WHICH 2-HYPONORMAL 2-VARIABLE WEIGHTED SHIFTS ARE SUBNORMAL?

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1 WHICH -HYPONORMAL -VARIABLE WEIGHTED SHIFTS ARE SUBNORMAL? RAÚL E. CURTO, SANG HOON LEE, AND JASANG YOON Astract. It is well known that a -hyponormal unilateral weighted shift with two equal weights must e flat, and therefore sunormal. By contrast, a -hyponormal -variale weighted shift which is oth horizontally flat and vertically flat need not e sunormal. In this paper we identify a large class S of flat -variale weighted shifts for which -hyponormality is equivalent to sunormality. OnemeasureofthesizeofS is given y the fact that within S there are hyponormal shifts which are not sunormal. 1. Statement of the Main Results The Lifting Prolem for Commuting Sunormals (LPCS) asks for necessary and sufficient conditions for a pair of sunormal operators on Hilert space to admit commuting normal extensions. It is well known that the commutativity of the pair is necessary ut not sufficient ([1], [18], [19], [0]), and it has recently een shown that the joint hyponormality of the pair is necessary ut not sufficient [13]. Our previous work ([9], [10], [13], [14], [15], [4], [5]) has revealed that the nontrivial aspects of LPCS are est detected within the class H 0 of commuting pairs of sunormal operators; we thus focus our attention on this class. The class of sunormal pairs on Hilert space will e denoted y H, and for an integer k 1theclassofk-hyponormal pairs in H 0 will e denoted y H k. Clearly, H H k H H 1 H 0 ; the main results in [13] and [9] show that these inclusions are all proper. It is then natural to look for suclasses of H 0 on which sunormality and k-hyponormality agree, that is, classes on which sunormality can e detected with a matricial test. In this paper we identify a large class S H 0 on which -hyponormality and sunormality agree, that is, S H = S H. Concretely, S consists of all -variale weighted shifts T (T 1,T ) H 0 such that α (k1,0) = α (k1 +1,0) and β (0,k ) = β (0,k +1) for some k 1 1andk 1, where α and β denote the weight sequences of T 1 and T, respectively. One measure of the size of S is given y the fact that hyponormality and sunormality do not agree on S, thatiss H S H 1. Thus, S consists of nontrivial shifts for which -hyponormality and sunormality are equivalent, ut for which hyponormality and -hyponormality are different; that is, S is small enough to ensure that - hyponormality implies sunormality, ut large enough to separate hyponormality from sunormality. Each hyponormal shift in S is flat, thatis,α (k1,k ) = α (1,1) and β (k1,k ) = β (1,1) for all k 1,k 1. As a result, each hyponormal shift in S elongs to the class TCof shifts whose core is of tensor form (cf. Definition.8); TC is a class that we have studied in detail in [11]. Previously, in [10] we had proved that there exist -variale weighted shifts in TC which are hyponormal ut not sunormal. By contrast, the -hyponormal shifts in S are automatically sunormal. We prove our main results y comining the 15-point Test for -hyponormality [9] with the Sunormal Backward Extension Criterion [13] and Smul jan s Test for positivity of operator matrices 000 Mathematics Suject Classification. Primary 47B0, 47B37, 47A13; Secondary 44A60, 47-04, 47A0, 8A50. Key words and phrases. Jointly hyponormal pairs, -hyponormal pairs, sunormal pairs, -variale weighted shifts, flatness. Research partially supported y NSF Grants DMS and DMS

2 [1]. As a first step, we prove a propagation result for -hyponormal -variale weighted shifts (Theorem.10). We then seek conditions to guarantee the sunormality of T (T 1,T ) H. Our main results follow; first, we need a definition. Definition 1.1. S := {T (T 1,T ) H 0 : α (k1,0) = α (k1 +1,0) and β (0,k ) = β (0,k +1) for some k 1 1 and k 1}. (Here α and β denote the weight sequences of T 1 and T, respectively; cf. Section ). Theorem 1.. S H = S H. The generic form of the weight diagram of a hyponormal -variale weighted shift in S is given in Figure (ii). Using that notation, we can sharpen Theorem 1. as follows. Theorem 1.3. Let T (T 1,T ) S, with weight diagram given y Figure (ii), and assume that T is -hyponormal. Then T is sunormal, with Berger measure given as Theorem 1.4. S H S H 1. µ = 1 {[ (1 x ) y (1 a )]δ (0,0) +y (1 a )δ (0, ) +( x a y )δ (1,0) + a y δ (1, )}. Remark 1.5. Hyponormality alone does not imply flatness. While it is true that in the presence of hyponormality the Six-point Test creates L-shaped propagation (i.e., α k+ε1 = α k = α k = α k+ε and β k = β k+ε1 ; cf [15, Proof of Theorem 3.3]), without horizontal propagation (as guaranteed y the quadratic hyponormality of T 1 )thisl-propagation does not result in vertical propagation, needed to eventually lead to flatness. The same phenomenon arises in one variale, where hyponormality is a very soft condition (α k α k+1 for all k 0), while -hyponormality is quite rigid. The work in [6] (extending the ideas in []) revealed that, for unilateral weighted shifts with two equal weights, -hyponormality and sunormality are identical notions. In two variales, however, the analogous result does not hold, as the present work shows. Acknowledgment. Some of the results in this paper were motivated y calculations done with the software tool Mathematica [3].. Notation and Preliminaries For α {α n } n=0 a ounded sequence of positive real numers (called weights), let W α shift(α 0,α 1, ):l (Z + ) l (Z + ) e the associated unilateral weighted shift, defined y W α e n := α n e n+1 (all n 0), where {e n } n=0 is the canonical orthonormal asis in l (Z + ). Two special cases of significant interest are U + := shift(1, 1, ) (the (unweighted) unilateral shift) and S a := shift(a, 1, 1, )(0<a<1). The moments of α are given as { 1 if k =0 γ k γ k (α) := α 0 α k 1 if k>0 It is easy to see that W α is never normal, and that it is hyponormal if and only if α 0 α 1. Similarly, consider doule-indexed positive ounded sequences α k,β k l (Z +), k (k 1,k ) Z + := Z + Z + and let l (Z + ) e the Hilert space of square-summale complex sequences indexed y Z +. We define the -variale weighted shift T (T 1,T )y T 1 e k := α k e k+ε1 T e k := β k e k+ε, }.

3 where ε 1 := (1, 0) and ε := (0, 1). Clearly, T 1 T = T T 1 β k+ε1 α k = α k+ε β k (all k Z + ). (.1) Trivially, a pair of unilateral weighted shifts W α and W β gives rise to a -variale weighted shift T (T 1,T ), if we let α (k1,k ) := α k1 and β (k1,k ) := β k (all k 1,k Z + ). In this case, T is sunormal (resp. hyponormal) if and only if so are T 1 and T ; in fact, under the canonical identification of l (Z + )withl (Z + ) l (Z + ), we have T 1 = I Wα and T = Wβ I,soTis also douly commuting. We now recall a well known characterization of sunormality for single variale weighted shifts, due to C. Berger (cf. [4, III.8.16]), and independently estalished y Gellar and Wallen [16]): W α is sunormal if and only if there exists a proaility measure ξ supported in [0, W α ], with W α suppξ, andsuchthatγ k (α) :=α 0 α k 1 = s k dξ(s) (k 1). We also recall the notion of moment of order k for a pair (α, β) satisfying (.1). Given k Z +, the moment of (α, β) oforderk is 1 if k =0 α (0,0) γ k γ k (α, β) :=... α (k 1 1,0) if k 1 1andk =0 β(0,0)... β (0,k 1) if k 1 =0andk 1. (.) α (0,0)... α (k 1 1,0) β (k 1,0)... β (k 1,k 1) if k 1 1andk 1 We remark that, due to the commutativity condition (.1), γ k can e computed using any nondecreasing path from (0, 0) to (k 1,k ). Moreover, T is sunormal if and only if there is a regular Borel proaility measure µ defined on the -dimensional rectangle R =[0,a 1 ] [0,a ](a i := T i )such that γ k = s k 1 t k dµ(s, t) (allk Z + )[17]. (.3) R Let H e a complex Hilert space and let B(H) denote the algera of ounded linear operators on H. For S, T B(H) let[s, T ] := ST TS. We say that an n-tuple T =(T 1,,T n )of operators on H is (jointly) hyponormal if the operator matrix [T, T] :=([Tj,T i]) n i,j=1 is positive on the direct sum of n copies of H (cf. [], [1]). The n-tuple T is said to e normal if T is commuting and each T i is normal, and T is sunormal if T is the restriction of a normal n-tuple to a common invariant suspace. Clearly, normal = sunormal = hyponormal. Moreover, the restriction of a hyponormal n-tuple to an invariant suspace is again hyponormal. The Bram-Halmos criterion states that an operator T B(H) is sunormal if and only if the k -tuple (T,T,,T k )is hyponormal for all k 1; we say that T is k-hyponormal when the latter condition holds. On the other hand, for a commuting pair T (T 1,T ) of operators on Hilert space we have Definition.1. (cf. [9]) A commuting pair T (T 1,T ) is called k-hyponormal if T(k) := (T 1,T,T1,T T 1,T,,Tk 1,T T1 k 1,,T k ) is hyponormal, or equivalently ([(T q T p 1 ),T n T1 m ])1 m+n k 0. 1 p+q k Clearly, sunormal = (k +1)-hyponormal = k-hyponormal for every k 1, and of course 1- hyponormality agrees with the usual definition of joint hyponormality (as aove). In [9] we otained the following multivariale version of the Bram-Halmos criterion for sunormality, which provided an astract answer to the LPCS, y showing that no matter how k-hyponormal the pair T might e, it may still fail to e sunormal. Theorem.. ([9, Theorem.3]) Let T (T 1,T ) e a commuting pair of sunormal operators on ahilertspace H. The following statements are equivalent. 3

4 (i) T is sunormal. (ii) T is k-hyponormal for all k Z +. In the single variale case, there are useful criteria for k-hyponormality ([6], [8]); for -variale weighted shifts, a simple criterion for joint hyponormality was given in([5]). The following characterization of k-hyponormality for -variale weighted shifts was given in [9, Theorem.4]. Theorem.3. Let T (T 1,T ) e a -variale weighted shift with weight sequences α {α k } and β {β k }. The following statements are equivalent. (a) T is k-hyponormal. () M u (k) :=(γ u+(m,n)+(p,q) )0 m+n k 0 for all u Z +. (For a sunormal pair T, thematrix 0 p+q k M u (k) is the truncation of the moment matrix associated to the Berger measure of T.) The following special cases of Theorem.3 will e essential for our work. Lemma.4. ([5])(Six-point Test; cf. Figure 1(i)) with weight sequences α and β. Then Let T (T 1,T ) e a -variale weighted shift, [T, T] 0 M k (1) 0(all k Z +) ( α k+ε 1 α k α k+ε β k+ε1 α k β k α k+ε β k+ε1 α k β k βk+ε βk ) 0 (all k Z +). (k 1,k +4) β k1,k +3 (k 1,k +3) α k1,k +3 β k1,k + β k1 +1,k + (k 1,k +) (k 1,k +) α k1,k + α k1 +1,k + β k1,k +1 β k1,k +1 β k1 +1,k +1 β k1 +,k +1 (k 1,k +1) α k1,k +1 (k 1 +1,k +1) α k1,k +1 α k1 +1,k +1 α k1 +,k +1 β k1,k β k1 +1,k β k1,k β k1 +1,k β k1 +,k β k1 +3,k α k1,k α k1 +1,k (k 1,k ) (k 1 +1,k ) (k 1 +,k ) (i) α k1,k α k1 +1,k α k1 +,k α k1 +3,k (k 1,k ) (k 1 +1,k ) (k 1 +,k ) (k 1 +3,k ) (k 1 +4,k ) (ii) Figure 1. Weight diagrams used in the Six-point Test and 15-point Test, respectively. 4

5 Lemma.5. ([9]) (15-point Test; cf. Figure 1(ii)) If T (T 1,T ) is -variale weighted shift with weight sequence α {α k } and β {β k }, then T is -hyponormal if and only if γ k1,k γ k1 +1,k γ k1,k +1 γ k1 +,k γ k1 +1,k +1 γ k1,k + γ k1 +1,k γ k1 +,k γ k1 +1,k +1 γ k1 +3,k γ k1 +,k +1 γ k1 +1,k + M (k1,k )() γ k1,k +1 γ k1 +1,k +1 γ k1,k + γ k1 +,k +1 γ k1 +1,k + γ k1,k +3 γ k1 +,k γ k1 +3,k γ k1 +,k +1 γ k1 +4,k γ k1 +3,k +1 γ k1 +,k 0, + γ k1 +1,k +1 γ k1 +,k +1 γ k1 +1,k + γ k1 +3,k +1 γ k1 +,k + γ k1 +1,k +3 γ k1,k + γ k1 +1,k + γ k1,k +3 γ k1 +,k + γ k1 +1,k +3 γ k1,k +4 for all k Z +. We now recall the following notation and terminology from [13]: (i) given a proaility measure µ on X Y R + R +,with 1 t L1 (µ), the extremal measure µ ext 1 (which is also a proaility measure) on X Y is given y dµ ext (s, t) :=(1 δ 0 (t)) dµ(s, t); t 1 t L 1 (µ) and (ii) given a measure µ on X Y, the marginal measure µ X is given y µ X := µ π 1 X, where π X : X Y X is the canonical projection onto X. Thus, µ X (E) =µ(e Y ), for every E X. We now list some results which are needed in the proof of Theorem 3.1. ( ) A B Lemma.6. (cf. [1], [7, Proposition.]) Let M e a operator matrix, where B C A and C are square matrices and B is a rectangular matrix. Then C 0 M 0 there exists W such that B = CW A W CW. For Lemma.7, Definition.8 and Theorem 3.1 the following two suspaces of l (Z + ) will e needed: M := {e k : k 1} and N := {e k : k 1 1}. Lemma.7. ([13]) (Sunormal ackward extension of a -variale weighted shift) Consider the -variale weighted shift whose weight diagram is given in Figure (i), and let T M denote the restriction of T to M. Assume that T M is sunormal with associated measure µ M, and that W 0 := shift(α 00,α 10, ) is sunormal with associated measure ν. Then T is sunormal if and only if (i) 1 t L1 (µ M ); (ii) β00 ( 1 t L 1 (µ M ) ) 1 ; (iii) β00 1 t L 1 (µ M ) (µ M) X ext ν. Moreover, if β00 1 t L 1 (µ M ) =1, then (µ M) X ext = ν. In the case when T is sunormal, Berger measure µ of T is given y dµ(s, t) =β00 1 t d(µ M ) ext (s, t)+(dν(s) β 1 00 L 1 (µ M ) t d(µ M ) X ext (s))dδ 0(t). L 1 (µ M ) J. Stampfli showed in [] that for sunormal weighted shifts W α a propagation phenomenon occurs that forces the flatness of W α whenever two equal weights are present. Later, R.E. Curto proved in [6] that a hyponormal weighted shift with three equal weights cannot e quadratically hyponormal without eing flat: If W α is quadratically hyponormal and α n = α n+1 = α n+ for some n 0, then α 1 = α = α 3 =. Y. Choi [3] improved this result, that is, if W α e quadratically hyponormal and α n = α n+1 for some n 1, then W α is flat, thatis,w α shift(α 0,α 1,α 1, ). In Theorem.10 we show that similar propagation phenomena occur for -hyponormal -variale weighted shifts T. In two variales, the flatness of T is captured y the so-called core of T, c(t) (cf. Definition.8 elow), as follows: T is flat when c(t) is a -variale weighted shift of tensor form. 5

6 Definition.8. (i) The core of a -variale weighted shift T is the restriction of T to M N,in symols, c(t) :=T M N. (ii) A -variale weighted shift T is said to e of tensor form if T = (I W α,w β I). When T is sunormal, this is equivalent to requiring that the Berger measure e a Cartesian product ξ η. (iii) The class of all -variale weighted shift T H 0 whose core is of tensor form will e denoted y TC, that is, TC:= {T H 0 : c(t) is of tensor form}. We now recall that a -variale weighted shift T is said to e horizontally flat when α (k1,k ) = α (1,1) for all k 1,k 1; and we call T vertically flat when β (k1,k ) = β (1,1) for all k 1,k 1. We say T is flat if T is horizontally and vertically flat, and that T is symmetrically flat if T is flat and α (1,1) = β (1,1). The next result shows the extent to which propagation holds in the presence of (joint) hyponormality. Proposition.9. ([15]) LetT e a commuting hyponormal -variale weighted shift whose weight diagram is given in Figure (i). Then (i) If α k+ε1 = α k for some k, thenα k+ε = α k and β k+ε1 = β k ; (ii) If T 1 is quadratically hyponormal, if T is sunormal, and if α (k1,k )+ε 1 = α (k1,k ) for some k 1,k 0, thent is horizontally flat. On the other hand, under the assumption of -hyponormality (and without assuming the sunormality of either T 1 or T ), we can prove that T is flat whenever α k+ε1 = α k and β m+ε = β m for some k and m. We do this using the 15-point Test (Lemma.5). Theorem.10. Let T e a -hyponormal -variale weighted shift. If α (k1,k )+ε 1 = α (k1,k ) for some k 1,k 1, thent is horizontally flat. If instead, β (k1,k )+ε = β (k1,k ) for some k 1,k 1, then T is vertically flat. Proof. Without loss of generality, assume α (k1,k )+ε 1 = α (k1,k ) = 1. Then y the aove mentioned result of Y. Choi and Proposition.9(i), we can see that T 1 is of tensor form when restricted to the suspace {e m : m 1 1andm k }. If k = 1 we are done, so assume k and consider the matrix γ k1,k γ k1 +1,k γ k1,k 1 γ k1 +,k γ k1 +1,k 1 γ k1,k γ k1 +1,k γ k1 +,k γ k1 +1,k 1 γ k1 +3,k γ k1 +,k 1 γ k1 +1,k M (k1,k )() = γ k1,k 1 γ k1 +1,k 1 γ k1,k γ k1 +,k 1 γ k1 +1,k γ k1,k +1 γ k1 +,k γ k1 +3,k γ k1 +,k 1 γ k1 +4,k γ k1 +3,k 1 γ k1 +,k, γ k1 +1,k 1 γ k1 +,k 1 γ k1 +1,k γ k1 +3,k 1 γ k1 +,k γ k1 +1,k +1 γ k1,k γ k1 +1,k γ k1,k +1 γ k1 +,k γ k1 +1,k +1 γ k1,k + which we know is positive semidefinite since T is -hyponormal (Lemma.5). It suffices to prove that α k ε = 1. Let us focus on the principal sumatrix M 0 determined y rows and columns 1, 3 and 5. From (.) we have 1 βk ε βk ε α k ε M = βk ε βk ε βk ε βk ε βk ε 0. (.4) βk ε α k ε βk ε βk ε βk ε βk ε For notational convenience, let a := βk ε, := βk ε and c := α k ε. Thus, M 1 a ac ( a a a a a a a 0 N := ) c a a ac a a c a a c 0. 6

7 If a = then we must necessarily have c =1,thatis,α k ε = 1, as desired. Assume, therefore, that a<. It follows that N 0 det N 0 (a a )(a a c ) (a a c) 0 a 3 (c 1) 0 c =1, that is, α k ε = 1, as desired. 3. Proofs of the Main Results We are now ready to prove our main results, which we restate for the reader s convenience. Theorem 3.1. S H = S H. Proof. By Proposition.9 and Theorem.10, we can assume, without loss of generality, that α (k1,k ) = α (1,0) =1(allk 1 1,k 0) and β (k1,k ) = β (0,1) 1(allk 1 0,k 1). For notational convenience, we let a := α (0,1) < 1, := β (0,1) 1, x := α (0,0) < 1, and y := β (0,0) < 1. We summarize this information in Figure (ii). (0, 3)... (0, 3)... β 0 γ1,3 γ 1, γ,3 γ, T (0, ) α 0 α 1 α T (0, ) a 1 1 β 01 γ1, γ 1,1 γ, γ,1 (0, 1) α 01 α 11 α 1 (0, 1) a 1 1 β 00 γ1,1 γ 1,0 γ,1 γ,0 y ay x ay x α 00 α 10 α 0 (0, 0) (1, 0) (, 0) (3, 0) x 1 1 (0, 0) (1, 0) (, 0) (3, 0) T 1 (i) T 1 (ii) Figure. Weight diagrams of the -variale weighted shifts in Lemma.7 and Theorem 3.1, respectively. Let C k := {(a,, x, y) :T H k } (k =1,, ). Clearly, C C C 1. We first descrie concretely the set C in terms of necessary and sufficient conditions on the four parameters a,, x and y that guarantee the -hyponormality of the pair T. We then estalish that C C. Finally, in Theorem 3.3 we will show that C C 1. It is straightforward to verify that T M = (I shift(a, 1, 1, ),U+ I) andt N = (I U +,shift( ay x,,, ) I); thus, T M and T N are sunormal. As a consequence, to descrie C 7

8 we only need to apply the 15-point Test (Lemma.5) at k =(0, 0), that is, we need to guarantee that M (0,0) () 0. We thus have: T H M (0,0) () 0. Now, since the moments γ k (k Z +) associated with T are 1 if k 1 =0andk =0 x if k 1 1andk =0 γ k = y (k 1) if k 1 =0andk 1 a y (k 1) if k 1 1andk 1, it follows that 1 x y x a y y x x a y x a y a y M (0,0) () y a y y a y a y 4 y x x a y x a y a y a y a y a y a y a y a 4 y y a y 4 y a y a 4 y 6 y M := 1 x y a y x x a y a y y a y y a y a y a y a y a y 0 0 (since the sixth row is a multiple of the third row, and the second and fourth rows are identical). If we now interchange the second and third rows and columns, we see that the positivity of M is determined y the positivity of ( ) ( 1 y x a y ) y y a y a y ( x a y a y a y ) ( x a y a y a y ) ( A := Thus, from Lemma.6 (with C = B and W = I), we have M 0 A B 0. Now oserve that B 0 ay x and ( 1 x y A B = (1 a ) y (1 a ) y (1 a ) B ). ) B 0. B Since the (1, 1)-entry of A B is always positive, the positivity of A B is completely determined y its determinant; that is, A B 0 det(a B) 0 y (1 a ) (1 x ). It follows that T H ay x and y (1 a ) (1 x ). (3.1) We thus see that C = {(a,, x, y) :0<x<1, 0 <y<1, ay x, andy (1 a ) (1 x )}. (3.) 8

9 We will now prove that C C. Let (a,, x, y) C. Let z := ay x. Case 1. If z =, thent N (I U +,U + I), and a straightforward application of Lemma.7 in the s direction shows that T is sunormal if and only if x y if and only if a 1(since x = ay), which is true. Thus, (a,, x, y) C. Case. Assume now that z<1. Since T N (I U +,S z/ I) is sunormal with Berger measure µ N δ 1 [(1 z )δ 0 + z δ ], we can think of T as a ackward extension of T N (in the s direction) and apply Lemma.7. Note that 1 s L 1 (µ N ) =1,sod(µ N ) ext (s, t) (1 δ 0 (s)) 1 s dµ N (s, t) = dδ 1 (s)[(1 z )dδ 0 (t)+ z dδ (t)] and α 1 00 s L 1 (µ N ) (µ N ) Y ext = x [(1 z )δ 0 + z δ ]. Thus, y Lemma.7, T is sunormal α 1 00 s L 1 (µ N ) (µ N ) Y ext η 0 (where η 0 denotes the Berger measure of shift(y,,, )) x [(1 z )δ 0 + z δ ] (1 y )δ 0 + y δ (3.3) x (1 z ) (1 y )and x z y y (1 a ) (1 x ), as in the last condition in (3.). Therefore, T is sunormal, and the proof is complete. Theorem 3.. Let T (T 1,T ) S, with weight diagram given y Figure (ii), and assume that T is -hyponormal. Then T is sunormal, with Berger measure given as µ = 1 {[ (1 x ) y (1 a )]δ (0,0) (3.4) +y (1 a )δ (0, ) +( x a y )δ (1,0) + a y δ (1, )}. Proof. We apply the main result in [11], in the special form needed here; cf. [11, Proposition 3.1]. For a -variale sunormal weighted shift with weight diagram given y Figure (ii), the Berger measure is µ = ϕ (δ 0 δ )+y 1 t δ 0 ( ψ δ )+ξ x δ, (3.5) L 1 (ψ) where ψ =(1 a )δ, ϕ = ξ x y 1 a δ 0 a y δ 1, d ψ(t) 1 := dψ(t) andξ t 1 t x is the Berger L 1 (ψ) measure of S x. A straightforward calculation shows that ψ = δ, so that (3.5) ecomes µ = {[ (1 x ) y (1 a ) ]δ 0 + x a y δ 1 } (δ 0 δ ) +[(1 x )δ 0 + x δ 1 ] δ, which easily leads to the desired formula (3.4). We conclude this section with a proof of Theorem 1.4, which we reformulate in terms of C 1 and C. Toward this end, we need a description of C 1 := {(a,, x, y) :T H 1 }: y the Six-point Test (Lemma.4), T is hyponormal if and only if M (0,0) (1) 1 x y x x a y 0. y a y y Theorem 3.3. C C 1. 9

10 Proof. Since x<1, M (0,0) (1) 0 if and only if det M (0,0) (1) 0, that is, if and only if P 1 := y ( x a 4 y x 4 +a x y x y ) 0. Let x> and let a := x 1. It follows that 1 a =(1 x ), so that P := (1 x ) y (1 a )= (1 x ) y (1 x )=( y )(1 x ), and it suffices to choose y such that <y<to make P < 0, and thus reak -hyponormality (cf. (3.1)). On the other hand, P 1 P x y + y 4 a (x a ) = y ( x x 4 y + x y ) = ( x y )(1 x ), which can e made nonnegative y taking x close to 1. This shows that C C 1. References [1] M. Arahamse, Commuting sunormal operators, Ill. Math. J. (1978), [] A. Athavale, On joint hyponormality of operators, Proc.Amer.Math.Soc. 103(1988), [3] Y. Choi, A propagation of the quadratically hyponormal weighted shifts, Bull. Korean Math. Soc. 37(000), [4] J. Conway, The Theory of Sunormal Operators, Mathematical Surveys and Monographs, vol. 36, Amer. Math. Soc., Providence, [5] R. Curto, Joint hyponormality: A ridge etween hyponormality and sunormality, Proc. Symposia Pure Math. 51(1990), [6] R. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13(1990), [7] R. Curto and L. Fialkow, Solution of the truncated complex moment prolem with flat data, Memoirs Amer. Math. Soc., no. 568, Amer. Math. Soc., Providence, [8] R. Curto, S.H. Lee and W.Y. Lee, A new criterion for k-hyponormality via weak sunormality, Proc. Amer. Math. Soc. 133(005), [9] R. Curto, S.H. Lee and J. Yoon, k-hyponormality of multivariale weighted shifts, J. Funct. Anal. 9(005), [10] R. Curto, S.H. Lee and J. Yoon, Hyponormality and sunormality for powers of commuting pairs of sunormal operators, J. Funct. Anal. 45(007), [11] R. Curto, S.H. Lee and J. Yoon, Reconstruction of the Berger measure when the core is of tensor form, Oper. Theory Adv. Appl., to appear. [1] R. Curto, P. Muhly and J. Xia, Hyponormal pairs of commuting operators, Oper. Theory Adv. Appl. 35(1988), 1-. [13] R. Curto, J. Yoon, Jointly hyponormal pairs of sunormal operators need not e jointly sunormal, Trans. Amer. Math. Soc. 358(006), [14] R. Curto and J. Yoon, Disintegration-of-measure techniques for multivariale weighted shifts, Proc. London Math. Soc. 9(006), [15] R. Curto and J. Yoon, Propagation phenomena for hyponormal -variale weighted shifts, J. Operator Theory, to appear. [16] R. Gellar and L.J. Wallen, Sunormal weighted shifts and the Halmos-Bram criterion, Proc. Japan Acad. 46(1970), [17] N.P. Jewell and A.R. Luin, Commuting weighted shifts and analytic function theory in several variales, J. Operator Theory 1(1979), [18] A. Luin, Weighted shifts and product of sunormal operators, Indiana Univ. Math. J. 6(1977), [19] A. Luin, Extensions of commuting sunormal operators, Lecture Notes in Math. 693(1978), [0] A. Luin, A sunormal semigroup without normal extension, Proc. Amer. Math. Soc. 68(1978), [1] J.L. Smul jan, An operator Hellinger integral (Russian), Mat. S. 91(1959), [] J. Stampfli, Which weighted shifts are sunormal?, Pacific J. Math. 17(1966), [3] Wolfram Research, Inc. Mathematica, Version 4., Wolfram Research Inc., Champaign, IL,

11 [4] J. Yoon, Disintegration of measures and contractive -variale weighted shifts, Integral Equations Operator Theory, to appear. [5] J. Yoon, Schur product techniques for commuting multivariale weighted shifts, J. Math. Anal. Appl. 333(007), Department of Mathematics, The University of Iowa, Iowa City, Iowa 54 address: URL: Department of Mathematics, Chungnam National University, Daejeon, , Korea address: Department of Mathematics, The University of Texas - Pan American, Edinurg, Texas address: jasangyoon@hotmail.com 11