Module tensor product of subnormal modules need not be subnormal
MMODULE TENSOR PRODUCT OF SUBNORMALMODULES NEED NOT BE SUBNORMAL
AKASH ANAND AND SAMEER CHAVAN
Abstract.
Let κ : D × D → C be a diagonal positive definite kerneland let H κ denote the associated reproducing kernel Hilbert space ofholomorphic functions on the open unit disc D . Assume that zf ∈ H whenever f ∈ H . Then H is a Hilbert module over the polynomialring C [ z ] with module action p · f (cid:55)→ pf . We say that H κ is a subnor-mal Hilbert module if the operator M z of multiplication by the coor-dinate function z on H κ is subnormal. In [Oper. Theory Adv. Appl,32: 219-241, 1988], N. Salinas asked whether the module tensor product H κ ⊗ C [ z ] H κ of subnormal Hilbert modules H κ and H κ is again sub-normal. In this regard, we describe all subnormal module tensor prod-ucts L a ( D , w s ) ⊗ C [ z ] L a ( D , w s ), where L a ( D , w s ) denotes the weightedBergman Hilbert module with radial weight w s ( z ) = 1 sπ | z | − s ) s ( z ∈ D , s > . In particular, the module tensor product L a ( D , w s ) ⊗ C [ z ] L a ( D , w s ) isnever subnormal for any s ≥
6. Thus the answer to this question is no. Introduction
Let H be a reproducing kernel Hilbert space of holomorphic functionsdefined on the unit disc D such that zf ∈ H whenever f ∈ H . Thusthe linear operator M z of multiplication by the coordinate function z on H is bounded. This allows us to realize H as a Hilbert module over thepolynomial ring C [ z ] with module action given by( p, f ) ∈ C [ z ] × H (cid:55)−→ p ( M z ) f ∈ H . Following [9], we say that the Hilbert module H is contractive if the op-erator norm of the multiplication operator M z is at most 1 . Further, wesay that H is subnormal if M z is subnormal, that is, M z has a normal ex-tension in a Hilbert module containing H (refer to [7] for a comprehensiveaccount on subnormal operators). By Agler’s Criterion [1, Theorem 3.1], H is a contractive subnormal Hilbert module if and only if for every f ∈ H ,φ f ( n ) = (cid:107) z n f (cid:107) ( n ∈ N ) is completely monotone for every f ∈ H . Recallfrom [5] that φ : N → (0 , ∞ ) is completely monotone if m (cid:88) j =0 ( − j (cid:18) mj (cid:19) φ ( n + j ) ≥ m, n ∈ N . Remark 1.1 :
Note that if φ is a completely monotone sequence then so is ψ m for any m ∈ N , where ψ m ( n ) = φ ( m + n ) ( n ∈ N ). Mathematics Subject Classification.
Primary 46E20; Secondary 46M05, 47B20.
Key words and phrases. positive definite kernels, module tensor product, subnormality. a r X i v : . [ m a t h . F A ] A ug ODULE TENSOR PRODUCT OF SUBNORMAL MODULES 2
We further note that, as a consequence of Hausdorff’s solution to theHausdorff’s moment problem [5, Chapter 4, Proposition 6.11], H is a con-tractive subnormal Hilbert module if and only if for every unit vector f ∈ H , {(cid:107) z n f (cid:107) } n ∈ N is a Hausdorff moment sequence , that is, there exists a uniqueprobability measure µ f supported in [0 ,
1] such that (cid:107) z n f (cid:107) = (cid:90) [0 , t n dµ f ( n ∈ N ) . In this text, we are primarily interested in the following one parameterfamily of subnormal Hilbert modules.
Example 1.2 :
For a real number s > , consider the Hilbert space L a ( D , w s )of holomorphic functions defined on the open unit disc D which are square in-tegrable with respect to the weighted area measure w s dA with radial weightfunction w s ( z ) = 1 sπ | z | − s ) s ( z ∈ D ) . Then L a ( D , w s ) is a Hilbert module over the polynomial ring C [ z ] (refer to[10] for the basic theory of weighted Bergman spaces). Since L a ( D , w s ) is aclosed subspace of L ( D , w s ), L a ( D , w s ) is a subnormal Hilbert module. Fur-ther, since the measure w s dA is rotation-invariant, the monomials { z n } n ∈ N are orthogonal in L a ( D , w s ). Further, (cid:107) z n (cid:107) L a ( D ,w s ) = 1 sn + 1 ( n ∈ N ) . (1.1)In particular, L a ( D , w s ) is a reproducing kernel Hilbert space associatedwith the diagonal reproducing kernel κ s ( z, w ) = s (1 − zw ) + 1 − s − zw ( z, w ∈ D ) . Remark 1.3 :
Note that L a ( D , w ) is the unweighted Bergman space L a ( D ).It is worth noting that the definition of L a ( D , w s ) , s > s = 0 . Indeed, L a ( D , w ) may be identified with the Hardy space H ( D ) ofthe open unit disc D .In [11], N. Salinas studied the notion of module tensor product of Hilbert A -modules for a complex unital algebra A . Recall that for Hilbert A -modules H and K , H ⊗ A K is obtained by tensoring Hilbert modules H and K , and then dividing out by the natural action of A on H and K . In particular, in [11, Corollary 3.6], it is shown that for the polynomialalgebra C [ z ] and for Hilbert modules H κ and H κ associated with so-calledsharp, diagonal kernels κ and κ (of finite rank) respectively, the moduletensor product H κ ⊗ C [ z ] H κ is C [ z ]-isomorphic to the Hilbert space H κ κ associated with the diagonal kernel κ κ . In case of scalar valued diagonalkernels κ and κ , the moment sequences {(cid:107) z n (cid:107) H κ } n ∈ N , {(cid:107) z n (cid:107) H κ } n ∈ N , {(cid:107) z n (cid:107) H κ κ } n ∈ N are related by the following relations: (cid:107) z n (cid:107) H κ κ = 1 n (cid:88) k =0 (cid:107) z k (cid:107) H κ (cid:107) z n − k (cid:107) H κ ( n ∈ N ) . (1.2) ODULE TENSOR PRODUCT OF SUBNORMAL MODULES 3
In [11, Remark 3.7], he asked whether H κ ⊗ C [ z ] H κ is a subnormal Hilbertmodule for subnormal Hilbert modules H κ and H κ associated with di-agonal scalar-valued kernels κ and κ respectively ? In view of (1.2) andthe discussion following Remark (1.1), this is equivalent to the followingquestion: Question 1.4.
Whether {(cid:107) z n (cid:107) H κ κ } n ∈ N is a Hausdorff moment sequencefor Hausdorff moment sequences {(cid:107) z n (cid:107) H κ } n ∈ N and {(cid:107) z n (cid:107) H κ } n ∈ N ? The same question in the context of hyponormal Hilbert modules wassettled in the affirmative in [4] around the same time (other variations ofQuestion 1.4 have also been investigated, for example, see [3, Proposition6], [6, Theorem 1.1], and [8, Theorem 7.1]). For subnormal Hilbert modules,it was believed that the answer is no. Indeed, this is true as shown in thefollowing theorem.
Theorem 1.5.
Let s and s be positive real numbers and let H κ denote themodule tensor product L a ( D , w s ) ⊗ C [ z ] L a ( D , w s ) of the weighted BergmanHilbert modules L a ( D , w s ) and L a ( D , w s ) . Then / (cid:107) z n (cid:107) H κ is a degree polynomial in n ∈ N , say, p . If ˜ p denotes the analytic extension of p to thecomplex plane, then we have the following:(1) if ˜ p has all real roots then H κ is a subnormal Hilbert module if andonly if all roots of ˜ p lie in L ;(2) if ˜ p has a non-real complex root z then H κ is a subnormal Hilbertmodule if and only if z lies in the closure of L − .where, for a real number r, L r denotes the open left half plane L r := { z ∈ C : real part of z is less than r } . Corollary 1.6.
Let s and s be positive real numbers and let H κ denote themodule tensor product L a ( D , w s ) ⊗ C [ z ] L a ( D , w s ) of the weighted BergmanHilbert modules L a ( D , w s ) and L a ( D , w s ) . Let s and p denote the sum andproduct of s , s respectively. Then we have the following:(1) If (3 s − p ) ≥ p then H κ is subnormal if and only if s > p .(2) If (3 s − p ) < p then H κ is a subnormal Hilbert module if andonly if s ≥ p . Remark 1.7 : If s ≥ p then L a ( D , w s ) ⊗ C [ z ] L a ( D , w s ) is always subnormal.Also, if 3 s ≤ p then L a ( D , w s ) ⊗ C [ z ] L a ( D , w s ) is never subnormal.The following is immediate from the preceding remark. Corollary 1.8.
For a positive number s, let H κ denote the module tensorproduct L a ( D , w s ) ⊗ C [ z ] L a ( D , w s ) of the weighted Bergman Hilbert module L a ( D , w s ) with itself. Then we have the following:(1) If s ≤ then H κ is always subnormal.(2) If s ≥ then H κ never subnormal. The proofs of Theorem 1.5 and Corollary 1.6 will be presented in thenext section. Let us illustrate these results with the help of some instructiveexamples of different flavor.
ODULE TENSOR PRODUCT OF SUBNORMAL MODULES 4 (a) (cid:8) ( s , s ) : (3 s − p ) ≥ p and 3 s > p (cid:9) (b) (cid:8) ( s , s ) : (3 s − p ) < p and s ≥ p (cid:9) Figure 1.
Regions of subnormality of the module tensorproduct L a ( D , w s ) ⊗ C [ z ] L a ( D , w s ) in s - s plane, where s and p denote the sum and product of s , s respectively; spe-cific cases investigated in Example 1.9 have been shown aspoints in s - s plane. Example 1.9 :
Let H κ , ˜ p be as given in Theorem 1.5 and let D m = m (cid:88) j =0 ( − j (cid:18) mj (cid:19) (cid:107) z j (cid:107) H κ ( m ∈ N ) . Let s and p denote the sum and product of s , s respectively. Then we havethe following:(1) The case in which ˜ p has real roots, that is, (3 s − p ) > p (seeFigure 1(a)):(a) If s = 15 , s = 10 then 3 s < p . In this case, the roots of ˜ p are − , / , / D < . (b) If s = 1 , s = 1 then s > p . In this case, the roots of ˜ p are − , − , − D m ≥ m ∈ N . The later part may beconcluded from Corollary 1.6(1).(c) If s = 3 / , s = 25 then 3 s > p . In this case, roots are − , − ± √ /
25 and D m ≥ m ∈ N . Once again,the later part may be concluded from Corollary 1.6(1).(2) The case in which ˜ p has a complex root, that is, (3 s − p ) < p (see Figure 1(b)):(a) If s = 6 , s = 6 , then s < p . In this case, the roots of ˜ p are − , ± i/ √ D < . (b) If s = 8 , s = 12 , then s < p . In this case, the roots of ˜ p are − , (3 ± i √ /
16 and D < . (c) If s = 2 , s = 2 then s = p . In this case, the roots of ˜ p are − , − ± i/ √ D m ≥ m ∈ N . The later part maybe concluded from Corollary 1.6(2).
ODULE TENSOR PRODUCT OF SUBNORMAL MODULES 5
In parts (1)(a) and (2)(a) of Example 1.9, for sufficiently large values of m ,the multiplication operator M z on H κ is not m -hypercontractive in the senseof J. Agler [1]. Moreover, in these examples, m = 75 ,
74 are the smallestintegers for which m -hypercontractivity of M z fails. It also indicates thatdirect verification of the non-subnormality of M z is tedious.2. Proof of Theorem 1.5
Recall that, for a real number r, L r denotes the open left half plane L r = { z ∈ C : real part of z is less than r } . In the proof of Theorem 1.5, we need the following lemma.
Lemma 2.1. If a ∈ ( −∞ , and a , a ∈ L , then n − a )( n − a )( n − a ) = (cid:90) [0 , t n w ( t ) dt ( n ∈ N ) , where the weight function w : [0 , → (0 , ∞ ) is given as follows:(1) If a , a , a are distinct real numbers, then w ( t ) = t − a − ( a − a )( a − a ) + t − a − ( a − a )( a − a ) + t − a − ( a − a )( a − a ) ( t (cid:54) = 0) . (2) If a = a and a is a real number not equal to a , then w ( t ) = 1( a − a ) ( t − a − − t − a − ) − a − a t − a − log t ( t (cid:54) = 0) . (3) If a = a = a , then w ( t ) = 12 t − a − (log t ) ( t (cid:54) = 0) . (4) If a = a + ib, a = a − ib are complex numbers with b > then w ( t ) = 1( a − a ) + b (cid:16) t − a − − (cid:112) ( a − a ) + b b t − a − sin( b log t + θ ) (cid:17) ( t (cid:54) = 0) , where θ denotes the principal argument of a − a .Proof. The first three parts are special cases of [2, Theorem 3.1]. To see thelast part, note that 1( n − a )( n − a )( n − a )= a − a ) + b (cid:16) n − a + a − a ib n − a − a − a ib n − a (cid:17) . Further, the term in brackets can be rewritten as (cid:90) t n − a − dt + a − a ib (cid:90) t n − a − dt − a − a ib (cid:90) t n − a − dt. However, a − a ib t − a − − a − a ib t − a − = − (cid:112) ( a − a ) + b b (cid:16) e i ( b log t + θ ) − e − i ( b log t + θ ) i (cid:17) t − a − , ODULE TENSOR PRODUCT OF SUBNORMAL MODULES 6 where θ denotes the principal argument of a − a . It is now easy to see that w ( t ) has the desired form. (cid:3) Remark 2.2 :
Note that in all the above cases, (cid:82) [0 , w ( t ) dt ∈ (0 , ∞ ) . Proof of Theorem 1.5.
We have already recorded in (1.1) that (cid:107) z n (cid:107) L a ( D ,w sj ) = 1 s j n + 1 for n ∈ N and j = 1 , . It now follows from (1.2) that (cid:107) z n (cid:107) H κ = 1 n (cid:88) k =0 ( s k + 1)( s ( n − k ) + 1) ( n ∈ N ) . However, n (cid:88) k =0 ( s k + 1)( s ( n − k ) + 1) = 16 ( n + 1)( s s n + (3( s + s ) − s s ) n + 6)= 1 α α ( n + 1)( n − α )( n − α ) , (2.3)where α , α are given by α := − γ + (cid:112) γ − s s s s , α := − γ − (cid:112) γ − s s s s (2.4)with γ := 3( s + s ) − s s . It follows that (cid:107) z n (cid:107) H κ = α α ( n + 1)( n − α )( n − α ) ( n ∈ N ) . This shows that 1 / (cid:107) z n (cid:107) H κ is a degree 3 polynomial p in n ∈ N . Let ˜ p denote the analytic extension of p to the complex plane. Assumenow that one of α and α belongs to C \ L . By (2.3) and (2.4), however,both α and α must belong to C \ L . We contend that the sequence (cid:110) (cid:107) z n (cid:107) H κ = α α ( n + 1)( n − α )( n − α ) (cid:111) n ∈ N is not a Hausdorff moment sequence. On the contrary, suppose that thereexists a probability measure µ on [0 ,
1] such that (cid:107) z n (cid:107) H κ = (cid:90) [0 , t n dµ ( n ∈ N ) . Let n be the smallest integer bigger than the maximum of real parts of α and α . Consider now the completely monotone sequence {(cid:107) z n + n (cid:107) H κ } n ∈ N (Remark 1.1), and note that by the preceding lemma, (cid:107) z n + n (cid:107) H κ = α α ( n + n + 1)( n + n − α )( n + n − α ) = (cid:90) [0 , t n w ( t ) dt for some non-negative integrable function w : [0 , → (0 , ∞ ) . On the otherhand, (cid:107) z n + n (cid:107) H κ = (cid:82) [0 , t n t n dµ ( t ) . By the determinacy of the Hausdorffmoment problem [5], we must have t n dµ ( t ) = w ( t ) dt. ODULE TENSOR PRODUCT OF SUBNORMAL MODULES 7
We now apply the previous lemma to a := − n − a := α − n and a := α − n to conclude that t − n w ( t ) takes one of the following expressions(for some scalars c , c , c , θ ):(1) If a , a are distinct real numbers, then t − n w ( t ) = c + c t − α − + c t − α − . (2) If a = a , then t − n w ( t ) = c (1 − t − α − ) + c t − α − log t. (3) If a = c + ib, a = c − ib are complex numbers with b > t − n w ( t ) = c t − a − + c t − α − sin (cid:16) b log t + θ (cid:17) , where θ is the principal argument of a − a .One way to get a contradiction is to show that the integral of t − n w ( t ) over[0 ,
1] is not convergent. However, to avoid computations, one can alterna-tively arrive at the contradiction through an interpolation result on com-pletely monotone sequences [12]. Toward this, observe that {(cid:107) z n (cid:107) H κ } n ∈ N isminimal in the sense that µ ( { } ) = 0 , where we used the convention that ∞· . By [2, Proposition 4.1], all the zeros of ˜ p ( z ) = ( z + 1)( z − α )( z − α )must lie in L , which contradicts the assumption that at least one of α and α belongs to C \ L . The remaining part in (1) is immediate from [2, The-orem 3.1].To see (2), assume that ˜ p has a non-real complex root α = a + ib ∈ L with b >
0. In view of the preceding discussion, it suffices to show that H κ is a subnormal Hilbert module if and only if α lies in the closure of L − .By (4) of the preceding lemma, w ( t ) = α α (1 + a ) + b (cid:16) − (cid:112) (1 + a ) + b b t − a − sin( b log t + θ ) (cid:17) = | α | (1 + a ) + b (cid:16) t a +1 − sin( b log t + θ )sin θ (cid:17) t − a − , (2.5)where θ denotes the principal argument of a + 1 + ib . If a = − w ( t ) = | α | b (1 − sin( b log t + θ )) ≥ t ∈ (0 , (cid:107) z n (cid:107) H κ is aHausdorff moment sequence. Assume now that α = a + ib lies in L − , thatis, a < − . Note that π/ < θ < π. Consider the function F ( t ) := t a +1 − sin( b log t + θ )sin θ ( t > , and note that F (cid:48) ( t ) ≤ t (cid:16) a + 1 − b cos( b log t + θ )sin θ (cid:17) . Let t := e − θ/b . Then, for t ∈ [ t , , we have F (cid:48) ( t ) ≤ t (cid:16) a + 1 − b cos θ sin θ (cid:17) = 0 . ODULE TENSOR PRODUCT OF SUBNORMAL MODULES 8
Hence F ( t ) ≥ F (1) = 0 for all t ∈ [ t , . For t ∈ (0 , t ) , we have F ( t ) ≥ e − θ ( a +1) /b − θ ≥ e − a +1) /b − (cid:114) (cid:16) a + 1 b (cid:17) ≥ e − a +1) /b − (cid:18) (cid:16) a + 1 b (cid:17) (cid:19) , which is clearly positive. It is now clear from (2.5) that (cid:107) z n (cid:107) H κ is a Hausdorffmoment sequence.Assume next that − < a < . Then 0 < θ < π/ . Choose m ∈ Z suchthat t a +1 m <
12 sin θ , where t m := e (2 mπ + π/ − θ ) /b . Then, by (2.5), w ( t m ) = = | α | (1 + a ) + b (cid:16) t a +1 m − θ (cid:17) t − a − m ≤ − | α | (1 + a ) + b t − a − θ < . By the continuity of w , w < t m . It follows that (cid:107) z n (cid:107) H κ is not a Hausdorff moment sequence. This completes the proof. (cid:3) Finally, we note that the conclusions in Corollary 1.6 follow from Theorem1.5 and (2.4).
Acknowledgment . The authors gratefully acknowledge Gadadhar Misrafor drawing their attention to Salinas’s question that had remained unre-solved thus far. They thank him for his constant encouragement throughoutthe preparation of this manuscript.
References [1] J. Agler, Hypercontractions and subnormality,
J. Operator Theory (1985), 203-217.[2] A. Anand and S. Chavan, A Moment Problem and Joint q-isometry Tuples, ComplexAnal. Oper. Theory (2015). doi:10.1007/s11785-015-0516-1[3] A. Athavale, Some operator-theoretic calculus for positive definite kernels,
Proc.Amer. Math. Soc. (1991), 701-708.[4] M. Badri and P. Szeptycki, Cauchy products of positive sequences. Proceedings ofthe Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987),
RockyMountain J. Math. (1990), 351-357.[5] C. Berg, J. P. R. Christensen, and P. Ressel, Harmonic Analysis on Semigroups ,Springer-Verlag, Berlin 1984.[6] C. Berg and A. J. Dur´an, Some transformations of Hausdorff moment sequences andharmonic numbers,
Canad. J. Math. (2005), 941-960.[7] J. B. Conway, The Theory of Subnormal Operators , Math. Surveys Monographs, vol36, Amer. Math. Soc. Providence, RI 1991.[8] R. Curto, S. H. Lee and J. Yoon, Subnormality for arbitrary powers of 2-variableweighted shifts whose restrictions to a large invariant subspace are tensor products,
J. Funct. Anal. (2012), 569-583.[9] R. Douglas, Operator theory and complex geometry,
Extracta Math. (2009),135-165.[10] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces , GraduateTexts in Mathematics, 199. Springer-Verlag, New York, 2000.[11] N. Salinas,
Products of kernel functions and module tensor products , Topics in op-erator theory, 219-241, Oper. Theory Adv. Appl, , Birkh¨auser, Basel, 1988. ODULE TENSOR PRODUCT OF SUBNORMAL MODULES 9 [12] D. Widder,
The Laplace Transform , Princeton University Press, London 1946.
Indian Institute of Technology Kanpur, Kanpur- 208016, India
E-mail address : [email protected] E-mail address ::