , z2 ∈ WCJ (A) are such that z2 − z1 = s c1 a1 , with s a non-zero real number. We consider the affine constraint (a2 a12 + c2 a21 + c2 a2 a22 ) = (z1 ),

(3.9)

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on the hypersurface F (a11 , a12 , a21 , a22 ) = 0 under the conditions a12 , a21 , a22 ≥ 0, a12 +a22 ≥ 1, a21 +a22 ≥ 1. The affine constraint reduces to a trivial condition if and only if the equations (a2 ) = 0,

(c2 ) = 0,

(c2 a2 ) = 0

hold. If these equations hold, WCJ (A) lies on a straight line, and the connectedness of the group SU(2, 1) ensures that [z1 , z2 ] ⊂ WCJ (A). So, we assume that one of the real numbers (a2 ), (c2 ), (c2 a2 ) is non-zero. By Lemma 2.1 there exists z0 ∈ L0 ∪ L1 ∪ L2 ∪ L3 ∪ L4 satisfying (z0 ) = (z1 ). This implies that there exists a point (a11 , a12 , a21 , a22 ) satisfying the affine constraint and a12 a21 a22 (a12 + a22 − 1)(a21 + a22 − 1) = 0. Thus, the two parts of the graph of the 2-valued function a11 on the domain (3.4) under the constraint (3.9) is connected. As a continuous image of this connected set, the set {z ∈ WCJ (A) : (z) = (z1 )} = {z ∈ WCJ (A) : z = z1 + s for some s ∈ IR},

(3.10)

is connected. Therefore, [z1 , z2 ] ⊂ WCJ (A). This case corresponds to the halfline L1 . By interchanging the roles of c1 , c2 and a1 , a2 , similar conclusions are valid for L2 , L3 and L4 . Next, we prove the second half of the assertion (1) of Theorem 1.1. We consider the following interval of the real line { (z) : z ∈ WCJ (A), z = z1 + s for some s ∈ IR}.

(3.11)

We claim that if α ∈ IR is the least upper bound of (3.11), then α is the maximum of this set. If the cj or the aj are colinear, the claim follows from Theorem 5.1 of [NBP]. Suppose, now, that the cj and the aj , j = 1, 2, 3, are non-colinear. We assume that the real number α is the least upper bound of (3.11), but it is not an element of the set. We show that this leads to a (n) contradiction. We may assume that there exists a sequence A(n) = (aij ) ∈ (n) (n) (n) (n) OJ (2, 1) such that the 4-tuples (a11 , a12 , a21 , a22 ) satisfy (3.9) and αn = (n) (n) (n) (n)

(a11 + a2 a12 + c2 a21 + c2 a2 a22 ) → α as n → ∞. Next, we apply a projective algebraic geometrical method. Consider the hyperplanes of the 4-dimensional real affine space with coordinates (a11 , a12 , a21 , a22 )

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(a2 a12 + c2 a21 + c2 a2 a22 ) = (z1 ) = β

(3.12)

(a11 + a2 a12 + c2 a21 + c2 a2 a22 ) = α.

(3.13)

The hypersurface F (a11 , a12 , a21 , a22 ) = 0 and the above two hyperplanes have (n) a common point at infinity. The sequence A(n) = (aij ) has a subsequence (n ) A(nk ) = (aij k ) such that the corresponding sequence of 4-tuples converges to a point at infinity on the 4-dimensional real projective space. The ratio a11 : a12 : a21 : a22 representing the point at infinity satisfies the simultaneous homogeneous equations F0 (a11 , a12 , a21 , a22 ) = a211 a222 + a212 a221 − 2a11 a12 a21 a22 = (a11 a22 − a12 a21 )2 = 0, (3.14) (3.15) (a2 a12 + c2 a21 + c2 a2 a22 ) = 0,

(a11 + a2 a12 + c2 a21 + c2 a2 a22 ) = 0,

(3.16)

where (3.14) may be replaced by a11 a22 − a12 a21 = 0.

(3.17)

We claim that the system of linear equations (3.15), (3.16), (3.17) does not have a non-zero real solution. Solving in a11 , a21 the homogeneous linear system (a2 )a12 + (c2 )a21 + (c2 a2 )a22 = 0, a11 + (a2 )a12 + (c2 )a21 + (c2 a2 )a22 = 0, and substituting the solution into a11 a22 − a12 a21 , we obtain a11 a22 − a12 a21 =

(a2 ) [( (c2 )a22 + a12 )2 + (c2 )2 a222 ] = 0, (c2)

a contradiction, and the claim follows. Thus, α is the maximum of the set (3.10). By interchanging the roles of c1 , c2 and a1 , a2 , similar assertions for L2 , L3 , L4 hold. Proposition 3.1 Suppose that neither the diagonal entries of C = diag(c1 , c2 , c3 ) ∈ M3 (C) nor those of A = diag(a1 , a2 , a3 ) ∈ M3 (C) are colinear. Suppose that the convex hull of L1 , L2 , L3 , L4 defined by (2.3), (2.4), (2.5), (2.6) is the whole complex plane, that is, the convex hull of the four points

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(c1 − c3 )(a1 − a3 ),

(c1 − c3 )(a2 − a3 ),

(c2 − c3 )(a1 − a3 ),

(c2 − c3 )(a2 − a3 ),

contains the origin as an interior point. Then there exist θ1 , θ2 ∈ IR such that: (1) WCJ (A) is a closed subset of C and contains the closed sectorial region {c1 aj1 + c2 aj2 + c3 a3 + r exp(−iθ) : 0 ≤ r < ∞, θ1 ≤ θ ≤ θ2 }; (2) WCJ (A) is contained in the region {c1 aj1 + c2 aj2 + c3 a3 + r exp(−iθ) : 0 ≤ r < ∞, θ1 ≤ θ ≤ θ2 }∪ {c1 aj1 + c2 aj2 + c3 a3 + r exp(−iθ) : 0 ≤ r ≤ R, θ2 ≤ θ ≤ θ1 + 2π}, for (j1 , j2 ) = (1, 2) or (j1 , j2 ) = (2, 1) with π < θ2 − θ1 < 2π and 0 < R < ∞. Proof. According to Remark 3.1, we assume c3 = a3 = 0, c1 = 1, a1 = 1, 0 < arg(c2 ) ≤ π, 0 < arg(a2 ) ≤ π and 0 < arg(c2 ) + arg(a2 ) < 2π. By the noncolinearity of the cj and of the aj , we may also assume (c2 ) > 0, (a2 ) > 0, and (c2 a2 ) < 0. A general point of WCJ (A) may be represented in the form z = a11 + a2 a12 + c2 a21 + c2 a2 a22 , where the entries of A = (aij ) ∈ OJ (2, 1) satisfy the equation F (a11 , a12 , a21 , a22 ) = 0. If the function

(z) = a11 + (a2 )a12 + (c2 )a21 + (c2 a2 )a22 ,

(3.18)

under the affine constraint (z) = (a2 )a12 + (c2 )a21 + (c2 a2 )a22 = β,

(3.19)

for β < (c2 a2 ), has an upper bound, by Theorem 1.1 (2) this upper bound is necessarily the maximum. (n)

We assume that there exists a sequence A(n) = (aij ) ∈ OJ (2, 1) such that (3.19) is satisfied and (n)

(n)

(n)

(n)

(zn ) = a11 + (a2 )a12 + (c2 )a21 + (c2 a2 )a22 → +∞

(3.20)

for a fixed β < (c2 a2 ). We show that this assumption leads to a contradiction. (n) By replacing the sequence A(n) = (aij ) by an appropriate subsequence, we may (n) assume that the sequence A(n) = (aij ) converges to a point at infinity on the

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4-dimensional real projective space. Since the sign of (zn ) is essential for our discussion, we use the 4-dimensional ”oriented real projective” space {IR5 \{(0, 0, 0, 0, 0)}}/ ∼ where (x1 , x2 , . . . , x5 ) ∼ (y1 , y2, . . . , y5 ) if and only yj = axj (j = 1, 2, . . . , 5) for some a > 0. This space is topologically equivalent to the closed ball {(x1 , x2 , x3 , x4 ) ∈ IR4 : x21 + x22 + x23 + x24 ≤ 1}. We denote by (∞)

(∞)

(∞)

(∞)

(a11 : a12 : a21 : a22 ) (n)

(∞)

the limit of the sequence A(n) = (aij ) where aij = 0 for some i, j ∈ {1, 2}. This limit satisfies the system of two linear equations (∞)

(∞)

(∞)

(a2 )a12 + (c2 )a21 + (c2 a2 )a22 = 0, (∞)

(∞)

(∞)

(∞)

a11 + (a2 )a12 + (c2 )a21 + (c2 a2 )a22 −  = 0, for some  ≥ 0 and the quadratic equation (∞) (∞)

(∞) (∞)

a11 a22 − a12 a21 = 0. (∞)

(∞)

We solve the system of the two linear equations in a11 , a21 and substitute this solution into the quadratic equation. Thus, we have (∞) (∞)

(∞) (∞)

a11 a22 − a12 a21 =

(a2 ) (∞) (∞) (∞)2 (∞) [( (c2 )a22 + a12 )2 + (c2 )2 a22 ] +  a22 = 0. (c2)

(∞)

(∞)

Therefore, a22 = 0, a12 = 0. Hence (∞)

(∞)

(∞)

(c2 )a21 = −(a2 )a12 − (c2 a2 )a22 = 0, (∞)

(∞)

and so a21 = 0. Thus, a11 = . (∞) (∞) (∞) (∞) Considering a22 = 0, a12 = 0, a21 = 0, a11 > 0, we find (n)

a22

(n)

a11

→ 0,

(3.21)

as n → ∞. Then (3.19) implies that (n)

(n)

(n)

(n)

0 ≤ (a2 )a12 + (c2 )a21 = β − (c2 a2 )a22 ≤ −(c2 a2 )a22 . We set η = (c2 a2 ) − β > 0. From the above inequality we get

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(n)

a22 ≥ 1 + (n)

(n)

0 ≤ a12 ≤ A1 a22 ,

η , −(c2 a2 ) (n)

(n)

0 ≤ a21 ≤ A2 a22 ,

where A1 , A2 are positive constants independent of n. Having in mind that a11 ≤ (a22 − 1)−2 [a12 a21 a22 + (a12 + a22 − 1)(a21 + a22 − 1) √ + a12 a21 a22 (a12 + a22 − 1)(a21 + a22 − 1)], we obtain (n)

(n)

a11 ≤ Aη a22

for some positive constant Aη and β ≤ (c2 a2 ) − η, contradicting (3.20). Since η > 0 is arbitrary, we conclude that the supremum of (3.18) is finite for an arbitrary β < (c2 a2 ). If cj , aj satisfy the assumptions of Lemma 2.1, the corresponding point z ∈ WCJ (A) to the above maximum lies on the half-line L2 or on the deltoid Γ. Set Ω1 = {1 + c2 a2 + r exp(iθ) : r ≥ 0, 0 ≤ θ ≤ arg(c2 ) + arg(a2 ) < 2π}, Ω2 = {1 + c2 a2 + r exp(iθ) : r ≥ 0, arg(c2 ) + arg(a2 ) ≤ θ ≤ 2π}. We show that Ω1 ⊂ WCJ (A) and an arc of a deltoid similar to Γ appears in ∂WCJ (A) ∩ Ω2 . Consider the J-unitary matrix U(s, t, −s). The corresponding point z(s, t) of WCJ (A) is given by z(s, t) = a11 (s, t) + a2 a12 (s, t) + c2 a21 (s, t) + c2 a2 a22 (s, t) (t > 0, 0 ≤ s ≤ 2π), where aij (s, t) is the square of the (i, j) entry of U(s, t, −s). Changing the variables as cosh t = 1 + T , cos2 s = S, z(s, t) ∈ WCJ (A) is represented as z(s, t) = z˜(S, T ) = (1 + c2 a2 ) + 2T {c2 a2 + (1 − c2 a2 )S} + T 2 {c2 + (1 − c2 )S}{a2 + (1 − a2 )S}, for T ≥ 0, 0 ≤ S ≤ 1, being

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z˜(0, T ) = (1 + c2 a2 ) + c2 a2 (2T + T 2 ),

z˜(1, T ) = (1 + c2 a2 ) + (2T + T 2 ).

For sufficiently small fixed T > 0, z˜(S, T ) behaves as (1+c2 a2 )+2T {c2 a2 +(1− z (S, T )) is an increasing function of S. Having in mind c2 a2 )S}. Hence, arg(˜ ˜ similar to Γ this fact, Theorem 1.1 (1) and Lemma 2.1, we find that a deltoid Γ ˜ appears in the interior of Ω2 , appears in ∂WCJ (A) ∩ Ω2 . If a singular point of Γ ˜ for some j ∈ {1, 2}. there exists a point on Lj at which Lj is the tangent of Γ ˜ ∩ Ω2 is contained in ∂W J (A) and another part In this case, a certain part of Γ C J ˜ of Γ ∩ Ω2 is not contained in WC (A). For T > 0 sufficiently large and fixed, the function arg(˜ z (S, T )) is a decreasing function of S and |˜ z (S, T ) − (1 + c2 a2 )| is not less than T 2 min{|c2 + (1 − c2 )S| |a2 + (1 − a2 )S| : 0 ≤ S ≤ 1} − 2T (1 + 2|c2a2 |). By this fact and Theorem 1.1 (1), Ω1 is contained in WCJ (A). Thus, Proposition 3.1 follows. We give an example to illustrate Proposition 3.1. Suppose that c1 = a1 = exp(iθ),

c2 = a2 = exp(−iθ),

c3 = a3 = 1, 0 < θ < π/2.

Set C = A = diag(c1 , c2 , c3 ). Then, WCJ (A) contains the sectorial region {2 cos(2θ) + 1 + r exp(iη) : r ≥ 0, −(π − θ) ≤ η ≤ π − θ}. The singular point 3 of the deltoid Γ belongs to WCJ (A). The two other singular points 3 exp(2iπ/3), 3 exp(−2iπ/3) belong to WCJ (A) if and only if 0 < θ ≤ π/3. In the case 0 < θ < π/3, WCJ (A) has two non-C (1) boundary points. Proposition 3.2 Suppose that the convex hull of L1 , L2 , L3 , L4 defined by (2.3), (2.4), (2.5), (2.6) is a closed half-plane in C, that is, the diagonal entries of the diagonal matrices C and A, cj and aj , respectively, satisfy cj − c3 = 0, aj − a3 = 0, j = 1, 2, and max{arg(ci − c3 )(aj − a3 )} − min{arg((ci − c3 )(aj − a3 ))} = π, for i, j ∈ {1, 2}. If the arguments of (ci − c3 )(aj − a3 ), i, j = 1, 2, are different, then WCJ (A) coincides with the closed half-plane conv(L1 ∪ L2 ∪ L3 ∪ L3 ).

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Proof. As in the proof of Proposition 3.1, we assume that c3 = a3 = 0, c1 = a1 = 1, c2 a2 < 0 and (c2 ) > 0, (a2 ) > 0. Taking into account (2.7) and considering a12 ≥ 1, a21 ≥ 1, we have (a2 )a12 + (c2 )a21 + (c2 a2 )a22 ≥ (a2 + c2 ) + (c2 a2 )a22 ≥ (a2 + c2 ) ≥ (c2 a2 ). If min{a12 , a21 } < 1, then a22 ≥ 1 − min{a12 , a21 }, and so (a2 )a12 + (c2 )a21 + (c2 a2 )a22 ≥ (a2 + c2 ) min{a12 , a21 } + (c2 a2 )[1 − min{a12 , a21 }] ≥ (c2 a2 ) min{a12 , a21 } + (c2 a2 )[1 − min{a12 , a21 }] = (c2 a2 ). Hence (z) = a12 (a2 ) + a21 (c2 ) ≥ 0, and so WCJ (A) is contained in the closed upper half-plane. We prove that the converse inclusion holds. Consider the generic case: 0 = arg(c1 ) = arg(a1 ) < arg(c2 ) = arg(a2 ) < arg(c2 a2 ) = π. For definiteness, we assume 0 < arg(a2 ) < π/2 and arg(c2 ) = π − arg(a2 ). The upper half-plane is divided into three closed regions D1 = conv(L1 ∪ L0 ∪ L3 ),

D2 = conv(L3 ∪ L4 ),

D3 = conv(L2 ∪ L0 ∪ L4 ).

Each point of D1 lies on a line segment 1 joining two points of L1 ∪ L0 ∪ L3 such that 1 is parallel to L4 . Similarly, each point of D3 lies on a line segment 3 joining two points of L2 ∪ L0 ∪ L4 such that 3 is parallel to L3 . Each point of D2 lies on a line segment 2 joining two points of L3 ∪ L4 such that 2 is parallel to L1 . The case 0 < arg(c2 ) < arg(a2 ) may be treated similarly. Thus, Proposition 3.2 follows from Theorem 1.1(2). The assertion of Proposition 3.2 in the exceptional case arg(a2 ) = arg(c2 ) = π/2 may be proved using the point z˜(S, T ) in the proof of Proposition 3.1. Now, we complete the proof of the assertion (2) of Theorem 1.1. In [NBP], it was proved that WCJ (A) is closed if the ci or the aj are colinear. So, now we assume that ci − c3 = 0, aj − a3 = 0, i, j = 1, 2. In Proposition 3.1 (1) of [NBP] it was proved that WCJ (A) is closed if (c2 a2 ) > 0. By Proposition

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3.1, WCJ (A) is closed if (c2 a2 ) < 0, and by Proposition 3.2 this set is closed if (c2 a2 ) = 0. Thus, the assertion (2) of Theorem 1.1 follows.

4. A necessary and sufficient condition for WCJ (A) to be convex As shown in [NBP] the convexity of WCJ (A) follows from the colinearity of the eigenvalues of C or of A. However, if C = A = diag(exp(2iπ/3), exp(−2iπ/3), 1), we still have convexity since WCJ (A) = {r exp(iθ) : r ≥ 0, −π/3 ≤ θ ≤ π/3}.

In this section we prove Theorem 1.2. According to the Remark 3.1, we may assume that c3 = a3 = 0, c1 = a1 = 1 and (c2 ) > 0, (a2 ) > 0. By Propositions 3.1 and 3.2, to prove Theorem 1.2 it is sufficient to show the following. Theorem 4.1 Let J = diag(1, 1, −1), A = diag(1, a2 , 0), C = diag(1, c2 , 0), (a2 ) > 0, (c2 ) > 0, so that σ1 = 1 + a2 c2 , σ2 = a2 + c2 . WCJ (A) is convex if and only if σ2 ∈ Σ = {σ1 +z : z ∈ C, 0 ≤ arg z ≤ arg(a2 c2 )} and arg(a2 c2 ) ≤ π. Proof. (⇐) Let  2 = {σ1 + a2 c2 s, 0 ≤ s ≤ ∞}, L  1 = {σ1 + s, 0 ≤ s ≤ ∞}, L L  3 = {σ2 + a2 s, 0 ≤ s ≤ ∞}, L  4 = {σ2 + c2 s, 0 ≤ s ≤ ∞}, L  0 = {σ1 s + σ2 (1 − s), 0 ≤ s ≤ 1}. Observe that min(arg(a2 ), arg(c2 )) ≥ 0 and max(arg(a2 ), arg(c2 )) ≤ arg(c2 a2 ). Thus, if σ2 ∈ Σ = {σ1 + z : z ∈ C, 0 ≤ arg z ≤ arg(a2 c2 )}, the half-rays L1 , L2 do not intercept the half-rays L3 , L4 . Notice also that Σ ⊆ WCJ (A) by Theorem 1.1 (1). By Lemma 3.1, WCJ (A) ⊆ conv({L1 ∪ L2 ∪ L3 ∪ L4 }) and, under the hypothesis, conv({L1 ∪ L2 ∪ L3 ∪ L4 }) ⊆ Σ. (⇒) (By contradiction.) By Proposition 3.1, WCJ (A) is not convex if arg(a2 c2 ) > π. So, we may assume that arg(a2 c2 ) ≤ π. Suppose that a2 + c2 does not belong to conv(L1 ∪ L2 ). As in the case that a2 + c2 belongs to the convex hull conv(L1 ∪ L2 ), the closed convex cone conv(L1 ∪ L2 ) is contained in WCJ (A). Exchanging the roles of c1 , c2 and a1 , a2 ,

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the closed convex cone conv(L3 ∪ L4 ) is also contained in WCJ (A). We consider z1 in the intersection of L3 ∪ L4 and L1 ∪ L2 . The convex hull of 1 + c2 a2 , a2 + c2 , z1 , is a closed triangle T , which is contained in WCJ (A) by Theorem 1.1 (1). The union

J WC (A) = conv(L1 ∪ L2 ) ∪ conv(L3 ∪ L4 ) ∪ T,

is not convex. Its closed convex hull contains the two closed half-lines {a2 + c2 + t : t ≥ 0},

{a2 + c2 + tc2 a2 : t ≥ 0}.

J (A). If WCJ (A) is One of these half-lines is not contained in the union WC convex, it must constitute a part of the boundary of WCJ (A). But this halfline does not belong to the list of the candidates of a part of the boundary of WCJ (A) given in Lemma 2.1. Thus, WCJ (A) is not convex and Theorem 4.1 follows.

Remark 4.1 The statement of Theorem 4.1 is equivalent to (i) π ≥ arg(a2 c2 ) ≥ arg(σ2 − σ1 ) ≥ 0, (ii) π ≥ arg(a2 c2 ) ≥ arg(a2 − 1) + arg(c2 − 1) − π ≥ 0. Example 4.1 Suppose that C = diag(1, 1 + i, 0) and A1 = diag(1, 3/2 + √ 3i/2, 0), A2 = diag(1, 3/2 − 3i/2, 0). Although WCJ (A2 ) is convex, WCJ (A1 ) is not convex. In contrast to the above situation, the convexity of WCJ (diag(a1 , a2 , a3 )) for J = I3 , simply denoted by WC (A), is equivalent to the convexity of √

WC (diag(a1 , a2 , a3 )) for any complex diagonal matrix C = diag(c1 , c2 , c3 ). A necessary and sufficient condition for WC (A) to be convex was given in [AP] and equivalent conditions in terms of the triangles c1 c2 c3 and a1 a2 a3 was found in [NW].

5. Whole plane and the convex hull Proposition 5.1 Suppose that J = diag(1, 1, −1) and C = diag(c1 , c2 , c3 ), A = diag(a1 , a2 , a3 ) ∈ M3 (C). Then WCJ (A) is the whole complex plane if and only if the cj , aj satisfy the following conditions:

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(i) ci = cj for 1 ≤ i < j ≤ 3 and ai = aj for 1 ≤ i < j ≤ 3, (ii) either the entries c1 , c2 , c3 or the entries a1 , a2 , a3 are colinear, (iii) the convex hull of the four complex numbers (ci − c3 )(aj − a3 ), i, j = 1, 2, contains the origin as an interior point in C.

Proof. The ”if ” part follows from Theorem 5.1 (2) of [NBP]. We prove the ”only if ” part. We assume that one of the conditions (i), (ii), (iii) does not hold. If (i), (ii) hold and (iii) does not hold, WCJ (A) is contained in a closed half-plane by Theorem 5.1 (1) of [NBP]. So, we assume that c3 = a3 = 0, c1 = a1 = 1, (c2 ) > 0, (a2 ) > 0. In the case (c2 a2 ) > 0, either WCJ (A) is not convex or WCJ (A) is contained in the closed cone conv(L1 ∪ L2 ), being in both cases a proper subset of C. In the case (c2 a2 ) ≤ 0, WCJ (A) is a proper subset of C by Propositions 3.1 and 3.2. Now, we give a necessary and sufficient condition for the inclusion of WCJ (A) in a closed half-plane. If the entries c1 , c2 , c3 (or the entries a1 , a2 , a3 ) are colinear, WCJ (A) is convex and is characterized by L0 and Lj ( j = 1, 2, 3, 4). So, we consider that ci = cj and ai = aj , for 1 ≤ i < j ≤ 3, and that neither c1 , c2 , c3 nor a1 , a2 , a3 are colinear. Proposition 5.2 Let J = diag(1, 1, −1), C = diag(c1 , c2 , c3 ), A = diag(a1 , a2 , a3 ) ∈ M3 (C). Suppose that neither the cj nor the aj are colinear. Then, WCJ (A) is contained in a closed half-plane of C if and only if the condition (iii) of Proposition 5.1 does not hold. Proof. (⇒) Under the assumption of noncolinearity of the eigenvalues of A and C, the supremum of { (z exp(−iφ)) : z ∈ WCJ (A)} is infinity or the supremum is attained at z = c1 a1 + c2 a2 + c3 a3 or z = c1 a2 + c2 a1 + c3 a3 , for each φ ∈ [0, 2π]. If the four points (ci − c3 )(aj − a3 ) (i, j = 1, 2) lie in an open half-plane defined by a line through the origin, then WCJ (A) is contained in a half-plane and one of the σ-points c1 a1 +c2 a2 +c3 a3 , c1 a2 +c2 a1 +c3 a3 lies on the boundary of WCJ (A). In the case |arg((c2 − c3 )/(c1 − c3 ))| + |arg((a2 − a3 )/(a1 − a3 ))| = π, J WC (A) is a closed half-plane, and one of the σ-points, c1 a1 + c2 a2 + c3 a3 or c1 a2 + c2 a1 + c3 a3 , lies on the boundary of WCJ (A).

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(⇐) (By contradition.) If the convex hull of the four points (c1 −c3 )(aj −a3 ) ( i, j = 1, 2) contains the origin as an interior point, then the convex hull of Lj (j = 1, 2, 3, 4) is the whole plane. But WCJ (A) is a proper closed subset of C. In this case the supremum of the values { (z exp(−iφ)) : z ∈ WCJ (A)} is infinity for every φ ∈ [0, 2π]. References [A] T. Ando, Linear operators on Krein spaces, Mimeographed note, Hokkaido Univ., 1979. [AP] Y. H. Au-Yeung and Y. H. Poon, ”3×3 orthostochastic matrices and the numerical ranges”, Linear Algebra Appl., 27 (1979), pp. 69-79. (1986), 249-257. [Bo] J. Bogn´ar, Indefinite inner product spaces, Springer-Verlag, 1974, Berlin, New York. [BLPS1] N. Bebiano, R. Lemos, J. da Providˆencia and G. Soares, ”On generalized numerical ranges of operators on an indefinite inner product space”, to appear in Linear and Multilinear Algebra. [BLPS2] N. Bebiano, R. Lemos, J. da Providˆencia and G. Soares, ”On the geometry of numerical ranges in spaces with indefinite inner product”, to appear in Linear Algebra and its Appl. [BR] J.P. Blaizot and G. Ripka, Quantum theory of finite systems, The MIT Press, Cambridge, Massachussets, USA, 1986. [GLR] I. Gohberg, P. Lancaster and L. Rodman, Matrices and indefinite scalar products, Birkh¨auser Verlag, 1983, Basel, Boston, Stuttgart. [H] S. Helgason, Differential Geometry , Lie Groups, and Symmetric Spaces, Academic Press, 1978, Orland, San Diego, New York. [Ny] Y. Nakagami, Tomita’s spectral analysis in Krein spaces, Publ. R.I.M.S, Kyoto Univ. 22(1986), 637-658. [Nn] N. Nakanishi, ”Indefinite-metric quantum field theory”, Prog. Theoret. Phys. Suppl. 51 (1972), 1-95.

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[NBP] H. Nakazato, N. Bebiano and J. da Providˆencia, ”J- orthostochastic matrices of size 3 × 3 and numerical ranges of Krein space operators”, submitted. [NW] H. Nakazato and K. Wakaizumi, ”An alternating condition for certain generalized numerical range to be convex”, Sci. Rep. Hirosaki Univ, 43(1996), 1-7 [V] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Springer Verlag, 1974, New York, Berlin, Heidelberg, Tokyo. Received: October 6, 2006