Numerical Ranges of the Product of Operators

Home , Ning (surname)

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First of all, for two positive contraction operators A, B B(H), it is known that ∈ W (AB) x + iy : 1/8 x 1, 1/4 y 1/4 ; ⊆ { − ≤ ≤ − ≤ ≤ } see [1], [2], [3]. To see this1, we use the positive deﬁnite ordering that X Y , and show that ≥ (a) I/8 (AB + BA)/2 I and (b) I/4 (AB BA)/(2i) I/4. − ≤ ≤ − ≤ − ≤ For (a), it is clear that AB + BA 2 so that 2I AB + BA 2I for two positive contractions k k≤ − ≤ ≤ A and B. Furthermore, note that A + B A2 B2 0, and hence − − ≥ 0 (A + B I/2)2 = A2 + B2 + (AB + BA)+ I/4 A B ≤ − − − = (AB + BA)+ I/4 + (A2 A) + (B2 B) (AB + BA)+ I/4. − − ≤ For (b), since A I/2 1/2 and B I/2 1/2, we have k − k≤ k − k≤ i(AB BA) = (A I/2)(B I/2) (B I/2)(A I/2) k − k k − − − − − k 2 (A I/2)(B I/2) 1/2. ≤ k − − k≤ Moreover, i(AB BA) is self-adjoint and then condition (b) holds. − Suppose λ [0, 1]. Denote by (λ) the elliptical disk with foci 0, λ, minor axis with end points ∈ E λ (λ i λ(1 λ))/2, and major axis with end points (λ √λ)/2. Then W 0 = (λ). ± − ± pλ(1 − λ) 0 E We havep the following result in [4].

Theorem 1.1. Let P,Q M be non-scalar orthogonal projections. Then ∈ n W (PQ) = conv (λ) . {∪λ∈σ(PQ)E } One can obtain the above result using the following canonical for a product of projections P,Q ∈ Mn; see [1, 7] and its references.

Proposition 1.2. Suppose P,Q M are non-scalar projections and U M is unitary such that ∈ n ∈ n U ∗(P + iQ)U is a direct sum of (I + iI ) I iI 0 , and p p ⊕ q ⊕ r ⊕ s c2 + i c s C = j j j , j = 1,...,k, j c s s2 j j j where c (0, 1),s = 1 c2. Then U ∗P QU will be a direct sum of I 0 and j ∈ j − j p ⊕ q+r+s q c2 0 Cˆ = j , j = 1,...,k. j c s 0 j j In the next two sections, we will consider containment regions for the numerical range of the product of a pair of positive contractions, and extend the results to a more general class of matrices, namely, those matrices with numerical ranges contained in line segments. We will consider the inﬁnite dimensional version of Theorem 1.1 and its generalization in Section 4.

1Li would like to thank Professor Fuzheng Zhang for showing him this proof. NUMERICALRANGESOFTHEPRODUCTOFOPERATORS 3

2. Positive Contractions

In this section, we extend Theorem 1.1 to obtain a containment region of W (AB) for two S positive contractions A, B M , and determine the conditions for = W (AB). We begin with ∈ n S some technical lemmas. We will denote by λ (X) λ (X) the eigenvalues of a Hermitian 1 ≥···≥ n matrix X M . ∈ n Lemma 2.1. Suppose T = T T T M and µ = µ + iµ C satisfying T = = 1 ⊕···⊕ m ⊕ 0 ∈ n 1 2 ∈ 1 · · · T M are non-scalar matrices, m ∈ 2 ∗ 2µ1 = λ1(T1 + T1 ) > λ1(T0). Then up to a unit multiple, there is a unique unit vector x˜ C2 such that x˜∗T x˜ = µ + iµ . ∈ 1 1 2 Moreover, if x Cn such that x∗Tx = µ + iµ , then ∈ 1 2 x = v x˜ 0 = (v x˜t, . . . , v x˜t, 0,..., 0)t ⊗ ⊕ n−2m 1 m where v = (v , . . . , v )t Cm is a unit vector. 1 m ∈ 2 ∗ Proof. Obviously, µ ∂W (T1). Hence there is a unit vectorx ˜ C such thatx ˜ T1x˜ = µ1 + iµ2. ∈ 2 ∈ n−2m Now, suppose x = x1 xm x0, where, x1,...,xm C and x0 C , is a unit vector ∗ ⊕···⊕ ⊕ ∈ ∈ such that x Tx = µ1 + iµ2. Then

m m 2µ = x∗(T + T ∗)x = x∗(T + T ∗)x 2µ x 2 + λ (T + T ∗) x 2. 1 j j j j ≤ 1 k jk 1 0 0 k 0k Xj=0 Xj=1 Thus, x = 0 and (T +T ∗)x = 2µ x for j = 1,...,m. and x = v x˜ with v C for j = 1,...,m. 0 j j j 1 j j j j ∈ Let v = (v , . . . , v )t. Then x = v x˜ 0 , and v = x / x˜ = 1 as asserted. 1 m ⊗ ⊕ n−2m k k k k k k

Lemma 2.2. Let P,Q M be non-scalar orthogonal projections. Suppose that there is a support- ∈ n ing line L of W (PQ) satisfying L W (PQ)= µ (λˆ) with λˆ σ(PQ) and λˆ (0, 1). Suppose ∩ { }⊆E ∈ ∈ that µ / (λ) for all other λ σ(PQ) and that PQx,x = µ for some unit vector x. Then there is ∈E ∈ h i a unitary matrix V M with the ﬁrst two columns v , v such that span v , v = span x,PQx , ∈ n 1 2 { 1 2} { } and λˆ λˆ λˆ2 1 0 V ∗PV = − P ′ and V ∗QV = Q′. λˆ λˆ2 p1 λˆ ⊕ 0 0 ⊕ − − ! p Proof. Suppose PQ has the canonical form described in Proposition 1.2, and U is the unitary such that U ∗P QU = C C I 0 , P = P P I P ′ and Q = Q Q I Q′, 1 ⊕···⊕ k ⊕ r ⊕ s 1 ⊕···⊕ k ⊕ r ⊕ 1 ⊕···⊕ k ⊕ r ⊕ where P ′Q′ = 0 and C = P Q for 1 i k. We may further assume that C ,...,C satisfy s i i i ≤ ≤ 1 m W (C )= = W (C )= (λˆ) such that µ / W (C ) for all other j m + 1,...,k . Thus, 1 · · · m E ∈ j ∈ { } c2 0 c2 cs 1 0 C = = C = , P = = P = and Q = = Q = 1 · · · m cs 0 1 · · · m cs s2 1 · · · m 0 0 with λˆ = c2 and s2 = √1 c2. Because L W (PQ)= µ , there is t [0, 2π) such that eitµ+e−itµ¯ − ∩ { } ∈ is the largest eigenvalue of eitPQ+e−itQP ; e.g., see [5, Chapter 1]. Now, set Pˆ +iQˆ = U ∗(P +iQ)U 4 HONGKEDU,CHI-KWONGLI, KUO-ZHONGWANG,YUEQINGWANG, NING ZUO andx ˆ = U ∗x so thatx ˆ∗PˆQˆxˆ = µ. Then T = eitPˆQˆ will satisfy the hypothesis of Lemma 2.1. It follows that xˆ = U ∗x = v x˜ 0 and PˆQˆxˆ = (U ∗P QU)(U ∗x)= v y˜ 0 y,ˆ 0 ⊗ ⊕ n−2m 0 ⊗ ⊕ n−2m ≡ where v Cm with v = 1 andy ˜ = C x˜. For any y span x,ˆ yˆ , we have y = v y 0 0 ∈ k 0k 1 ∈ { } 0 ⊗ 0 ⊕ n−2m for some y span x,˜ y˜ , and then 0 ∈ { } (1) PˆQyˆ = v C y 0 , Pyˆ = v P y 0 , Qyˆ = v Q y 0 . 0 ⊗ 1 0 ⊕ n−2m 0 ⊗ 1 0 ⊕ n−2m 0 ⊗ 1 0 ⊕ n−2m Let V ′ M be a unitary such that the span of the ﬁrst two columns of V ′ contains the set ∈ n x,PQx , and let Vˆ = (ˆv ,..., vˆ ) = U ∗V ′. Then Vˆ is a unitary and span vˆ , vˆ = span x,ˆ yˆ . { } 1 n { 1 2} { } From (1), we obtain that

Vˆ ∗PˆQˆVˆ = C′ C′, Vˆ ∗PˆVˆ = P ′ P ′, and Vˆ ∗QˆVˆ = Q′ Q′, 1 ⊕ 1 ⊕ 1 ⊕ where C′ = C and P ′,Q′ are two 2-by-2 orthogonal projections. Since c2 = λˆ (0, 1) and 1 ∼ 1 1 1 ∈ P ′Q′ = C′ , Q′ = 0 , I . There is a unitary Rˆ M such that 1 1 1 1 6 2 2 ∈ 2 1 0 p p Rˆ∗Q′ Rˆ = and Rˆ∗P ′Rˆ = 11 12 . 1 0 0 1 p p 12 22 p 0 Hence 11 = Rˆ∗C′ Rˆ = C , and then c2 = p , p = eiθcs for some θ [0, 2π). Let p 0 1 ∼ 1 11 12 ∈ 12 1 0 R = Rˆ I , and V = UVRˆ . Then 0 eiθ ⊕ n−2 λˆ λˆ λˆ2 1 0 V ∗PV = R∗Vˆ ∗PˆVRˆ = − P ′ and V ∗QV = R∗Vˆ ∗QˆVRˆ = Q′ λˆ λˆ2 p1 λˆ ⊕ 0 0 ⊕ − − ! as asserted. p

Theorem 2.3. Let A, B M be two non-scalar positive contractions. Then ∈ n W (AB) conv (λ) . ⊆ {∪λ∈σ(AB)E } ′′ The set equality holds if and only if there is a unitary matrix U such that U ∗AU = A′ A ,U ∗BU = ′′ ′′ ′′ ⊕ B′ B such that A′,B′ are orthogonal projections such that W (A B ) W (A′B′)= W (AB). ⊕ ⊆ Proof. Let A √A A2 0 B 0 √B B2 − − Aˆ = √A A2 I A 0 and Bˆ = 0 0 0 .  n    0− − 0 0 √B B2 0 I B − n −     Then AB 0 A√B B2 − T = AˆBˆ = √A A2B 0 (A A2)(B B2)   −0 0− 0 − p   satisﬁes σ(AˆBˆ)= σ(AB) 0 and ∪ { } W (AB) W (T ) = conv (λ) . ⊆ {∪λ∈σ(AˆBˆ)E } NUMERICALRANGESOFTHEPRODUCTOFOPERATORS 5

Now, suppose that W (AB) = conv (λ) = W (T ). Then {∪λ∈σ(AB)E } W (AB) = conv S (λ) , { ∪λ∈σ(AB)\S E } where S = σ(AB) 0, 1 . Obviously, σ(AB) S = if and only if W (AB) [0, 1]. ∩ { } \ ∅ ⊆ If W (AB) [0, 1], then AB is a Hermitian matrix so that AB = B∗A∗ = BA. Hence A ⊆ ∗ ∗ and B commute, and there is a unitary U such that A = U Λ1U and B = U Λ2U, where Λ1 = diag(a1, . . . , an) and Λ2 = diag(b1, . . . , bn). Then W (AB) = W (Λ1Λ2) = [α0, α1], where α0 = min1≤i≤n aibi and α1 = max1≤i≤n aibi. Hence we have the desired conclusion. Next, suppose that W (AB) is not in [0, 1]. This is σ(AB) S = . Let λ σ(AB) S be such \ 6 ∅ 1 ∈ \ that ∂ (λ ) ∂W (AB) contains an arc. Then there exists µ ∂ (λ ) ∂W (AB) with µ / (λ) for E 1 ∩ ∈ E 1 ∩ ∈E all other λ σ(AB). Let x Cn be a unit vector with x∗ABx = µ. Since ∂W (AB)= ∂W (T ), ∈ 1 ∈ 1 1 there is θ [0, 2π) satisfying 2Re(eiθ1 µ) = max σ(eiθ1 T + e−iθ1 T ∗). Letx ˆ = x 0 . Thenx ˆ is 1 ∈ 1 1 ⊕ 2n 1 an eigenvector of eiθ1 T + e−iθ1 T ∗ corresponding to eiθ1 T so that

(eiθ1 AB + e−iθ1 BA)x = 2Re(eiθ1 µ)x , eiθ1 A A2Bx = 0, and e−iθ1 B B2Ax = 0. 1 1 − 1 − 1 Hence T xˆ = ABx 0 . By Lemma 2.2, therep is a unitary Uˆ and Uˆ = Up I such that the 1 1 ⊕ 2n 1 1 1 ⊕ 2n span of the ﬁrst two columns of U contains the set x ,ABx , 1 { 1 1}

∗ λ1 λ1(1 λ1) ∗ 1 0 Uˆ AˆUˆ1 = − Aˆ1 and Uˆ BˆUˆ1 = Bˆ1, 1 λ (1 λ ) 1 λ ⊕ 1 0 0 ⊕ 1 − 1 p − 1 where p ′ ∗ ′ ∗ A1 C1 0 B1 0 D1 Aˆ1 = C1 I A 0 and Bˆ1 = 0 0 0 .  0− 0 0   D 0 I B  1 − Thus,    

∗ λ1 λ1(1 λ1) ′ ∗ 1 0 ′ U AU1 = − A , U BU1 = B , 1 λ (1 λ ) 1 λ ⊕ 1 1 0 0 ⊕ 1 1 − 1 p − 1 p ∗ ′ ′ λ1 0 ′ ′ and U1 ABU1 = C1 A1B1, where C1 = . Then A1,B1 are positive contractions, ⊕ λ1(1 λ1) 0 − Aˆ1, Bˆ1 are orthogonal projections, and p conv (λ) = conv (λ ) (λ) . {∪λ∈σ(AB)\S E } {E 1 ∪λ∈σ(A1B1)\S E } Now, suppose W (AB)= W (T ) = conv S k W (C ) for k distinct matrices C ,...,C M { ∪j=1 j } 1 k ∈ 2 such that for j = 1,...,k, λ σ(AB) S, ∂ (λ ) ∂W (AB) contains an arc, and W (C )= (λ ). j ∈ \ E j ∩ j E j Since the argument in the preceding paragraph is true for any (λ ) for j = 1,...,k, there is an E j orthonormal set v , . . . , v Cn and a unitary V = V V M , where the ﬁrst 2k columns { 1 2k}⊆ 1 ⊕ 2 ∈ 3n of V equals v , . . . , v , such that V ∗TV = C C T . Thus, V ∗AVˆ = A A A˜ , 1 1 2k 1 ⊕···⊕ k ⊕ 0 1 ⊕···⊕ k ⊕ 0 V ∗BVˆ = B B B˜ . Consequently, V ∗AV = A A A , and V ∗BV = B 1 ⊕···⊕ k ⊕ 0 1 1 1 ⊕···⊕ k ⊕ 0 1 ⊕···⊕ B B . k ⊕ 0 Evidently, 0 σ(A B ) 0, 1 S. If1 / S, then W (AB)= W (A B ) so that the conclusion ∈ 1 1 ∩ { }⊆ ∈ 1 1 of the theorem holds with (A′,B′) = (A ,B ). Suppose that 1 S. Then 1 σ(A B ) because 1 1 ∈ ∈ 0 0 σ(C ) = 0, λ with λ (0, 1). Because A ,B are positive contractions, there is a unitary j { j } j ∈ 0 0 6 HONGKEDU,CHI-KWONGLI, KUO-ZHONGWANG,YUEQINGWANG, NING ZUO

U satisfying U ∗A U = [1] A′ and U ∗B U = [1] B′ . Let U = (V (I U )) V . Then 0 0 0 0 ⊕ 0 0 0 0 ⊕ 0 1 2k ⊕ 0 ⊕ 2 (U ∗AU, U ∗BU) = (A′ A′′,B′ A′′) with (A′,B′) = (A [1], A [1]), and the desired conclusion ⊕ ⊕ 1 ⊕ 2 ⊕ follows.

3. Essentially Hermitian matrices

Recall that a matrix A M is an essentially Hermitian matrix if eit(A (tr A)I /n) is Hermitian ∈ n − n for some t [0, 2π). ∈ It is known and not hard to show that the following conditions are equivalent for A M . ∈ n (a) A is essentially Hermitian

(b) W (A) is a line segment in C joining two complex numbers a1, a2. (c) A is normal and all its eigenvalues lie on a straight line. The results in the previous section can be extended to essentially Hermitian matrices. We begin with the following result which follows readily from Proposition 1.2.

Proposition 3.1. Suppose A, B M are normal matrices with σ(A) = a , a and σ(B) = ∈ n { 1 2} b , b . Then A = (a a )P + a I and B = (b b )Q + b I , where P and Q are orthogonal { 1 2} 1 − 2 2 n 1 − 2 2 n projections, and there is a unitary matrix U such that U ∗(P + iQ)U is a direct sum of (I + iI ) p p ⊕ I iI 0 , and q ⊕ r ⊕ s c2 + i c s j j j , j = 1,...,k, c s s2 j j j where c (0, 1),s = 1 c2. Consequently, U ∗ABU is a direct sum of a diagonal matrix D with j ∈ j − j σ(D) a b , a b , a bq, a b and ⊆ { 1 1 1 2 2 1 2 2} a c2 + a s2 (a a )c s b 0 C = 1 j 2 j 1 − 2 j j 1 , j = 1,...,k. j (a a )c s a s2 + a c2 0 b 1 − 2 j j 1 j 2 j 2 Suppose A, B M satisfy the hypotheses of the above proposition. Then ∈ n W (AB) = conv k W (C ) W (D) . {∪j=1 j ∪ } Evidently, W (D) is the convex hull of the diagonal entries of D. Here note that some or all of the entries a b , a b , a b , a b may absent in D. By the result on the numerical range of 2 2 matrix, 1 1 1 2 2 1 2 2 × we have the following proposition.

Proposition 3.2. Let a , a , b , b C with a = a , b = b , c (0, 1), s = √1 c2, and 1 2 1 2 ∈ 1 6 2 1 6 2 ∈ − a c2 + a s2 (a a )cs b 0 C = 1 2 1 2 1 . (a a )cs a s2−+ a c2 0 b 1 − 2 1 2 2 Then W (C) is the elliptical disk (a , a , b , b ; γ) with foci γ γ2 a a b b and length of minor E 1 2 1 2 ± − 1 2 1 2 axis p 2 γˆ 2 + ( b 2 + b 2) a a 2c2s2 2 γˆ2 + b b (a a )2c2s2 1/2, { | | | 1| | 2| | 1 − 2| − | 1 2 1 − 2 |} where γ = tr C = [(a b + a b )c2 + (a b + a b )s2] and γˆ = [(a b a b )c2 + (a b a b )s2]/2. 1 1 2 2 1 2 2 1 1 1 − 2 2 2 1 − 1 2 NUMERICALRANGESOFTHEPRODUCTOFOPERATORS 7

Several remarks in connection to Proposition 3.2 are in order. (1) If (a , a ) = (b , b ) = (1, 0), then (a , a , b , b ; γ)= (γ) deﬁned in Section 2. 1 2 1 2 E 1 2 2 2 E (2) The center of W (C) in Proposition 3.2 always lies in the line segment with end points a b + a b and a b + a b , and these two points are diﬀerent if a = a and b = b . 1 1 2 2 1 2 2 1 1 6 2 1 6 2 (3) Suppose a , a , b , b are given such that a = a , b = b . Every γ in the interior of the 1 2 1 2 1 6 2 1 6 2 line segment with end points a b + a b and a b + a b uniquely determine c (0, 1) and 1 1 2 2 1 2 2 1 ∈ s = √1 c2 so that one can construct the matrix C (based on a , a , b , b , γ) such that − 1 2 1 2 W (C)= (a , a , b , b ; γ). E 1 2 1 2 (4) Let A, B M satisfy the hypothesis of Proposition 3.1. Then for every λ (σ(AB) S) ∈ n ∈ \ with S = σ(AB) a b , a b , a b , a b , there is λ˜ σ(AB) such that λλ˜ = a a b b and ∩ { 1 1 1 2 2 1 2 2} ∈ 1 2 1 2 ˜ 2 2 λ + λ = (a1b1 + a2b2)cj + (a1b2 + a2b1)sj . Such a pair of eigenvalues correspond to the eigenvalues of C . If a a b b = 0, then λ (σ(C ) S) will ensure that λ = 0 so that j 1 2 1 2 ∈ j \ 6 λˆ = 0. Otherwise, λˆ = a1a2b1b2/λ. As a result, we can always assume that λˆ = a1a2b1b2/λ

and γ = λ + a1a2b1b2/λ. By the above remarks and Propositions 3.1, 3.2, we have the following.

Theorem 3.3. Suppose A, B M are non-scalar normal matrices with σ(A) = a , a and ∈ n { 1 2} σ(B)= b , b . Let S = σ(AB) a b , a b , a b , a b . Then { 1 2} ∩ { 1 1 1 2 2 1 2 2} W (AB) = conv (a , a , b , b ; λ + (a a b b )/λ) S . {∪λ∈(σ(AB)\S)E 1 2 1 2 1 2 1 2 ∪ } We can use the dilation technique to study the numerical range of the product of essentially Hermitian matrices. Let A˜ be an essentially Hermitian matrix such that W (A˜) is a line segment joining a , a C. Then A˜ = a I + (a a )A for a positive contraction A. Then A˜ has a dilation 1 2 ∈ 2 n 1 − 2 of the form P˜ = a I + (a a )P with 2 2n 1 − 2 A √A A2 P = − . √A A2 I A − − Then P˜ is normal with σ(P˜)= a , a so that W (P˜)= W (A). Now, if B˜ = b I + (b b )B is { 1 2} 2 n 1 − 2 another essentially Hermitian matrix, then B˜ has a dilation Q˜ = b I + (b b )Q, where 2 2n 1 − 2 B √B B2 Q = − , √B B2 I B − − such that Q˜ is normal with σ(Q˜)= b , b and W (B˜)= W (Q˜). { 1 2} Using this observation and arguments similar to those in the proof of Theorem 2.3, we have the following.

Theorem 3.4. Suppose A, B M are essentially Hermitian matrices such that A = a I + (a ∈ n 2 n 1 − a )A and B = b I + (b b )B for two positive contractions A ,B . Let A˜ = a I + (a a )P 2 1 2 n 1 − 2 1 1 1 2 3n 1 − 2 and B˜ = b I + (b b )Q 2 3n 1 − 2 A A A2 0 B 0 B B2 1 1 − 1 1 1 − 1 P = A A2 I A 0 and Q = 0 0 0 .  1 1 pn 1   p  0− − 0 0 B B2 0 I B p 1 1 n 1    − −  p 8 HONGKEDU,CHI-KWONGLI, KUO-ZHONGWANG,YUEQINGWANG, NING ZUO

Then W (AB) W (A˜B˜), where W (A˜B˜) can be determined by Theorem 3.3. The set equality ⊆ holds if and only if there is a unitary U such that UAU ∗ = A A ,UBU ∗ = B B satisfying 1 ⊕ 2 1 ⊕ 2 σ(A )= a , a , σ(B )= b , b , and W (A B ) W (A B )= W (AB). 1 { 1 2} 1 { 1 2} 2 2 ⊆ 1 1

4. Extension to infinite dimensional spaces

We can extend the results in the previous sections to B(H), where H is infinite dimensional. Note that for a pair of non-scalar orthogonal projections P,Q B(H), there is a unitary U such ∈ that U ∗(P + iQ)U is a direct sum of (1 + i)I I iI 0, and ⊕ ⊕ ⊕ C2 + iI C√I C2 − , C√I C2 I C2 − − where C is a positive contraction; see [1, 7] and their references. Consequently, PQ is a direct sum of I 0 and ⊕ C2 0 T = . C√I C2 0 − Note that T can be approximated by a sequence of operators of the form C2 0 T = m , m = 1, 2,..., m C I C2 0 m − m p 2 where Cm has finite spectrum and therefore can be assumed to be the direct sum of cj IHj on some subspace H for j = 1,...,c2 with c (0, 1). It is easy to see that j km j ∈ W (T ) = conv (λ) . m {∪λ∈σ(Cm)E } In fact, the same conclusion holds if σ(C)= σ(T ) is a discrete set, equivalently, σ(PQ) is a discrete set. In other words, if P,Q B(H) are non-scalar orthogonal projections such that σ(PQ) is a ∈ finite or countably infinite, then W (PQ) = conv (λ) . {∪λ∈σ(PQ)E } This result was proved in [1, Theorem 1.3], for separable Hilbert spaces. One readily sees that the proof works for general Hilbert space. In general, one can approximate T by Tm and obtain the following result concerning the closure of W (PQ); see [1, Theorem 1.2].

Proposition 4.1. Let P,Q B(H) be non-scalar orthogonal projections. Then ∈ W (PQ)= conv (λ) . {∪λ∈σ(PQ)E } The closure signs on both sides can be removed if σ(PQ) is a discrete set.

One can show that the results in Sections 2-3 hold in the inﬁnite dimension setting.

Theorem 4.2. Let A, B B(H) be non-scalar positive semi-deﬁnite contractions. Then ∈ W (AB) conv (λ) . ⊆ {∪λ∈σ(AB)E } The closure signs can be removed if σ(AB) is a discrete set. The set equality holds if A, B are orthogonal projections. NUMERICALRANGESOFTHEPRODUCTOFOPERATORS 9

Theorem 4.3. Let A, B B(H) be such that W (A) is the line segment joining a , a , and W (B) ∈ 1 2 is the line segment joining b , b , where a = a and b = b . Then 1 2 1 6 2 1 6 2 W (AB) conv (a , a , b , b ; λ + λˆ) S , ⊆ {∪λ∈(σ(AB)\S)E 1 2 1 2 ∪ } where S = σ(AB) a b , a b , a b , a b , λˆ = a a b b /λ, and (a , a , b , b ; λ + λˆ) is deﬁned ∩ { 1 1 1 2 2 1 2 2} 1 2 1 2 E 1 2 1 2 as in Theorem 3.3. The closure signs can be removed if σ(AB) is a discrete set. The set equality holds if σ(A)= a , a and σ(B)= b , b . { 1 2} { 1 2} In Theorems 4.2 and 4.3, we only have suﬃcient conditions for the set inclusions become set equalities. The problems of characterizing the set equality cases are open.

Acknowledgment The authors would like to thank Professor Ngai-Ching Wong for catalyzing the collaboration. The research of Li was supported by the USA NSF grant DMS 1331021, the Simons Foundation Grant 351047, and the NNSF of China Grant 11571220. The research of K.Z. Wang was supported by the Ministry of Science and Technology of the Republic of China under project MOST 104-2918-I-009-001. This research of Y. Wang and N. Zuo was partially supported by the Schol- arship Program of Chongqing University of Science & Technology, the Doctoral Research Fund of Chongqing University of Science & Technology (CK2010B09) and the National Natural Science Foundation of China (11571211).

References

[1] H. Klaja, The numerical range and the spectrum of a product of two orthogonal projections, Journal of Mathematical Analysis and Applications, 411 (2014), 177–195. [2] Y.Q. Wang, H.K. Du, A characterization of maximum norms of commutators of positive contractions, Journal of Mathematical Analysis and Applications, 348(2) (2008), 990–995. [3] T. Ogasawara, A Theorem on operator algebras, J. Sci. Hiroshima Univ. Ser. A., 18(1955), 307–309. [4] K. Gustafson and D.K.M. Rao, Numerical range. The ﬁeld of values of linear operators and matrices, New York: Springer; 1997. [5] R.A. Horn and C.R. Johnson, Topics in matrix analysis, Cambridge: Cambridge University Press; 1991. [6] P.R. Halmos, A Hilbert space problem book, 2nd ed., New York: Springer; 1982. [7] I. Spitkovsky, On polynomials of two projections, Electronic Journal of Linear Algebra, Volume 15, pp. 154-158.

(H. Du) College of Mathematics and Information Science, Shaanxi Normal University, Xian 710062, China. [email protected] (C.K. Li) Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, USA. [email protected]. (K.Z. Wang) Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan. [email protected]. (Y. Wang) Department of Mathematics and Physics, Chongqing University of Science and Tech- nology, Chongqing 401331, China. [email protected] (N. Zuo) Department of Mathematics and Physics, Chongqing University of Science and Tech- nology, Chongqing 401331, China. [email protected]