A note on polynomial convexity of the union of finitely many totally-real planes in C2

J. Math. Anal. Appl. 418 (2014) 842–851

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

A note on polynomial convexity of the union of finitely many totally-real planes in C2 ✩ Sushil Gorai Statistics and Mathematics Unit, Indian Statistical Institute Bangalore, R.V.C.E. Post, Bangalore 560 059, India

a r t i c l e

i n f o

Article history: Received 20 May 2013 Available online 13 March 2014 Submitted by E.J. Straube Keywords: Polynomial convexity Totally real Simultaneous triangularization

a b s t r a c t In this paper we discuss local polynomial convexity at the origin of the union of finitely many totally-real planes through 0 ∈ C2 . The planes, say P0 , . . . , PN , satisfy a mild transversality condition that enables us to view them in Weinstock’s normal form, i.e., P0 = R2 and Pj = M (Aj ) := (Aj + iI)R2 , j = 1, . . . , N , where each Aj is a 2 × 2 matrix with real entries. Weinstock has solved the problem completely for pairs of transverse, maximally totally-real subspaces in Cn ∀n  2. Using a characterization of simultaneous triangularizability of 2 × 2 matrices over the reals, given by Florentino, we deduce a sufficient condition for local polynomial convexity of the union of the above planes at 0 ∈ C2 . Weinstock’s theorem for C2 occurs as a special case of our result. © 2014 Published by Elsevier Inc.

1. Introduction and statement of results  := {z ∈ Cn : |p(z)|  For a compact subset K of Cn the polynomially convex hull of K is defined by K  supK |p|, p ∈ C[z1 , . . . , zn ]}. We say that K is polynomially convex if K = K. A closed subset E of Cn is called locally polynomially convex at p ∈ E if E ∩ B(p; r) is polynomially convex for some r > 0 (here, B(p; r) denotes the open ball in Cn with centre p and radius r). In general, it is very difficult to determine whether a given compact subset of Cn , n  2, is polynomially convex. However, we know that any compact subset of a totally-real subspace of Cn is polynomially convex. It is also true that the uniform algebra on K generated by holomorphic polynomials coincides with C(K), the algebra of all continuous functions on K, where K is a compact subset of a totally-real subspace of Cn . The natural next step would be to consider compact subsets of unions of totally-real subspaces in Cn . An approximation result by O’Farrell, Preskenis and Walsh [5] (see also [1, Theorem 3]) says that the uniform algebra on a compact subset K of a union of totally-real subspaces generated by holomorphic polynomials coincides with C(K) provided the compact K is polynomially convex. Hence, this gives us a motivation to study polynomial convexity of compact subsets ✩

This work is supported in part by an INSPIRE Faculty Award (IFA-MA-02) and by a CSIR-UGC fellowship (09/079(2063)). E-mail address: [email protected].

http://dx.doi.org/10.1016/j.jmaa.2014.03.011 0022-247X/© 2014 Published by Elsevier Inc.

S. Gorai / J. Math. Anal. Appl. 418 (2014) 842–851

843

of unions of totally-real subspaces. It is not hard to see that the problem of determining the polynomial convexity of a connected compact subset of a union of totally-real subspaces in Cn reduces to understanding the local polynomial convexity of such a union at 0 ∈ Cn . Weinstock [10] studied the local polynomial convexity at 0 ∈ Cn of the union of two maximally totally-real subspaces of Cn intersecting transversely at 0 ∈ C2 . In view of the above mentioned approximation result, it is a natural question to ask what happens if we consider the union of finitely many totally-real subspaces in Cn intersecting transversely at the origin. In C2 , Thomas [9] gave an example of a one-parameter family of triples of totally-real planes, intersecting at 0 ∈ C2 , showing that local polynomial convexity of each pairwise union at the origin does not imply the local polynomial convexity of the union at the origin. On the other hand, Thomas [8, Proposition 10] also demonstrated examples of N -tuples of totally-real planes containing 0 ∈ C2 , for each N  2, whose union is locally polynomially convex at the origin. Also see [1] for a discussion about local polynomial convexity at 0 ∈ C2 of countable and uncountable unions of totally-real planes. In a recent work [3], a sufficient condition for the local polynomial convexity at 0 ∈ C2 of a union of three totally-real planes, that is an open condition in the space of triples of totally-real planes containing 0 ∈ C2 was demonstrated (also see the remarks at the end of this section). In this paper, we shall study N -tuples of totally-real planes, N  2, intersecting at 0 ∈ C2 that satisfy a certain overarching condition. Under this condition, we shall give a complete characterization of local polynomial convexity at the origin of unions of finitely many totally-real planes intersecting at 0 ∈ C2 . This paper is motivated by the methods introduced in [10]. We now state Weinstock’s result. Let us consider P0 ∪ P1 , where P0 and P1 are two transverse totally-real n-dimensional subspaces of Cn . Applying a C-linear change of coordinate, we can assume that P0 = Rn . The same change of coordinate gives us P1 = (A + iI)Rn for some A ∈ Rn×n (see [10] for details). Result 1.1 (Weinstock). Let P0 and P1 be two totally-real subspaces of Cn of maximal dimension intersecting only at 0 ∈ Cn . Denote the normal form for this pair as: P0 : Rn , P1 : (A + iI)Rn . P0 ∪ P1 is locally polynomially convex at the origin if and only if A has no purely imaginary eigenvalue of modulus greater than 1. It turns out that many of Weinstock’s ideas in [10] are the “right” ones to follow when one considers the union of finitely many totally-real subspaces containing 0 ∈ C2 . One of the ideas is to focus on a certain normal form for the given union of subspaces. Consider a finite collection of maximally totally-real subspaces P0 , P1 , . . . , PN of C2 , satisfying P0 ∩ Pj = {0}, j = 1, . . . , N . By exactly the same arguments as in [10], we can find a C-linear change of coordinate under which we get Weinstock’s normal form for P0 , P1 , . . . , PN : P0 : R2 Pj : M (Aj ) = (Aj + iI)R2 ,

j = 1, . . . , N,

(1.1)

where Aj ∈ R2×2 , j = 1, . . . , N . The crux of Weinstock’s proof is that when we transform Rn ∪ M (A) by a matrix S ∈ GL(n, R) acting as a C-linear map, M (A) transforms to M (SAS −1 ). This allows Weinstock to reduce his analysis to the case when A is in a sparse canonical form (in fact, the Jordan normal form when A has real eigenvalues). Following this idea, we may examine what happens if {A1 , . . . , AN }, as given in (1.1), can be simultaneously conjugated over R to, say, upper triangular matrices. This idea is the basis of the hypothesis in our theorem below. In order to state our theorem we need the following definition.

844

S. Gorai / J. Math. Anal. Appl. 418 (2014) 842–851

Definition 1.2. (See Florentino, [2].) A matrix sequence A = {A1 , . . . , AN }, Aj ∈ Rn×n , is said to be reduced if there are no commuting pairs among its terms, i.e. Aj Ak −Ak Aj = 0 for all 1  j < k  N . A subsequence B ⊂ A is called a reduction of A if B is reduced and is obtained from A by deleting some of its terms. The reduced length of A is the greatest k ∈ Z+ such that there exists a reduction B ⊆ A of cardinality k. We now state our result. We shall denote AB − BA as [A, B]. Theorem 1.3. Let P0 , . . . , PN be distinct totally-real planes in C2 containing the origin. Assume (1) P0 ∩ Pj = {(0, 0)} for all j = 1, 2, . . . , N . Hence, let Weinstock’s normal form for {P0 , . . . , PN } be P 0 : R2 Pj : M (Aj ) = (Aj + iI)R2 ,

j = 1, . . . , N,

where Aj ∈ R2×2 . Let L be the reduced length of {A1 , . . . , AN }. Assume further that for some maximal reduced subset B ⊂ {A1 , . . . , AN }, (2) det[Aj , Ak ] = 0, j = k, 1  j, k  N , and, additionally, T r(ABC − CBA) = 0 ∀A, B, C ∈ B if L = 3. Under these conditions: N (a) If each Aj has only real eigenvalues, then j=0 Pj is polynomially convex at the origin. (b) If there exists a j, 1  j  N , such that Aj has non-real eigenvalues, then the spectrum of Ak is of the form {λk , λk } ∀k = 1, . . . , N . Write λk = sk + itk , Vk = (s2k + t2k − 1, 2sk ), k = 1, . . . , N , and N V0 = (1, 0). Then, j=0 Pj is locally polynomially convex at the origin if and only if there exists no pair (l, m), l = m, 0  l, m  N , satisfying Vl = cVm for some constant c > 0. It turns out that known methods become tractable if the Aj ’s in the above theorem are upper-triangular. Therefore, in view of our earlier discussion, the requirements of proving the above theorem are: • {A1 , . . . , AN } is simultaneously triangularizable; • The common conjugator must be a matrix with real entries. These requirements are met by a result of Florentino (see Result 2.4) which gives a necessary and sufficient condition for simultaneous triangularizability over R. Our condition (2) above is precisely Florentino’s criterion. Let us make some remarks before going in the technicalities of the proof. Remark 1.4. We note that under condition (2) of the above theorem, local polynomial convexity of pairwise n unions of P0 , . . . , PN at the origin implies local polynomial convexity of j=0 Pj at 0 ∈ C2 . Remark 1.5. Condition (2) in Theorem 1.3 describes the pairs of matrices that lie in a real algebraic variety in R2×2 ×R2×2 , namely {(A, B) ∈ R2×2 ×R2×2 : det(AB −BA) = 0}. This condition is not strict; restricting to N = 2, i.e., when the number of planes is three, an open set of triples of totally-real planes whose union is locally polynomially convex at 0 ∈ C2 is given in [3] (see Section 4 for an example of a one parameter family of such triples of totally-real planes).

S. Gorai / J. Math. Anal. Appl. 418 (2014) 842–851

845

Remark 1.6. Theorem 1.3 subsumes a result of P. Thomas [8, Proposition 10] which states that, for each N ∈ N, there is an N -tuple of totally-real planes passing through 0 ∈ C2 such that the union of planes is locally polynomially convex at the origin. This work is a part of the author’s doctoral thesis and Theorem 1.3 formed a part of a preprint, which included later developments that is now the article [3]. Because of the thematic differences between Theorem 1.3 and the objectives of the main theorem of [3], the former is being presented as a separate article. 2. Technical preliminaries Let us begin by stating a lemma from Weinstock’s paper [10] that allows us to conjugate the matrices arising from Weinstock’s normal form by real nonsingular matrices. Lemma 2.1. Let T be an invertible linear operator on Cn whose matrix representation with respect to the standard basis is an n × n matrix with real entries. Then T maps M (A) ∪ Rn onto M (T AT −1 ) ∪ Rn . Next, we state two lemmas from the literature that we shall use in the proof of Theorem 1.3. The first, by Kallin [4] (also see [1]), deals with the polynomial convexity of the union of two polynomially convex sets. The second, which is a version of a lemma from Stolzenberg’s paper [6, Lemma 5] (also see Stout’s book [7]), gives a criterion for polynomial convexity of a compact K in terms of the existence of a special function in the uniform algebra on K generated by the holomorphic polynomials. Lemma 2.2 (Kallin). Let K1 and K2 be two compact polynomially convex subsets in Cn . Suppose L1 and L2 are two compact polynomially convex subsets of C with L1 ∩ L2 = {0}. Suppose further that there exists a holomorphic polynomial P satisfying the following conditions: (i) P (K1 ) ⊂ L1 and P (K2 ) ⊂ L2 ; and (ii) P −1 {0} ∩ (K1 ∪ K2 ) is polynomially convex. Then K1 ∪ K2 is polynomially convex. For given compact subset K of Cn , P(K) will denote the uniform algebra on K generated by holomorphic polynomials. Lemma 2.3 (Stolzenberg). Let X ⊂ Cn be compact. Assume P(X) contains a function f such that f (X) has empty interior and C \ f (X) is connected. Then, X is polynomially convex if and only if f −1 {w} ∩ X is polynomially convex for each w ∈ f (X). The version of Lemma 2.3 that we stated above originates in a remark following [7, Theorem 1.2.16] in Stout’s book. The next result, due to Florentino [2], concerns the simultaneous triangularizability of a family of matrices over the field of real numbers. Though the following theorem is actually valid over any integral domain, we shall state it over the field of real numbers, which is the field relevant to our proof. Result 2.4. (See Florentino, [2].) Let A = (A1 , . . . , An ), Aj ∈ R2×2 , have reduced length l  n. Let A ⊂ A be a maximal reduction and, without loss of generality, let A = (A1 , . . . , Al ). Then: (i) If l = 3, then A is triangularizable if and only if each Ak is triangularizable, det[Aj , Ak ] = 0, j, k  l, and T r(ABC − CBA) = 0 for all A, B, C ∈ A .

S. Gorai / J. Math. Anal. Appl. 418 (2014) 842–851

846

(ii) If l = 3, then A is triangularizable if and only if each Ak is triangularizable and det[Aj , Ak ] = 0, j, k  l. We must clarify that, in the above result, the expression “Ak is triangularizable” means that Ak is similar to a real upper triangular matrix by conjugation with a real invertible matrix. Likewise, the expression “A is triangularizable” means that each member of A is triangularizable by conjugation by the same matrix. We refer the reader to Definition 1.2 for the definitions of reduction and reduced length. We can now appreciate the complexity of condition (2) in Theorem 1.3; the latter half of this condition is inherited from part (i) of Result 2.4. The case l = 3 is genuinely exceptional. Florentino shows in [2, Example 2.11] that the condition T r(A1 A2 A3 − A3 A2 A1 ) = 0 cannot, in general, be dropped. We shall need some lemmas from linear algebra. The first of these is basic. Lemma 2.5. Let A ∈ R2×2 and suppose A has non-real eigenvalues p ± iq. Then, there exists S ∈ GL(2, R) such that   p −q −1 S AS = . q p Lemma 2.6. Let A1 , . . . , AN ∈ R2×2 and assume that det[Aj , Ak ] = 0 ∀j = k. Assume also that A1 has non-real eigenvalues.Then, there exists S ∈ GL(2, R) such that   sj −tj −1 , j = 1, . . . , N, S Aj S = tj sj where sj , tj ∈ R, j = 1, . . . , N . Proof. Since A1 has non-real eigenvalues, by Lemma 2.5, there exists S ∈ GL(2, R) such that S −1 A1 S =  p −q    −1 Aj S = ac db for some j: q p , where p ± iq (q = 0) are the eigenvalues of A1 . Suppose S 1  j  N . Since the determinant remains invariant under conjugation by an invertible matrix, by hypothesis, det[S −1 A1 S, S −1 Aj S] = 0. A simple calculation gives us 

S

−1

A1 S, S

−1



Aj S =



−q(b + c) q(a − d)

q(a − d) q(b + c)

 .

Since the determinant of the latter is 0, from the fact that q = 0, we infer that b = −c, a = d. Thus, we  s −t  have S −1 Aj S = tjj sjj , where sj , tj ∈ R, j = 1, . . . , N . 2 Lemma 2.7. Let A1 , A2 ∈ R2×2 be two matrices such that det[A1 , A2 ] = 0 and A1 − A2 is invertible. Suppose A1 has non-real eigenvalues. Then • A2 either has non-real eigenvalues or is a scalar matrix; and • B := (A1 A2 + I)(A1 − A2 )−1 has complex conjugate eigenvalues. Proof. Since A1 has non-real eigenvalues and det[A1 , A2 ] = 0, appealing Lemma 2.6, we see that there exists S ∈ GL(2, R) such that   sj −tj , j = 1, 2. S −1 Aj S = tj sj   This shows that A2 either has non-real eigenvalues or is a scalar matrix. Conjugating by the matrix 1i 1i ,   we see that Aj ∼ λ0j λ0 , j = 1, 2. Hence, (A1 − A2 ) and (A1 A2 + I) can be conjugated by S to diagonal j

S. Gorai / J. Math. Anal. Appl. 418 (2014) 842–851

847

matrices with diagonal entries (λ1 − λ2 , λ1 − λ2 ) and (λ1 λ2 + 1, λ1 λ2 + 1) respectively. Hence, by examining S −1 BS, we see that the matrix B = (A1 A2 + I)(A1 − A2 )−1 has complex conjugate eigenvalues. 2 3. The proof of Theorem 1.3 We remind the reader that, in this section, the word “triangularizable” will refer to triangularization by a real matrix. Proof of Theorem 1.3. (a) Since all Aj , j = 1, . . . , N , have real eigenvalues, each Aj , j = 1, . . . , N , is triangularizable. We now appeal to Result 2.4 to get:  Aj ∼

μj 0

aj νj

 ,

μj , νj , aj ∈ R, j = 1, . . . , N,

by the conjugation with a common matrix S ∈ GL(2, R). In view of Lemma 2.1, we may assume that  Aj =

μj 0

aj νj

 ,

μj , νj , aj ∈ R, j = 1, . . . , N.

From this point, we will — for simplicity of notation — refer to each S(Pj ) as Pj , j = 0, . . . , N . Hence, Pj = M (Aj ) for the preceding choice of Aj , j = 1, . . . , N . We have M (Aj ) =



  (μj + i)x + aj y, (νj + i)y : x, y ∈ R ,

for all j = 1, . . . , N.

N Let K := ( j=0 Pj ) ∩ B(0, 1) and Kj := Pj ∩ K, j = 0, . . . , N . Since, for every j = 0, . . . , N , Kj is a compact subset of a totally real plane, Kj is polynomially convex. We shall use Lemma 2.3 to show the polynomial N convexity of K = j=1 Kj . Consider the polynomial F (z, w) = w. Clearly, there exists a real number R > 0 such that:



N



F (K) ⊂

(νj + i)y: y ∈ R, |y|  R



 ∪ y: y ∈ R, |y|  R .

(3.1)

j=1

Since each of the members in the union of the right hand side of (3.1) is a bounded real line segment in C, F (K) has no interior and C \ F (K) is connected. We shall now calculate F −1 {ζ} ∩ K, where ζ ∈ F (K). If ζ = 0, then ζ ∈ F (Kj 0 ) for some j 0  N . Hence, we get F −1 {ζ} ∩ K =



{((μj 0 + i)x + aj 0 ζ/(νj 0 + i), ζ): x ∈ R} ∩ K,

if 1  j 0  N,

{(x, ζ) ∈ C2 : x ∈ R} ∩ K,

if j 0 = 0.

(3.2)

We remark that ζ ∈ F (Kj 0 ) implies that ζ/(νj 0 + i) ∈ R. Also, νj 0 + i = 0 because, in the present case, νj 0 ∈ R. If ζ = 0, then we have  F

−1

{0} ∩ K =

N

   (μj + i)x, 0 ∈ C2 : x ∈ R







∪ (x, 0) ∈ C : x ∈ R 2

 ∩K

(3.3)

j=1

In view of (3.2), we have the set F −1 {ζ} ∩ K is a single line segment in C2 when ζ = 0. Hence, 

 F −1 {ζ} ∩ K = F −1 {ζ} ∩ K,

for ζ = 0.

(3.4)

S. Gorai / J. Math. Anal. Appl. 418 (2014) 842–851

848

From (3.3), we see that F −1 {0} ∩ K is a union of line segments in Cz × {0} intersecting only at the origin. Hence, 

 F −1 {0} ∩ K = F −1 {0} ∩ K.

(3.5)

Thus, in view of (3.4) and (3.5), we can appeal to Lemma 2.3 to get the polynomial convexity of K. (b) Without loss of generality, assume that A1 has non-real complex eigenvalues. From Lemma 2.6, it follows that there exists S ∈ GL(2, R) such that S −1 Aj S =



sj tj

−tj sj

 ,

sj , tj ∈ R, j = 1, . . . , N.

Again, by Lemma 2.1, there is no loss of generality to assume   sj −tj , sj , tj ∈ R j = 1, . . . , N. Aj = tj sj We will relabel S(Pj ) as Pj , j = 0, . . . , N , exactly as in (a). Then M (Aj ) =



  (sj + i)x − tj y, tj x + (sj + i)y : x, y ∈ R ,

j = 1, . . . , N.

Note that, in this case, det(Aj − Ak ) = (sj − sk )2 + (tj − tk )2 > 0,

(3.6)

and consequently, M (Aj ) ∩ M (Ak ) = {0} for j = k. Suppose there does not exist any constant c > 0 such that Vj = cVk ,

for some j = k, 0  j, k  N.

N N We shall show that K := ( j=0 Pj ) ∩ B(0, 1) = j=0 Kj is polynomially convex, where Kj := Pj ∩ K, 0  j  N . Each Kj is necessarily polynomially convex. In this case we shall use Kallin’s lemma (i.e. Lemma 2.2) to show that K is polynomially convex. Let us consider the polynomial F (z, w) := z 2 + w2 . We shall now look at the image of Kj under this map: F (K0 ) ⊂ {z ∈ C: z  0} = {αV0 : α  0},   2  2 F (sj + i)x − tj y, tj x + (sj + i)y = (sj + i)x − tj y + tj x + (sj + i)y      = s2j + t2j − 1 x2 + y 2 + 2isj x2 + y 2 . 

(3.7)

(3.8)

Hence,

 F (Kj ) ⊂ βVj ∈ R2 : β  0 ,

j = 1, . . . , N.

(3.9)

Since there does not exist any constant c > 0 such that Vj = cVk , for some j = k, 0  j, k  N , we have by Eqs. (3.7) and (3.9): F (Kl ) ∩ F (Km ) = {0}

for l = m.

(3.10)

Furthermore, F −1 {0} ∩ Kj = {0} for all j = 0, . . . , N.

(3.11)

S. Gorai / J. Math. Anal. Appl. 418 (2014) 842–851

849

From (3.10), (3.11), it follows that: • for each j, F (Kj ) lies in different line segment of C, each of which has an end at 0 ∈ C2 ; and • F −1 {0} ∩ K = {0}, which is polynomially convex. Since each F (Kj ), j = 0, . . . , N , is polynomially convex, the above shows that all the conditions of Kallin’s N N lemma are satisfied. Hence, K = j=1 Kj is polynomially convex; i.e. j=0 Pj is locally polynomially convex at the origin. We will prove the converse in its contrapositive formulation. Let there exist two numbers l, m such that l = m and for some constant c > 0, Vl = cVm . This implies det(Al + iI) = c.det(Am + iI).

(3.12)

Without loss of generality, let us assume that l = 2 and m = 1. From (3.12), we get    det (A1 − iI)(A2 + iI) = c.det A21 + I > 0.

(3.13)

Note that if we view (A1 − iI) as a C-linear transformation on C2 , then     (A1 − iI) M (A1 ) = A21 + I R2 = R2    (A1 − iI) M (A2 ) = A1 A2 + I + i(A1 − A2 ) R2  = (A1 A2 + I)(A1 − A2 )−1 + iI R2 ≡ (B + iI)R2 . The first equality follows from the fact that (A21 + I) is invertible (because of (3.13) and the fact that M (A1 ) is totally-real) and the invertibility of (A1 − A2 ) follows from (3.6). Now, from Lemma 2.7, we get that B has complex conjugate eigenvalues. We now write σ(B) = {μ, μ}, and μ = s + it, s, t ∈ R. From (3.6) and (3.13), we get:   det(B + iI) = s2 + t2 − 1 + 2is > 0. This implies s = 0 and |t| > 1. From an auxiliary result of Weinstock [10, Theorem 2], it follows that N R2 ∪ M (B) = (A1 − iI)(M (A1 ) ∪ M (A2 )) is not locally polynomially convex at (0, 0). Equivalently, j=0 Pj is not locally polynomially convex at (0, 0). 2 4. An example In this section we provide a one parameter family {(At1 , At2 ) ∈ R2×2 × R2×2 , t ∈ R \ {0}} such that i) for each t, the pair (At1 , At2 ) does not satisfy condition (2) in Theorem 1.3, i.e., det[At1 , At2 ] = 0; ii) for each t, P0 ∪ P1t ∪ P2t is locally polynomially convex at 0 ∈ C2 , where P0 = R2 , Pjt = M (Atj ), j = 1, 2. For that we need the following result [3, Corollary 1.3]: Result 4.1. Let P0 , P1 , and P2 be three totally-real planes containing 0 ∈ C2 . Assume P0 ∩ Pj = {0} for j = 1, 2. Hence, let Weinstock’s normal form for {P0 , P1 , P2 } be P0 : R2 Pj : M (Aj ) = (Aj + iI)R2 ,

j = 1, 2,

850

S. Gorai / J. Math. Anal. Appl. 418 (2014) 842–851

where Aj ∈ R2×2 . Let the pairwise unions of P0 , P1 , and P2 be locally polynomially convex at 0 ∈ C2 . Then P0 ∪ P1 ∪ P2 is locally polynomially convex at 0 ∈ C2 if either one of the following conditions holds: (i) det[A1 , A2 ] > 0 and detAj > 0, j = 1, 2, (ii) det[A1 , A2 ] < 0 and detAj < 0, j = 1, 2.  0  t  2t t  Example 4.2. For each t ∈ R \ {0}, we consider At1 = 2t , A2 = t 2t . The planes corresponding to the 0 4t one parameter family (At1 , At2 ) of pair of 2 × 2 real matrices are P 0 : R2     Pjt : M Atj = Atj + iI R2 ,

j = 1, 2.

Clearly each of the above planes is totally real, and each triple {P0 , P1t , P2t } is already in Weinstock’s normal form. We note that, for each t ∈ R \ {0}, both At1 and At2 have real eigenvalues. Hence, by Weinstock’s result (Result 1.1), R2 ∪ M (Atj ), j = 1, 2, is locally polynomially convex at 0 ∈ C2 . Viewing (At1 − iI) as a C-linear transformation, we observe that     2  At1 − iI M At1 = At1 + I R2 = R2       t A1 − iI M At2 = B t + iI R2 , 

where B t := (At1 At2 + I)(At1 − At2 )−1 . Then an easy computation shows us that, with the matrices At1 and At2 as above, B t has no purely imaginary eigenvalue. Again, by Result 1.1, R2 ∪ M (B t ) is locally polynomially convex at 0 ∈ C2 . Hence, for each t = 0, the pairwise unions of P0 , P1t , and P2t are locally polynomially convex at the origin. We now use Result 4.1 to show that P0 ∪ P1t ∪ P2t is locally polynomially convex at 0 ∈ C2 . We now compute: det[A1 , A2 ] = 4t4 > 0 detAt1 = 8t2 > 0,

detAt2 = 3t2 > 0.

Hence, by Result 4.1, P0 ∪ P1t ∪ P2t is locally polynomially convex at the origin. Since det[At1 , At2 ] > 0 for t = 0, (At1 , At2 ) does not satisfy condition (2) of Theorem 1.3. Acknowledgment I would like to thank Gautam Bharali for many helpful discussions during the course of this work. References [1] P.J. de Paepe, Eva Kallin’s lemma on polynomial convexity, Bull. Lond. Math. Soc. 33 (1) (2001) 1–10. [2] C.A.A. Florentino, Simultaneous similarity and triangularization of sets of 2 by 2 matrices, Linear Algebra Appl. 431 (9) (2009) 1652–1674. [3] S. Gorai, On the polynomial convexity of the union of three totally-real planes in C2 , Int. Math. Res. Not. IMRN 2013 (21) (2013) 4985–5001. [4] E. Kallin, Fat polynomially convex sets, in: Function Algebras, Proc. Internat. Sympos. on Function Algebras, Tulane University, 1965, Scott Foresman, Chicago, IL, 1966, pp. 149–152. [5] A.G. O’Farrell, K.J. Preskenis, D. Walsh, Holomorphic approximation in Lipschitz norms, Contemp. Math. 32 (1984) 187–194. [6] G. Stolzenberg, Uniform approximation on smooth curves, Acta Math. 115 (1966) 185–198.

S. Gorai / J. Math. Anal. Appl. 418 (2014) 842–851

851

[7] E.L. Stout, Polynomial Convexity, Birkhäuser, Boston, 2007. [8] Pascal J. Thomas, Enveloppes polynomiales d’unions de plans réels dans Cn , Ann. Inst. Fourier (Grenoble) 40 (2) (1990) 371–390. [9] Pascal J. Thomas, Unions minimales de n-plans réels d’enveloppe égale à Cn , in: Several Complex Variables and Complex Geometry, Part 1, Santa Cruz, CA, 1989, in: Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 233–244. [10] B.M. Weinstock, On the polynomial convexity of the union of two totally-real subspaces of Cn , Math. Ann. 282 (1) (1988) 131–138.