1 Tag THEORY © Copyright by INCREST, 1979
COMMUTING WEIGHTED SHIFTS AND ANALYTIC FUNCTION THEORY IN SEVERAL VARIABLES
NICHOLAS P JEWELL and A R LUBIN
1 INTRODUCTION
Given a separable complex Hilbert space H with orthonormal basis {e,} and a bounded sequence of complex numbers {w,}, a weighted shift operator T is a (bounded linear) operator which satisfies Te, = w,e,41 for all n T is called unilateral or bilateral according as the index n ranges over the non-negative integers or over all the integers An excellent introduction to the theory of such operators and an extensive bibliography can be found in the recent comprehensive survey article by A L Shields [12] It is shown there that each weighted shift is unitarily equivalent to multiplication by the function z on a weighted H? or L? space This jdentification has been the cornerstone of an extensive interplay between operator theory and analytic function theory and weighted shift operators have been a rich source of examples and counter-examples in both areas
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2 DEFINITIONS AND ELEMENTARY PROPERTIES
Let N be a fixed positive integer throughout We will use multi-index notation, ie., let J be a multi-index (i,, , ix) of integers We write J > 0 whenever i; 2 0,
i= 1, , N We also use the notation
II = lá T Tại [= al ivh
For J > 0 we write where z =(z,, .,Zy)¢C™ and
T! = Th Tix
whenever T = {7,, , Ty} is a family of N commuting operators We let e, = = (0, , 1, ., 0) be the multi-index having i; = 1 or 0 according as j =k or otherwise and 0 be the multi-index (0,0, .,0) whose every entry is zero For I the multi-index (/,, , iy), 7 + e, denotes the multi-index (4, .,%, +1, ., dy) Let {e;} be an orthonormal basis of a complex Hilbert space H and let {w,,;:7 = 1, ., N} be a bounded net of complex numbers such that
(*) WroiWtet = WrWiee,« for al L1<kl<N
&(H) denotes the algebra of all bounded linear operators on HH
DEFINITION A system of N-variable weighted shifts is a family of N operators, T = {T,, ., Ty} on H such that
Tye; = Wr, j€i+e, , G = 1, oe , N)
Clearly the condition (*) on the set {w,, ;} implies that T is a commuting family of operators The family, 7, is called a unilateral shift or bilateral shift according as [ ranges over {1:7 2 0} or all the multi-indices of integers
In the following we will restrict our attention primarily to systems of N-va- riable unilateral weighted shifts (which we will just call a unilateral shift) since these yield our main applications of the theory Some of the results which are stated only for unilateral shifts have analogous statements for the bilateral case and, for the most part, we leave it to the reader to investigate when this is possible So from now on, unless stated otherwise, we assume that T is a unilateral shift (Similarly, we could generalize further and omit condition (*) to define non-commuting shifts, but our applications all deal with the commutative case.)
PROPOSITION 1 If {A;} are complex numbers of modulus 1, then T is unitarity equivalent to the weighted shift S = {S,, ., Sy} with weights
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[We note that T= {7j, , Ty}, where each 7, acts on a Hilbert space H, is said to be unitarily equivalent to S = {S,, , Sy} where each S; acts on a Hilbert space K, if there exists a unitary operator U: H > K such that
U*S,U = T,, J=1, ,N.]
Proof Let U be the unitary operator defined by Ue, = /,;e;
COROLLARY 2 Suppose all the weights w,;; of T are non-zero Then T is unitarily equivalent to the unilateral shift S with weights given by
J [wy jl
Proof In Proposition 1 let 24) = 1 and 4;„, = A,w,,;/|w, | The corollary follows once we show that {/,: J > 0} is well-defined Let J=+g=h+y=1+ + & Then ° _ _ Whi Ate eM It tei Ay = Anse, = Dy TT TT ¿,.| [Wry ex, jl _ AW WI ex j — ÂM}, 77+ k _ Wis ex, i |wz, ¿| IW 2+a,x Are Wr+ey,k = J TT TT —— “tƑ,+ gự* |f⁄r+ s;, |
Hence, by induction (over |/|), {A,;: J 2 0} is well-defined
Note Provided all the weights are non-zero, Corollary 2 is valid for bilateral shifts also In this case we define {A,} inductively as follows: (note that Proposition 1 is also true for bilateral shifts),
Ay = 1, Apee, = Ary, llr jl, and Aye, = 2yWz-c,, l|Wr-e,, j|- As above 7, is well-defined for
J=h+e=l,+e, and J=1,—¢=1,— &;
Trang 4210 NICHOLAS P JEWELL and A R, LUBIN Then > — 7 41,3 Arne Wye, j _ Age, SO |tz,,;Í Iwye, jl _ Aare kk Wnt 7 —_ Wye, al |wy—e,, jl
— ZW ye, Wren, eT, ilW]—e,48;,k
|Wyee al [Wye Wr, M/[Wr—ecae,, kl
2 Wee, xl? WƑ, jÌYĐ/— e„ 3 e;, kÌ “I =
Wy —eq, el” |wy, jl Writ, +8;,k
= Wrioj WJƑ-a+e,k } Wreeteyk
a 5 TT TT TT “E+ Ey ee
lw, || [Wye te, xÌ Wye, +8;,k
_ — 4
= Ajy—t, F Aye
This calculation shows that {/,} is well-defined for all J
COROLLARY 3 Suppose 4 = (A,, ,4y) where |A,| = = |Ay| = 1 Then T= {T\, , Ty} and 2T= {AT ., AnTy}
are unitarily equivalent
Proof Let 4; = 4} 2x in Proposition 1
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PROPOSITION 6 T; is compact if and only if jw,,;)) 20 áas jÍ + co Tjc€ € G°(0 < p < 00) if and only if Y)\w;, |? < c
PROPOSITION 7 Let A be an operator on H having matrix (a,s) with respect to the basis {e;\, ie., ayy = (Aes, e;), and let S be a weighted shift with weight sequence {u,, ;} Then, for any j,1 <j < N,
: AS, = TA
if and only if
| Uy, j47,542, = 9 if =0 (J>0) Vy, j4t+e,,s+¢; = Mr, đi; otherwise
(If S and T are bilateral, then AS;=T,A if and only if Vy, Aree, ste; = Wr, 741, ; for all I,J.)
Proof Compare the action of AS;, T;A on e;
This proposition can be used to derive necessary and sufficient conditions for two shifts T and S to be similar or unitarily equivalent, e.g., two unilateral shifts, both with positive weights sets, w,,, and v;,,, are unitarily equivalent if and only if wy,; =07,; for all 7>O0andl1<j<N
EXAMPLES (1) Let L2(7”) denote the standard Lebesgue space of square sum- mable functions from the N-torus, T, into C Let H?(T) denote the standard Hardy space of L°(7%) functions with analytic extension to the N-polydisc Then {e, = z'} is an orthonormal basis for Hj®(7®) or L7») according as J > 0 or J is all multi- indices The system M = {M,,, , Mz,y} where M,, acts on H*(T%) or L*(7*) by multiplication by z;(1 <j < N) gives a system of N-variable weighted shifts, uni- lateral or bilateral, respectively, with weights w,,; = 1 for all J andj
(2) Let S% denote the unit sphere in C’ Let H?(S%) denote the standard Hardy space given by the closure in £2(S%) of the polynomials in the coordinate functions Z,.+.,Zy For each j,1 <j < N, let M,, act on H*(S%) by multiplication by Z, We can parametrize the sphere in such a way that the system M = {M,,, , Mzy}
is identified as a system of N-variable weighted shifts with weights given by
w= G+ IPMN + NYP
(see [7]) Note that M,, is the Toeplitz operator acting as H*S*) with symbol the jth coordinate function
(3) Let {e,: I 20} be an orthonormal basis for a Hilbert space H and let T= {T,, ., Ty} be a system of N-variable weighted unilateral shifts with weights
1 +
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Then T can be used as a universal model for a large class of commuting con- tractions in the sense that if S = {S,, , Sy} isa system of commuting contractions such that
N
3› lIS;|l? < 1,
j=l
then S is unitarily equivalent to a compression of T to the orthogonal complement of some joint invariant subspace [8] This yields analogues of some well-known theorems modelling (single) contractions on the adjoint of the standard unilateral shift For some related work on compressions of systems of unilateral shifts and their dilations see (2, 3]
Furthermore, it can be shown that each 7; is subnormal, i.e, (for each j, 1 < <j < N) there exists a Hilbert space K; 2 H and normal operators N;€ A(K;) with N;| H=T, The lifting problem asks whether T has a commuting subnormal extension, i.e., whether there exists a Hilbert space K > H and commuting normal operators M,, , My € B(K) such that
M,ÌH=T, 1<j<N
It was formerly unknown whether the lifting problem always had a solution, but 7 answers this negatively [8] Two additional examples of commuting subnor- mals without commuting normal extension follow; we note that, at present, all known examples of this phenomenon use weighted shifts In this context, Carl Cowen has recently described an analytic Toeplitz operator (which is thus subnormal) whose commutant does not dilate In fact its commutant contains a compact ope- trator See [5]
- (4) Let N = 2and fe,: J > 0} be an orthonormal basis for H Let T = {7), To} be a two-variable system of weighted unilateral shifts with weights
_ [2 if, =0, ».— J2" ;=0,đ=n
2! 1x0, “ 1 if i, #0
Then T, and 7; are both subnormal, but do not have a commuting normal extension In fact, T, does not have any bounded extension commuting with the minimal normal extension of 7¡ This example is due to M B Abrahamse [1], although it was not given in the context of weighted shifts
(5) Let N = 2 and {e;: I > 0} be an orthonormal basis for H Let T = {T,, T2} be a two-variable system of weighted shifts with weights
Wi=l if j,=0
W2=1 if =0
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Then 7, and 7, are both subnormal, and, in fact, are both quasinormal; also, each element of the two parameter semigroup {7’} is subnormal However, T does not have a commuting normal extension [10]
Although Examples 4 and 5 are of interest as counter-examples to some natural conjectures, shifts having some of their weights zero represent, in some sense, a degenerate case Hence, unless specified otherwise, we assume all weights w, ; are non-zero Then, as already noted, Corollary 2 implies that we may assume that all the weights are positive real numbers
3 WEIGHTED SEQUENCE SPACES
As in the single operator case, we now find that we may view a system of N-variable weighted unilateral shifts as multiplication operators on certain weighted sequence spaces DEFINITION Let {8,: J > 0} be aset of strictly positive numbers with fy= 1 Then, let H?(B) = [ƒŒ) = bịt: II/Hỗ = DUP Bi < œ} Clearly H®(ÿ) is a Hilbert space with the inner product <f.8> = ¥ fib Bi iz0
We note that the elements of H?(f) are considered as formal power series with- out regard to convergence at any point ze C®, {z!: J > 0} forms an orthogonal basis for H?(B) which is, in general, not orthonormal
Let M = {M,,, , M_,} denote the multiplication operators given by
M,,f(z) = 2f@ (j= 1,. ,N)
defined on the ‘“‘polynomials” in the coordinate functions, z,, of H°(f) Then M.,, which may not be bounded on H*(f), shifts the weighted basis {z!} of W*() and, as the following proposition shows, this is equivalent to a weighted shift acting on an orthonormal basis
PROPOSITION 8 The linear transformations M_, (j= 1, ., N) acting on H*(B) form a system which is unitarily equivalent to a system of injective weighted unilateral shift linear transformations with weights w,, , defined in terms of B as below Conversely, every system of injective weighted unilateral shifts with weights w,, ; is unitarily equi- valent to {M,,, ., M,,,} acting on some H(B)
Proof For the first half define w, ; = ¡;,/; On the other hand, given {T,, ., Ty} define B, by
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B, # 0 for all I since each T, is injective and, in fact, B; > 0 since we may assume that the shifts have positive weights In either case define U: H > H(B) by Ue, = By} z! Then U is unitary and
U*M,U=T,,j=1, ,N
Note that the above proposition holds only for commuting weighted shifts For the bilateral case, define again Mỹ; — Brze,l Br In the other direction, given {w, ;}, define Bo =1, Bite, = Br Mr, (which reduces to 7e; = ze; in the unilateral case) and Bree, = B;ÍW1-‹,¿ We must show that {Ø,} ¡s well-defned Thịs follows, since, if J=h+t¿=Ì; —t,—= Ï+T+ tị— bạ then
By = Brae, = Ũn,Wi,¡ = Br-eWi—eji = By Wi—e;, i/Wr—e;, is = By Wr, |Wyre ey i = BreedWi+es—e;,5 = Br, — & = Br-
COROLLARY 9 M,, is bounded (j= 1, ., N) if and only if
{BU + e;)/BŒ):I > 0}
is bounded for each j, 1 <S j < N
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Note that {8,} is uribounded, but My Mẹ are éach bounded operators “This is
possible since an
II, =2> 1
4 THE COMMUT ANT
Given a formal power series, g{z), in N variables, @ induces a map on H?(f) by formal power series multiplication f— pf We denote by H~(f) the set {p: of € H%B) for all fe H*(B)} and, for pe H™ () we denote by M, the map
taking f to of Since z  Hđ(B) we have oz° = g € H*(f) for any pe H(6), ie., H°* (8) 6 H*) So ° has the representation
oz) =.) 972"
/ —— 720
A linear operator A on H(f) can be represented by the matrix (A,;) with respect to the orthogonal basis z/, where
¢Az’, z I 271?
If A and B are “operators with corresponding’ matrices (A4) and (Bia), then AB is represented by the matrix whose G J)th entry is pa Al,xK Bx.z Ai; = PROPOSITION 10 (1) M, is a bounded map on H*(f); (2) Moy = My My (9, ÿ e H*(0) Proof (1) Note that g(z) 7 = ys @¡ ztÍ = — ers? 1 Thus ” _ 927, 2°) = Ox 2# (K 2 J)
Hence the matrix of M, is given by Arz= 7-7 (U2 2) 0 elsewhere This implies that M, is bounded since its matrix is everywhere defined @) For f c H0), Mẹyƒ= (œÚ) ƒ (note that gy is a well-defined element of H™()) Thus M,,f= PW) = My My;
This last proposition shows that H°(B) is a commutative algebra of bounded operators on H*(f) containing M,,, j = 1, ., N Hence the commutant, {M, j= =l, , N}', contains #®() We will show that equality holds
THEOREM 11 Jf A is a bounded operator on H*(B) which commutes with M,,(j=1, ,N), then A= M, for some po ¢ H® (B)
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Proòf Let @=4z9 Then ọ e H*(ÿ) and A4z#Z— AM,x z° where M,—={M;„, -; Äf,„} Thus 4zX =M,x Az°=zXo (K > 0) Thus Af = of for all polynomials ƒ For arbitrary ƒc H*{(ÿ) we approximate ƒ by polynomials ƒ„, and, by the algebraic properties of power series multiplication, we see that Af= of Thus ge H*(f) and 4 = Mẹ
COROLLARY 12 Suppose T is a system of injective unilateral shifts Then {T,, +, Ty}' is a maximal abelian subalgebra of B(H)
CoROLLARY 13 {7), °+°, Ty} have no common reducing subspace
EXAMPLES (1) Consider Example 2 of Section 2 The operators 7), ., Ty are the Toeplitz operators on H?(S%) with symbols given by the coordinate functions Theorem 11 shows that Te @(H%(S*)) commutes with T,, , Ty if and only if T = M, for some g € H@™ (f) Since H*(B) = H*(S%), it is easy to see that H™(B)= = HS"), i.e., functions which are boundary values of bounded analytic functions as the open unit ball in C’ This was first proved for general N in [7] Corollary 12 shows that the C*-algebra generated by {7,:j= 1, ., N} is irreducible; this was first proved by Coburn [4] using properties of the Szegé reproducing kernel for
H(S?) An alternative proof in the spirit of this paper is given in [7]
(2) In the case N = 1 it is easy to see that M,, does not have a square root For N > 1, since it is only {M,,, , Mz, }’ and not {M,,}' that is well-behaved,
it is not surprising that roots exist For f¢ H*(8) we write ƒ) = > zh (vn, : ‹ -›Zw—1)-
Define A by
AŒĐgŒì, ., Z„—1)) — Z8?! gà, -› Zn—) A(t} 8)=Z¡ ZnB
We choose f so that we can extend A to H*(f) by linearity and continuity (this is possible for many choices of £) Then A3ƒ= M,, ƒ We use the notation ||| = ||M,|| for g ¢ H(B) Note that loll, < ll@|Ì+ (since |l@llạ = l⁄¿Z°llạ < Malllz°lạ = lạ) COROLLARY 14 H°{) ís a commutative Banach algebra 5 THE SPECTRUM
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THEOREM 15, Let T be a system of N-variable weighted unilateral shifts (not necessarily injective) Then
o(T;) = (LEC: A] <r(T)} (j= 1, .,N)
Proof Consider 7, and assume the system is injective Suppose A ¢ o(7,) Then (7, — 4)™ exists and commutes with 7), ., Ty Hence (7, — 4) += Mẹ for some g € H®(f) by Theorem 11 Thus (z¡ — 2)@ = 1 which implies — —A®†Đjƒ= =iy=0 %1 0 otherwise Therefore, lp;8Œ + J)*| = |< Myz’, 27+7>| < MIB) BU + 9) Thus BU + JIB) < Ml lal? for all J with i, = = iy = 0 and for all J Using Proposition 4 we have J+k | Mx || = sup Wp Wore oe) Wa (ke, 11 SUP Bs + Kes) J>0 J>0 BY)
So | MEI < |, ||Al*+! Taking kth roots and letting k > oo, we have r(T,) < lÀI Thus o(7,) = {A : |A| < r(7,)} If 7, has some zero weights, then T, is a norm limit of operators of the form S, where {S;: 1 < j-< 4} is a system of injective unilateral shifts and hence the result follows
At this point it seems valuable to point out that information concerning {7), +, Ty} can be gleaned by regarding each 7; as a countable direct sum of one-va- riable weighted shift operators Let us return to our original orthonormal basis {e;: 2 0} and for simplicity consider the case N = 2; the case for general N holds analogously
Write X,, for the closed linear span of {e,,,:” 20} for each m > 0 and let Y, be the closed linear span of {e,,,:m 20} for each n 20 Clearly for each n,m > 0, X,, reduces 7, and Y, reduces T, Thus, we can write JT, = © T,|X,, and
m=0
T, = ®T,|Y, and each of the summands is a one-variable weighted shift with
n=0
respect to the corresponding basis
THEOREM 16 Let T: {T,, T,} be a system of injective two-variable weighted unilateral shifts Then T, and T, have empty point spectrum
Proof Suppose x is an eigenvector for T, corresponding to eigenvalue 4 We write
œ
x= V Xam Cam
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Then
œ
Xọo — 3 Xn0€n0
n=0
is an eigenvector for the injective shift, 7,|x», with eigenvalue 7 However, injective (one-variable) unilateral shifts have no eigenvalues [9] and so x = 0 Similarly,
OO
0= Xm py XnmÊnm
for each m and, therefore, x = 0 Thus 7¡, and similarly, T,, have empty point
spectrum
We can use the direct sum decomposition to improve Corollary 13 by dropping the commutativity assumption
PROPOSITION 17 Let {T,, T,} be a system of not necessarily commuting weighted unilateral shifts having no zero weights Then T,, T, have no common reducing subspace
Proof Consider H = ‘@ X,, With respect to this decomposition, we have m=0 TIX, ˆ 0} [0- 0 DI Sy 0” T, =" TAY and T,= | ` 5 0, tý] | 05%
where S; maps X; onto X;,, since all of the weights are non-zero Tết P be a pro- jection onto a common reducing a so that PT, = TỊP and P7, = 7P Writing the matriX P = (P„) with P* 4 and comparing entries of PT; „and 7;Ð
we se, since each Sỹ is onto, ‘that P,, = sự ii # j Thus, each P;; is a projection’ on X,, dnd P,, iT |X,.= (1X, ) Pa %'1) :An injective weighted (one- variable) unilateral shift i is irreducible and so P;, = Oor 7|X,_, for each 7 PT, =7,P implies that PS) 9 = = Si oPi-1, 12> 2 and hénce P = O or Taccording as P,, —0or 1X9, L€., there are no nonittivial comnion reducing subspaces
6 ANALYTIC STRUCTURE
For any w = (wy, ., wy) € CN, let 4, denote the linear functional of eva- luation at w defined on the polynomials by 4,,(p) = p(w)
DEFINITION w is said to be a bounded point evaluation (bpe) on H*(B) if 1, extends to a bounded linear functional on H°(B), i.e., there exists some c > 0 such that
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-If w.is a.bpe, then, by the Riesz theorem, there exists.some k,, € H?(B) such that 2„(ƒ) = Œ.,k„) for all fe H*(B) We call k,; the reproducing kernel for H*(B) at w Since (z7, k,) = w’ we see that we must have
ky) => iv 1B
PROPOSITION 18 w is a bpe if and only jƒ k„(z) € HB), i.e., if and only if Y Imi|?2 |wx|?/x/8(J}##< œ
J20
PROPOSITION 19 w is a bpe if and onl y if w, is in the point spectrum of 1;* for each j, 1 <j < N, and each w, corresponding to a common eigenvector in H*(B) If |wl = I Tjll for each j,1 <j < N, then w is not a bpe
Proof Ifwisa bpe, then, for fe H?Œ), we have
(f Tk) = Lf kw )=w, fk) = (Lik)
Thus 77*k,,=w,k,, and so w, is in the point spectrum of Tt for each j By Corollary 3, w, is in the point spectrum ‘of T} for each j with a common eigenvector
Conversely, suppose x is a non-zero vector in HB) and Tx = w,xforj=1, ,N Let A(f) = (f cx), for f¢ H*(B) where c is a non-zero constaint to be determined A is'a bounded linear functional on H*(B) Also
Gf) = &h cx) = (f.¢ Tix) = wjAlf)
So Uz!) = = 1220), a z 0 and so ^(z) # 0 Put c= 1/(Z, x) so that -A(z9) — = 1; Then Mp) = p(w) for ail polynomials p and so w isa bpe If |w,| = |7;|| for j=l, , N, then y jw, [24 oe [wl > 1 y ————————> fw, [Ps a ee [wry [27% — J20, a BUY cà J= x „ BƠ? / j=l N n=0,1, 2, = § inane - 00 n=0 J=l, 5N
since B(ne;) < |77l? < I7,”
PROPOSITION 20 If w is a “bpe ‘and = She! € HX), then the series Yi f,w" converges absolutely to 4,(f) Thus we can unambiguously denote i wf) = #0) Further this property characterizes a bọc
Proof Suppose w is a bpe and f= ¥) fz’ Let S, = - ` fyz7-be the nth partial
lJi<n
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Conversely, suppose the power series 3 /;w converges for all ƒe #*(Ø) Then we can define a linear functional on H(B) by A(f) = Si ffw’ and let ko = iJ,San wzJ8 Then Lk) = YY Hw > Af) as n- œ JJ:<n By the uniform boundedness principle k,= 3; w'z!JBj< H*() J>0
and thus w is a bpe
PROPOSITION 2] If the power series @(z) converges at w for all ọc H°®(), then
le(w)| < Moll
Further, if w is a bpe and f € H*(B), then
AAP) = Awl@)Aw(f)-
Also k,, is an eigenvector for all operators in {T#, ., Tx}'
Proof Since H™(B) is a commutative Banach algebra with identity and evalua- tion at w is a multiplicative linear functional, we have |g(w)| < ||M,|| from general Banach algebra theory Note that this holds for all bpe’s w, but in general will hold for a larger class of w’s For
ọ = Xo„z'c H°(),
les) < IIM„I/I7”,
and this can be used to compute which w’s give convergence of the power series @(w) In special cases The second statement follows from a formal power series argument and the third by reasoning similar to the proof of Proposition 19
we have
_ THEOREM 22 If y represents a bounded analytic function on the polydisc {z: |z;| < < |/n} then ọc H™() and
Moll < sup {]9(2)I : leit < Var Till}
Proof The key to the proof is an analogue of von Neumann’s inequality for commuting contractions, namely
Ip(T: - , 7„)J<sup {Ip(2\ : lz;| < a.IT:l} f9]
We note that this result is not in general best possible and we do not know if,
in fact, ||p(7,, ., Ty)l| < sup {|p(z)| : [z;] < 1} (which is in general false for commut-
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This establishes the theorem for g a polynomial For a general bounded ana- lytic function we approximate @ by its rectangular Cesaro sums which converge strongly as in the one-variable case [13, p 310]
Question Under what conditions is H®(f)=H™(D) for some D< CY, and in this case how do we describe D?
7 SUBNORMAL SHIFTS
The results for one-variable weighted shifts carry through to N-variable case almost without change and so we only quote the most important of these which generalizes Berger’s characterization of one-variable subnormal weighted shifts PROPOSITION 23 {T), ., Ty} has a commuting normal extension if and only if there exists a probability measure yt defined on the N-dimensional rectangle R = (0, a,] x [0, a] x x [0, ay], where a; = ||T;|], such that Ỉ tỉ" 1a1"du(f)=
R
= ( 1 du(t) = B3 for all J > 0
®
The proof is identical to the one-variable case given in [12]
8 ALGEBRAS GENERATED BY SHIFTS
Let /, be-the closure in @(4), in the weak operator topology, of the polyno- mials in 7,, , Ty It is clear that Z+ is contained in the commutant of {7¡, oly} which is equal to {M,:ge¢H™(B)} by Theorem 11 We wish to show that equality holds This follows from Theorem 26 below since a subspace of @(H) is closed in the weak operator topology if and only if it is closed in the strong operator topology For weTN = {(w, ., wy) :|w;| = 1,1 <j < N}, define a map on H%(f), f>f,, by SZ, Zy) = /(Mi, ses WyZn), 1.€., (fos = Wh Proposition 24, (1) If 9 € H*(B) and we TN’, then o,, € H®(B) and Lvwlloo = Illes
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Proof (1) Let ƒ€ H*(ÿ) Then (0@ƒ)„0„J„ and (f;,)»;=/5»¿-: Therefore (ƒ>)„=— f
and.so @u„r=0„(fS)„= (@Œz))„ €.H°()and so ø„€ H(ÿ) Also ï/„'g=¡#¡; and so
uf | = hfs S 1 Pice fia 18s 1 Pwibeo < Đ le — 1Ø») ie S 0u le:
(2) llee; — Oveslle = | > PKer+K - wk €;+xÌ]; =
= j ¥ q ~~ w~')ø¡_„e,)g—0 as wo a, 1, 1) ies
for fixed J Thus, using (1), the required continuity holds at (1, 1; ., 1) By transla- tion the result follows
~ This proposition allows US to define the vector-valued Riemann integral (oun) fds (when fe HB), gy € H™(8), p is continuous on T* and ds is normalized Lebesgue measure on 7%) as in-the; one-variable case
PROPOSITION 25 If p € H®(B) and p is of the form:
Pw) = 3 ;pkwf0w< 7X}
where only finitely many coefficients are different from zero, then
(7.7) ds = Myxy when (xp) (2) = Deyps 2’ € HUB):
Proof Identical to the one-variable case
Let Kj(t)=K;,(4) Ky,(ty) be the multiple (rectangular) Fejer kernel [13, p 303] where X,, is the usual one-variable Fejer kernel Then, for @ € H®(f), @*X; = [UŒ + l) Gre] So Sx(@) = Ø,(@) where — Sx(g) = » 9,2", r< THEOREM “26 If @c H°(f}, then (1) ơ,(g)< H0); (2) Iø;(@)Jl+ < ll@lz:
(3) ø,(@) > @ in the strong operator -topology Proof Identical to the one-variable case
9 REMARKS
Trang 17COMMUTING WEIGHTED SHIFTS 223
{T,, -,Ty} of finite codimension can be obtained [7] and this result can be extended to the general situation with a little work Similarly, results concerning reflexivity of sé, can be proved [7] and again these can be extended to the general situation with suitable hypotheses However, little extra is gained by looking at the proofs in the general situation and so we omit them here
In [11] O’Donovan gave a beautiful description of the C*-algebras generated by a single weighted shift in terms of certain covariance algebras when the shift is either essentially normal or has closed range It would seem of interest to extend these ideas to the N-variable case since a general result would extend results already known for particular examples (Example | — see [6], Example 2 — see [7]) This appears
to be a non-trivial problem
Research of first author supported by the Commonwealth Fund of New York in the form of
a Harkness fellowship Research of second author supported by NSF Grant MCS 78—01442
REFERENCES
ABRAHMSE, M.B., Commuting subnormal operators, I/linois J Math., 22 (1978), 171—176 CLARK, D N., On commuting contractions, J Math Anal Appl., 32 (1970), 590 —596 CLarK, D.N., Commutants that do not dilate, Proc Amer Math Soc., 35 (1972), 483 —486 Copurn, L A., Toeplitz operators on odd spheres, Lecture Notes in Mathematics, Springer-
Verlag, No 345, 7—12
5 Cowen, C.C., An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators, to appear
6 DoucLas, R.G.; Howe, R., On the C*-algebra of Toeplitz operators on the quarter-plane, Trans Amer Math Soc , 158 (1971), 203 —217
7, JEWELL, N.P., Multiplication by the coordinate functions on the Hardy space of the unit
sphere in C", Duke Math J., 44 (1977), 839-851
8 Lupin, A., Models for commuting contractions, Michigan Math J., 23 (1976), 161—165 9 Lusin, A., A von Neumann inequality for commuting contractions, Internat J Math Math Sci., 1 (1978), 133— 135 10 LUBIN, Á., A subnormai semi-group without normai extension, Proc Amer Math Soc., 68 (1978), 176—178 11 O’Donovan, D.P., Weighted shifts and covariance algebras, Trans Amer Math Soc.,208 (1975), 1—25
12 SHieLps, A.L., Weighted shift operators and analytic function theory, in Topics in operator
theory, Math Surveys 13, Amer Math Soc., Providence, R.I., 1974
13 ZYGMUND, A., Trigonometric series, Vol.1, 2nd ed., Cambridge Univ Press, New York, 1959
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NICHOLAS P JEWELL A R.LUBIN
Department of Mathematics, Department of Mathematics, Stanford University, Illinois Institute of Technology,