# Subnormality of Powers of Multivariable Weighted Shifts

Authors: Sang Hoon Lee; Woo Young Lee; Jasang Yoon
Journal: Journal of Function Spaces
 (2020)

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## Abstract

Given a pairT≡T1,T2 of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for the existence of a commuting pair N≡N1,N2 of normal extensions of T1 and T2; in other words, T is a subnormal pair. The LPCS is a longstanding open problem in the operator theory. In this paper, we consider the LPCS of a class of powers of 2-variable weighted shifts. Our main theorem states that if a “corner” of a 2-variable weighted shift T=Wα,β≔T1,T2 is subnormal, then T is subnormal if and only if a power Tm,n≔T1m,T2n is subnormal for some m,n≥1. As a corollary, we have that if T is a 2-variable weighted shift having a tensor core or a diagonal core, then T is subnormal if and only if a power of T is subnormal.


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## Body

## 1. Introduction

For a Hilbert space operator, a subnormal operator means an operator admitting a normal extension, i.e., an extension which is a normal operator. As a lifting problem of operators, many researchers of operator theory have considered necessary and sufficient conditions for a pair of subnormal operators on a Hilbert space to admit commuting normal extensions: more concretely, given a pairT=T1,T2 of commuting subnormal operators T1,T2 on a Hilbert space, find a necessary and sufficient condition for the existence of commuting normal extensions N1 and N2 of T1 and T2, respectively. This problem is referred to as the Lifting Problem for Commuting Subnormals (LPCS). A pair of subnormal operators admitting commuting normal extensions is called a subnormal pair.For a bounded linear operatorT on a complex Hilbert space H, it is well known that the subnormality of T implies the subnormality of powers Tmm≥2. However, its converse is not true in general; in fact, Stampfli [1, p. 378] showed that the subnormality of all powers Tmm≥2 does not necessarily imply the subnormality of T, even if T≡Wa is a unilateral weighted shift. It is also well known that the hyponormality (i.e., T∗,T≡T∗T−TT∗ is positive semidefinite) of T does not imply the hyponormality of T2 [2]. However, for a unilateral weighted shift Wa, the hyponormality of Wa (detected by the condition ak≤ak+1 for all k≥0 when a=ann=0∞) clearly implies the hyponormality of all powers Wamm≥1.On the other hand, Franks [3] showed that given a pair T=T1,T2 of commuting subnormal operators, if pT is subnormal for all 2-variable polynomials p∈ℂz1,z2 with degp≤5, then T is a subnormal pair. Clearly, if T=T1,T2 is a subnormal pair and if m,n≥1, then Tm,n≔T1m,T2n is also a subnormal pair. Motivated by Stampfli’s work [1], it is natural to ask whether the subnormality of Tm,n≔T1m,T2n for each m,n>1,1 implies the subnormality of T. For the 2-variable weighted shifts, we may consider these analogous results. The standard assumption on a pair T≡T1,T2 is that each component Ti is subnormal i=1,2. With this in mind, these analogous results are highly nontrivial. In the works [4–8], it was considered whether there is a 2-variable weighted shift T=Wα,β≔T1,T2 such that Tm,n is subnormal for some m,n>1,1, but T is not subnormal.Fora≡ann=0∞, a bounded sequence of positive real numbers (called weights), a weighted shiftWa:ℓ2ℤ+⟶ℓ2ℤ+ is defined by Waen≔anen+1(all n≥0), where enn=0∞ is the canonical orthonormal basis in ℓ2ℤ+. In this case, we write
(1)Wa=shifta0,a1,a2,⋯.Now, for0<x<y<1, consider the weighted shift
(2)Wa=shiftx,y,1,1,⋯.Then,Wa is hyponormal (detected by the condition an≤an+1 for all n) but not subnormal. However, all powers Wamm≥2 are subnormal. If y=1 in Wa=shiftx,y,1,1,⋯, then the following statements are equivalent:
(a)
Wa is subnormal(b)
Wam is subnormal for all m≥1(c)
Wam is subnormal for some m≥1In [5, 6], we have examined the above results for the class of 2-variable weighted shifts T=Wα,β. More concretely, for the class of 2-variable weighted shifts T=Wα,β with a core of tensor form, denoted TC [5], or with a core of diagonal form, denoted DC [6], we have shown that if T=Wα,β∈TC∪DC, then the following statements are equivalent:
(a)
T is subnormal(b)
Tm,n is subnormal for all m,n≥1(c)
Tm,n is subnormal for some m,n≥1In spite of the above facts for 1 or 2-variable weighted shifts and consideration of the recent works ([4–8]), we have guessed that there exists a class of 2-variable weighted shifts Wα,β such that Wα,β is not subnormal but Wα,βm,n is subnormal for all m,n>1,1, under a more general condition that a “corner” of Wα,β is subnormal. In this paper, we show that this guess is not right and that the above three statements are equivalent whenever a corner of Wα,β is subnormal. In the below, we will notice that TC∪DC is a very special corner of Wα,β.On the other hand, the reason why we take 2-variable weighted shifts for examining the subnormality of powers for pairs of operators is that 2-variable weighted shifts play an important role in detecting properties such as subnormality, via the Lambert-Lubin Criterion ([9, 10]): a commuting pair T1,T2 of injective operators acting on a Hilbert space H admits a commuting normal extension if and only if for every nonzero vector x∈H, the 2-variable weighted shift with weights
(3)αi,j≔T1i+1T2jxT1iT2jxandβi,j≔T1iT2j+1xT1iT2jx,has a normal extension.The organization of this paper is as follows. In Section2, we give preliminary notions and state the main theorem. In Section 3, we provide a proof of the main theorem.


## 2. Preliminaries and the Main Theorem

LetH be a complex Hilbert space and let BH denote the algebra of bounded linear operators on H. For S,T∈BH let S,T≔ST−TS. We say that an n-tuple T=T1,⋯,Tn of operators on H is (jointly) hyponormal if the operator matrix
(4)T∗,T≔T1∗,T1T2∗,T1⋯Tn∗,T1T1∗,T2T2∗,T2⋯Tn∗,T2⋮⋮⋱⋮T1∗,TnT2∗,Tn⋯Tn∗,Tn,is positive semidefinite on the direct sum of n copies of H (cf. [11, 12]). The n-tuple T is said to be normal if T is commuting and each Ti is normal, and T is subnormal if T is the restriction of a normal n-tuple to a common invariant subspace. For k≥1, a commuting pair T≡T1,T2 is said to be k-hyponormal ([13]) if
(5)Tk≔T1,T2,T12,T2T1,T22,⋯,T1k,T2T1k−1,⋯,T2k,is hyponormal, or equivalently
(6)Tk∗,Tk=T2qT1p∗,T2mT1n1≤p+q≤k1≤n+m≤k≥0.Clearly, normal ⇒ subnormal ⇒k-hyponormal for n-tuples of operators. The Bram-Halmos criterion states that an operator T∈BH is subnormal if and only if the k-tuple T,T2,⋯,Tk is hyponormal for all k≥1.LetWa be a weighted shift with weights a≡ann=0∞. The moments of a are given as
(7)γk≡γka≔1,ifk=0a02⋯ak−12,ifk≥1.It is easy to see thatWa is never normal and that it is hyponormal if and only if a0≤a1≤⋯. Similarly, consider double-indexed positive bounded sequences α≡αk,β≡βk∈ℓ∞ℤ+2 and k≡k1,k2∈ℤ+2 and let ℓ2ℤ+2 be the Hilbert space of square-summable complex sequences indexed by ℤ+2. Recall that ℓ2ℤ+2 is canonically isometrically isomorphic to ℓ2ℤ+⊗ℓ2ℤ+. We define the 2-variable weighted shiftWα,β≡T1,T2 by a pair of operators acting on the Hilbert space H≡ℓ2ℤ+2 given by
(8)T1ek1,k2≔αk1,k2ek1+1,k2,T2ek1,k2≔βk1,k2ek1,k2+1,.for each k1,k2∈ℤ+2. For all k≡k1,k2∈ℤ+2, we clearly have
(9)T1T2=T2T1⇔βk1+1,k2αk1,k2=αk1,k2+1βk1,k2.For a commuting 2-variable weighted shiftWα,β, the moment of Wα,β of order k∈ℤ+2 is
(10)γk≡γkWα,β≔1,ifk1=0andk2=0α0,02⋯αk1−1,02,ifk1≥1andk2=0β0,02⋯β0,k2−12,ifk1=0andk2≥1α0,02⋯αk1−1,02βk1,02⋯βk1,k2−12,ifk1≥1andk2≥1.We remark that, due to the commutativity condition (9), γk can be computed using any nondecreasing path from 0,0 to k1,k2. For a detailed discussion of the 2-variable weighted shifts, the reader may refer to ([4, 13, 14] [8]).We now recall a well-known characterization of subnormality for multivariable weighted shifts [15], due to C. Berger (cf. [2, III.8.16]) and independently established by Gellar and Wallen [16]) in the single variable case: Wα,β admits a commuting normal extension if and only if there is a probability measure μ, called the Berger measure of Wα,β, defined on the 2-dimensional rectangle R=0,c1×0,c2 (where ci≔Ti2 such that
(11)γk=∫Rtkdμt≔∫Rt1k1t2k2dμt1,t2,forallk∈ℤ+2.Observe thatU+≡shift1,1,1,⋯ and Sc≡shiftc,1,1,⋯c≤1 are subnormal, with Berger measuresδ1 and 1−c2δ0+c2δ1, respectively, where δp denotes the point-mass probability measure with support from the singleton set p.Throughout this paper, we writeH≡ℓ2ℤ+2≡∨ekk∈ℤ+2 and
(12)H0≔thesetofallcommutingpairsofsubnormaloperatorsonH,(13)MpH≔∨ek:k=k1,k2withk1≥0,k2≥p,(14)NqH≔∨ek:k=k1,k2withk1≥q,k2≥0.for p,q≥0. For a 2-variable weighted shift Wα,β, a corner of Wα,β is defined by
(15)Wα,βMpH∩NqHforsomep,q≥0,which is a restriction of Wα,β to the invariant subspace MpH∩NqH. The core of Wα,β, denoted by cWα,β, is defined by a corner with p=q=1, i.e.,
(16)cWα,β≔Wα,βM1H∩N1H.Thus, for 2-variable weighted shifts, the core is a special form of a corner. A 2-variable weighted shiftWα,β is said to be of tensor form if it is of the form I⊗Wa,Wb⊗I. If a tensor form Wα,β is subnormal, then the corresponding Berger measure is given by a Cartesian product ξ×η where ξ and η are the Berger measure of Wa and Wb, respectively (cf. [5]). Also, for strictly increasing weight sequences a≡ann=0∞, consider a 2-variable weighted shift Wα,β on ℓ2ℤ+2 given by the double-indexed weight sequences
(17)αk=βk≔ak1+k2fork=k1,k2withk1,k2≥0.This2-variable weighted shift Wα,β induced by a 1-variable weighted shift Wa is said to be of the diagonal form. If a diagonal form Wα,β is subnormal, then the corresponding Berger measure is given by a measure supported in the diagonal s,s∈ℝ2:s≥0 (cf. [6]). The class of all 2-variable weighted shifts Wα,β∈H0 whose core is of the tensor form will be denoted by TC; in symbols,TC≔Wα,β∈H0:cWα,βisoftensorform (see Figure 1(i)).Figure 1
Weight diagram of the 2-variable weighted shiftT∈TC and weight diagram of the 2-variable weighted shift T∈DC, respectively.Also, the class of all 2-variable weighted shiftsWα,β∈H0 whose core is of the diagonal form will be denoted by DC; in symbols,DC≔Wα,β∈H0:cWα,βisofdiagonalform(see Figure 1(ii)).In [5, 6], it was shown that if T=Wα,β∈TC∪DC, then Tm0,n0 is subnormal for some m0,n0≥1 if and only if T is subnormal. Now, it is natural to consider that given a 2-variable weighted shift T=Wα,β∈H0, whether or not Tm,n is subnormal if and only if T is subnormal. In other words, we ask the following:Problem 1 ([6, 8]).
Given a 2-variable weighted shiftT=Wα,β∈H0, assume that Tm,n is subnormal for all m,n>1,1. Does it follow that T is subnormal?For the class of 2-variable weighted shiftsWα,β, it is often the case that the powers are less complex than the initial pair; thus, it becomes especially significant to unravel the invariance of subnormality under the action Wα,β↦Wα,βm,nm,n≥1. The aim of this paper is to shed new light on some of the intricacies associated with LPCS and powers of commuting subnormals in H0.Our main theorem now states:Theorem 2.
LetT≡Wα,β∈H0. If a corner of T is subnormal, i.e., TMpH∩NqH is subnormal for some p,q∈ℤ+, then the following are equivalent:
(a)
T is subnormal(b)
Tm,n is subnormal for all m,n≥1(c)
Tm,n is subnormal for some m,n≥1As we observed before,TC∪DC is a special corner of Wα,β. Indeed, if T=Wα,β∈TC∪DC, then TM1H∩N1H is subnormal, so that T satisfies the condition of Theorem 2. Therefore, if T=Wα,β∈TC∪DC, then three conditions of Theorem 2 are equivalent. Thus, as immediate corollaries of Theorem 2, we can recapture the both main results of [5, Theorem 7.1] and [6, Theorem 3.2].Briefly stated, our key idea to prove the main results is as follows: (i) we split the ambient spaceH≡ℓ2ℤ+2 as an orthogonal direct sum H=⊕p=0m−1⊕q=0n−1Hp,qm,n; (ii) when TM1H is subnormal, we show that for some m≥1,Tm,1 is subnormal if and only if T is subnormal by using the backward extension of subnormality; (iii) when TM1H is subnormal, we show that for some m,n≥1T1,nm,1 is subnormal if and only if Tm,1 is subnormal by using (i) and the backward extension of subnormality; and (iv) by combining (ii) and (iii), we have that if TMpH∩NqH is subnormal for some p,q∈ℤ+, then Tm,n=Tm,11,n is subnormal for some m,n≥1 if and only if T is subnormal.


## 3. The Proof of the Main Theorem

We will first establish several auxiliary lemmas and then prove the main theorem (Theorem2).To study subnormality for powers of multivariable weighted shifts, we recall that, in one variable, them-th power of a weighted shift is unitarily equivalent to the direct sum of m weighted shifts. First, we need some terminology. Let ℓ2ℤ+=∨j=0∞ej. Given integers i and mm≥1,0≤i≤m−1, define Hm,i≔∨j=0∞emj+i; clearly, ℓ2ℤ+=⊕i=0m−1Hm,i. Following the notation in [17], for a weight sequence a≡ann=0∞, we let
(18)Wam:i≔shiftΠn=0m−1amj+i+nj=0∞,that is, Wam:i denotes the shift with the weight sequence given by the products of weights in adjacent packets of size m, beginning with ai⋯ai+m−1. For example, given a weight sequence a≡ann=0∞, we have Wa2:0=shifta0a1,a2a3,⋯, Wa2:1=shifta1a2,a3a4,⋯, Wa3:2=shifta2a3a4,a5a6a7,⋯, etc. For m≥1 and 0≤i≤m−1, we note that Wam:i is unitarily equivalent to WamHm,i. Therefore, Wam is unitarily equivalent to ⊕i=0m−1Wam:i. Thus, we have (cf. [17]) that if Wa is subnormal with the Berger measure μ, then Wam,i is subnormal with the Berger measure μm,i, where
(19)dμm,is=si/mγidμs1/mfor0≤i≤m−1,and furthermore,
(20)Wamissubnormal⇐Wam:iissubnormalfor0≤i≤m−1.Forh≥1, we let Lh≔∨en:n≥h denote the invariant subspace obtained by removing the first h vectors in the canonical orthonormal basis of ℓ2ℤ+. Thus, if Wa≡shifta0,a1,a2,⋯ is subnormal with Berger measure σ, then WaLh is subnormal for each h≥1, and the Berger measure σh of WaLh is given by
(21)dσhs=shγhdσs.Something similar happens in two variables, as we will see it below. For a 2-variable weighted shiftT=Wα,β=T1,T2∈H0 on H≡ℓ2ℤ+2, we observe a new direct sum decomposition for powers of 2-variable weighted shifts which parallels the decomposition used in [17] to analyze the subnormality for powers of (one-variable) weighted shifts. Specially, we split the ambient space H≡ℓ2ℤ+2≡∨ekk∈ℤ+2 as an orthogonal direct sum H≡⊕p=0m−1⊕q=0n−1Hp,qm,n, where for p=0,1,⋯,m−1, and q=0,1,⋯,n−1,(22)Hp,qm,nT≔∨emi+p,nj+q:i=0,1,2,⋯,j=0,1,2,⋯,,where T=Wα,β is a 2-variable weighted shift on H≡ℓ2ℤ+2. Then, each of Hp,qm,nT reduces T1m and T2n. Also, Tm,n is subnormal if and only if each Tm,nHp,qm,nT is subnormal. Similarly, for h,l≥0, consider TMhH∩MlH on MhH∩MlH and let
(23)Hp,qm,nTMhH∩MlH≔∨emi+p+h,nj+q+l:i=0,1,2,⋯,j=0,1,2,⋯.In a similar fashion to (13) and (14), we can define
(24)MhHp,qm,nT≔∨ek∈Hp,qm,nT:i≥0andj≥h,NlHp,qm,nT≔∨ek∈Hp,qm,nT:i≥landj≥0.We thus have:Lemma 3.
LetT=Wα,β be a 2-variable weighted shift. Then, for a fixed m≥1 and 0≤p,q≤m−1, we have
(25)Tm,1H0,0m,1TMpH0,0m,1T≅TMpHm,1H0,0m,1TMpH,Tm,1H0,0m,1TNqH0,0m,1T≅TNmqHm,1H0,0m,1TMmqHand for m,n≥1, 0≤p≤m−1,0≤q≤n−1, we have
(26)TM1H∩N1Hm,nHp,n−1m,nTM1H∩N1H≅Tm,nHp,0m,nTM1H∩N1HHp,0m,nT;TMqHm,nH0,0m,nTMqH≅Tm,nH0,qm,nT;TNpHm,nH0,0m,nTNpH≅Tm,nHp,0m,nT;TM1Hm,nH0,n−1m,nTM1H≅Tm,nH0,0m,nTM1H0,0m,nT;TN1Hm,nHp,n−1m,nTN1H≅Tm,nHp,0m,nTN1Hp,0m,nT,where ≅ means a unitary equivalence of two operators.Lemma 4.
LetT=Wα,β be a 2-variable weighted shift in H0. If T is subnormal with Berger measure μ, then Tm,n is subnormal for all m,n≥1. Furthermore, for each 0≤p≤m−1,0≤q≤n−1, the Berger measure of Tm,nHp,qm,nT is given by
(27)dμp,qm,ns,t=sp/mtq/nγp,qdμs1/m,t1/n.Proof.
Observe first that ifT=Wα,β is subnormal then Tm,n≅⊕p=0m−1⊕q=0n−1Tm,nHp,qm,nT, where each direct summand is a subnormal 2-variable weighted shift.
For the second assertion, observe(28)γi,jμp,qm,n=γmi+p,nj+qμγp,qμforp,q≥0.
Thus, fori,j≥0,
(29)∫sitjdμp,qm,ns,t=γi,jμp,qm,n=γmi+p,nj+qγp,q=1γp,q∫smitnj+qdμs,t=1γp,q∫sitjsp/mtq/ndμs1/m,t1/n,so that
(30)dμp,qm,ns,t=sp/mtq/nγp,qdμs1/m,t1/n,which gives the result.To detect the subnormality of2-variable weighted shifts, we introduce some definitions.
(i)
For a regular Borel measureμ on ℝ+, we say that μ is positive if μE≥0 for all Borel subset E⊆ℝ+, or equivalently, μ≥0 if and only if ∫fdμ≥0 for all f∈Cℝ+ such that f≥0 on ℝ+. Similarly, we say that dμs is positive (denoted by dμs≥0) if ∫fsdμs≥0 for all f∈Cℝ+ such that f≥0 on ℝ+. For positive two measures μ and ν on ℝ+, we say that μ≥ν on ℝ+ if μ−ν is positive(ii)
Letμ be a probability measure on X×Y, and assume that 1/t∈L1μ. The extremal measureμext (which is also a probability measure) on X×Y is given by
(31)dμexts,t≔1−δ0t1t1/tL1μdμs,t.(iii)
Given a measureμ on X×Y, the marginal measureμX is given by μX≔μ∘πX−1, where πX:X×Y⟶X is the canonical projection onto X. Thus, μXE=μE×Y for every E⊆X, or equivalently, dμXs=∫Ydμs,t.Lemma 5 [14, Proposition 3.10] (subnormal backward extension).
LetT=Wα,β be a 2-variable weighted shift, and assume that TM1H is subnormal with associated Berger measure η and that W0≔shiftα00,α10,⋯ is subnormal associated with Berger measure ξα0. Then, T is subnormal if and only if
(32)1t∈L1η,β0,02≤1tL1η−1,β0,021tL1ηηextX≤ξα0.
Moreover, ifβ0021/tL1η=1, then ηextX=ξα0. In the case when T is subnormal, the Berger measure μ of T is given by
(33)dμs,t=β0,021tL1ηdηexts,t+dξα0s−β0,021tL1ηdηextXsdδ0t.We also recall:Lemma 6 [18].
For a positive measureμ on Z≡X×Y≡ℝ+×ℝ+, let 1/t∈L1μ. Then, 1/t∈L1μY and
(34)1tL1μ=1tL1μY,where μY≔μ∘πY−1 and πY:Z⟶Y is the canonical projection onto Y.Given a 2-variable weighted shiftT=Wα,β∈H0, and given k1,k2≥0, we let
(35)Wk2≔shiftα0,k2,α1,k2,⋯,be the k2-th horizontal slice of T1 with associated Berger measure ξαk2; similarly, we let
(36)Vk1≔shiftβk1,0,βk1,1,⋯,be the k1-th vertical slice of T2 with associated Berger measure ηβk1. Clearly, W0 and V0 are the unilateral weighted shifts associated with the 0th row and 0-column in the weight diagram for T, respectively.Then, we have:Lemma 7.
LetT=Wα,β∈H0. If TMpH∩NqH is subnormal for some p,q∈ℤ+, then TM1H∩N1H is also subnormal.Proof.
It is enough to show that ifTM2H∩N1H or TM1H∩N2H is subnormal, then TM1H∩N1H is subnormal. Without loss of generality, assume that TM2H∩N1H is subnormal. Since Wα,β∈H0, we have that Vk1≡shiftβk10,βk11,⋯ is subnormal for all k1≥0. By Lemma 5, we have βk1,121/tL1ηβk1=1k1≥0. Let ζ be the Berger measure of TM2H∩N1H. By Lemma 6, since 1/tL1ηβ1=1/tL1ζ and β1,121/tL1ζ=1, it follows from Lemma 5 that ζextX=ξα1, that is,
(37)ζextX=ξα1⇒β1,121tL1ζζextX≤ξα1.
Moreover,1/t∈L1ζ and β1,12≤1/tL1ζ−1. Thus, by Lemma 5 again, we have that TM1H∩N1H is subnormal, as desired.Next, we have:Lemma 8.
AssumeT=Wα,β∈H0 and TM1H is subnormal with Berger measure η. Assume 1/t∈L1η. Let ξα0 be the Berger measure of subnormal shift W0≡α0,0,α1,0,⋯ which is in the zero level of T. We also let
(38)lψ≔ξα0−β0021tL1ηηextX.
IfTm,1H0,0m,1T is subnormal for some m≥1, then ψ≥0. Moreover, if Tm,1H0,0m,1T is subnormal, then its Berger measure is
(39)β0021tL1ηdηexts1/m,t+dψs1/mdδ0t.Proof.
We first claim that(40)dψs≥0⇔dψs1/m≥0foranym∈ℕ.
To see (40), we note that the positivity of ψ depends on the densities of ηextX and ξα0. There are three cases to consider.
(i)
IfηextX and ξα0 are finite atomic measures, then it is clear(ii)
IfηextX and ξα0 are continuous measures, then by a change of variables, letting u=s1/m, which goes in both directions because s1/m is an invertible function on the positive axis. That is, one can change the name of the variable from s1/m to u and then relabel u as s(iii)
IfηextX and ξα0 are any probability measures, then by the above arguments, we have the claim (40).
Now, suppose thatTm,1H0,0m,1T is subnormal with the Berger measure ϖ. Since Tm,1H0,0m,1T and Tm,1H0,0m,1TM1H0,0m,1T are subnormal, we thus reconstruct the subnormality of Tm,1H0,0m,1T as a backward extension of
(41)Tm,1H0,0m,1TM1H0,0m,1Tinthetdirection,by applying Lemma 5. We let ς be the Berger measure of
(42)Tm,1H0,0m,1TM1H0,0m,1T.
Sinceη is the Berger measure of TM1H, by Lemma 4, we have
(43)lγk1,k2ς=1β002·γk1,k2+1ϖ⇒∫sk1tk2dςs,t=1β002·∫sk1tk2+1dϖs,t⇒∫sk1tk2dη0,0m,1s,t=∫sk1tk2tβ002dϖs,tforallk1,k2.
Thus, we obtain(44)dη0,0m,1s,t=tβ002dϖs,t.
Also, by Lemma4 and (44), we have
(45)dηs1/m,t=tβ002dϖs,t.
Sinceς is the Berger measure of
(46)Tm,1H0,0m,1TM1H0,0m,1T,and ϖ is the Berger measure of Tm,1H0,0m,1T, by Lemma 4, (44) and (45), we have
(47)dςs,t=tβ002dϖs,t=dη0,0m,1s,t=dηs1/m,t.
We letηβ0 be the Berger measure of the following subnormal shift:
(48)shiftβ0,0,β0,1,β0,2,⋯.
Sinceς is the Berger measure of Tm,1H0,0m,1TM1H0,0m,1T, by Lemma 6, we have
(49)1tL1η=1tL1ηβ01=1tL1ς,dςexts,tX=dηexts1/m,tX.
Ifshiftα0,0,α1,0,⋯ in the zero level of T is subnormal with the Berger measure ξα0, then
(50)shiftα0,0⋯αm−1,0,αm,0⋯α2m−1,0,⋯,is subnormal with the Berger measure dξα0m,0s=dξα0s1/m by (18).
SinceT=Wα,β∈H0 and Tm,1H0,0m,1T is subnormal, it follows from Lemma 5, (22), (44), (47), and (49) that
(51)Tm,1H0,0m,1Tissubnormal⇒Lemma3.3β0021tL1ςdςexts,tX≤dξα0s1/m⇒3.18β0021tL1ηdηexts1/m,tX≤dξα0s1/m⇒dψs1/m≥0⇔3.14dψu≥0bylettingu=s1/m⇔ψ≥0.
This proves the first assertion. The second assertion is clear from Lemma4 and analyzing the moments of Tm,1H0,0m,1T.The following corollary is an immediate result of Lemma8.Corollary 9.
AssumeT=Wα,β∈H0 and TM1H is subnormal with Berger measure η. Let 1/t∈L1η. Then
(a)
T is subnormal ⇔ψ≥0(b)
For anym,n≥1, Tm,1 is subnormal ⇔Tn,1 is subnormal. Hence, in particular
(52)Tm,1issubnormalforsomem≥1⇔Tissubnormal.Proof.
(a) This follows from Lemma5. (b) If Tm,1 is subnormal for some m≥1, then Tm,1H0,0m,1T is subnormal. By Lemma 8, we have ψ≥0 and hence T is subnormal by (a). Clearly, if T is subnormal then Tn,1 is subnormal for all n≥1.Remark 10.
We remark that ifT=Wα,β∈H0 and TM1H is subnormal with Berger measure η, then 1/t∈L1η because V0≔shiftβ0,0,β0,1,⋯ is subnormal.We next have:Corollary 11.
LetT=Wα,β∈H0. If there exists p∈ℤ+ such that TMpH is subnormal, then Tm,1 is subnormal for some m≥1 if and only if T is subnormal.Proof.
It suffices to consider the case ofp=2. In the case, if Tm,1 is subnormal for some m≥1, then TM1Hm,1 is subnormal. Thus, by Corollary 9, TM1H is subnormal, and therefore, T is subnormal. The converse is clear.We now have:Lemma 12.
LetT=Wα,β∈H0 and let TM1H be subnormal. Then,
(53)lT1,nissubnormalforsomen≥1⇔Tissubnormal.Proof.
⇒ Let η be the Berger measure of TM1H and suppose that T1,n is subnormal for fixed n≥1. Then, T1,nH0,01,nTM1H0,01,nT is subnormal. Let τ be its Berger measure. By Lemma 3, we have
(54)TM1H1,nH0,n−11,nTM1H≅T1,nH0,01,nTM1H0,01,nT.
Hence, by Lemma4 and (54), we have
(55)dτs,t=tn−1/nγ0,n−1ηdηs,t1/n.
Observe(56)dτexts,t=1tL1τ−1t−1/nγ0,n−1ηdηs,t1/n,(57)dτexts,tX=1/tL1τ−1γ0,n−1η∫0T22nt−1/ndηs,t1/n.
We now characterize the subnormality ofT1,nH0,01,nT as a backward extension of
(58)T1,nH0,01,nTM1H0,01,nTinthetdirection,after applying Lemma 5. To do so, let shiftα0,0,α1,0,⋯ in the zero level of T be subnormal with the Berger measure ξα0. Then, by (57), we have that
(59)T1,nH0,01,nTissubnormal⇒Lemma3.3γ0,nT1tL1τdτexts,tX≤dξα0s⇒3.22β002∫0T22nt−1/ndηs,t1/n≤dξα0s⇒β002∫0T22u−1dηs,u≤dξα0sbylettingu=t1/n⇒β002∫0T22t−1dηs,t≤dξα0s⇒β0021tL1ηηextX≤ξα0.
By again Lemma5 and (59), therefore, T is subnormal.
(60)⇐Clear.By Corollary9 and Lemma 12, we have:Corollary 13.
LetT=Wα,β∈H0. If TM1H or TN1H is subnormal, then the following are equivalent:
(a)
Tm,1 is subnormal for some m≥1(b)
Tm,1 is subnormal for all m≥1(c)
T1,n is subnormal for some n≥1(d)
T1,n is subnormal for all n≥1(e)
T is subnormalThe following theorem is a core of our main result.Theorem 14.
IfTM1H is subnormal, then for some m,n≥1(61)T1,nm,1issubnormal⇔Tm,1issubnormal.Proof.
⇒. For fixed m,n≥1, suppose that T1,nm,1=Tm,n is subnormal. Then, W≔Tm,nHp,0m,nTM1Hp,0m,nT is also subnormal. Let η be the Berger measure of TM1H and let τp,0m,n be the Berger measure of W. For each 0≤p≤m−1, using Lemma 5, we characterize the subnormality of Tm,nHp,0m,nT as a backward extension of W (in the t direction).
First observe that from (26) in Lemma 3, for 0≤p≤m−1,
(62)TM1Hm,nHp,n−1m,nTM1H≅Tm,nHp,0m,nTM1Hp,0m,nT.
By Lemma4 and (62), we have
(63)dτp,0m,ns,t=sp/mtn−1/nγp,n−1ηdηs1/m,t1/n.
Observe(64)dτp,0m,nexts,t=1tL1τp,0m,n−1sp/mt−1/nγp,n−1ηdηs1/m,t1/n(65)dτp,0m,nexts,tX=1/tL1τp,0m,n−1γp,n−1η∫0T22nsp/mt−1/ndηs1/m,t1/n.
LetWk2=shiftα0,k2,α1,k2,⋯ be the k2-th horizontal slice of T with Berger measure ξαk2 for k2≥0. By (65) and a similar way to (59), for 0≤p≤m−1, we have that
(66)Tm,nHp,0m,nTissubnormal⇒Lemma3.33.2βp,02⋯βp,n−12·1tL1τp,0m,ndτp,0m,ns,textX≤sp/mγpW0dξα0s1/m⇒checkingmomentsβp,02·γp,n−1ηγpW11tL1τp,0m,ndτp,0m,ns,textX≤sp/mγpW0dξα0s1/m⇒3.26βp,02γpW11tL1τp,0m,n1tL1τp,0m,n−1∫0T22nsp/mt−1/ndηs1/m,t1/n≤sp/mγpW0dξα0s1/m⇒βp,02∫0T22nsp/mt−1/nγpW1dηs1/m,t1/n≤sp/mγpW0dξα0s1/m⇒βp,02∫0T22u−1sp/mγpW1dηs1/m,u≤sp/mγpW0dξα0s1/mbylettingu=t1/n⇒βp,02∫0T22t−1sp/mγp,0ηdηs1/m,t≤sp/mγpW0dξα0s1/m⇒3.26βp,021tL1ηp,0m,1ηp,0m,1extX≤ξα0m,p,where ξα0m,p is the Berger measure of the subnormal shift:
(67)shiftαp,0⋯αp+m−1,0,αp+m,0⋯αp+2m−1,0,⋯.
By Lemma5, (66), and a similar way to the proof of Lemma 12, for 0≤p≤m−1, we have
(68)T1,nm,1issubnormal⇒Tm,nHp,0m,nTissubnormal⇒βp,021tL1ηp,0m,1ηp,0m,1extX≤ξα0m,p⇒Tm,1issubnormal.
Therefore, by (68), we get the result.
⇐ Since T1,nm,1=Tm,n=Tm,11,n, it follows that T1,nm,1 is subnormal whenever Tm,1 is subnormal.Corollary 15.
LetT=Wα,β∈H0. If TN1H is subnormal, then the following are equivalent:
(a)
Tm,11,n is subnormal for some m,n≥1(b)
T1,n is subnormal(c)
T is subnormalWe are ready to give a proof of our main theorem (Theorem2). For convenience, we restate Theorem 2:Theorem 16.
LetT≡Wα,β∈H0. If a corner of T is subnormal, i.e., TMpH∩NqH is subnormal for some p,q∈ℤ+, then the following are equivalent:
(a)
T is subnormal(b)
Tm,n is subnormal for all m,n≥1(c)
Tm,n is subnormal for some m,n≥1Proof.
(a)⇒ (b): This is clear from the functional calculus. (b) ⇒ (c): Clear. (c) ⇒ (a): Suppose TMpH∩NqH is subnormal for some p,q∈ℤ+ and Tm,n is subnormal for some m,n≥1. By Lemma 7, we have TM1H∩N1H is subnormal. Also, TN1Hm,n is subnormal. We thus have
(69)TN1Hm,n=TN1H1,nm,1issubnormal⇒Theorem3.12TN1Hm,1issubnormal⇒Corollary3.123.11TN1Hissubnormal.
Therefore, by Corollary15, we can see that T is subnormal.We conclude by revealing examples to illustrate Theorem2. Their proofs are given from a straightforward calculation. We will omit their proofs.Example 1.
LetT=T1,T2≡Wα,β∈H0 and let TM1H be subnormal with Berger measure
(70)dηs,t=tdsdt+16dδ0,1/3s,t+16dδ1/2,1s,t+16dδ1,1/2s,t.
Also, let(71)dξα0s≔12ds+16dδ0s+16dδ1/2s+16dδ1s,be the Berger measure of W0≡shiftα0,0,α1,0,⋯. We then have:
(a)
T is subnormal(b)
T2,1 is subnormal(c)
T1,2 is subnormal
(72)0<β0,0≤13.Example 2.
LetT=T1,T2≡Wα,β∈H0 and let TM1H∩N1H be subnormal with Berger measure
(73)dμcores,t=2stdsdt+14dδ1,1/2s,t+14dδ1/2,1s,t.Also, let(74)dξα0s≔12ds+16dδ0s+16dδ1/2s+16dδ1s,be the Berger measures on 0,1 of W0≡shiftα0,0,α1,0,⋯ and
(75)dηβ0t≔y2dt+1−11y26dδ0t+2y23dδ1/2t+y26dδ1t,be the Berger measures on 0,1 of V0≡shiftβ0,0,β0,1,⋯. We then have:
(a)
T is subnormal(b)
T2,2 is subnormal
(76)α0,1,β0,0∈0,13×0,111.



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