Tag THEORY © Copyright by INCREST, 1979 COMMUTING WEIGHTED SHIFTS AND ANALYTIC FUNCTION THEORY IN SEVERAL VARIABLES NICHOLAS P.. It is shown there that each weighted shift is unitarily
Trang 11 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
In this paper we begin to extend the theory of single (i.e., one-variable) weighted shifts to systems of (N-variable) weighted shifts (which we define below) and we show an analogous identification between such systems and multiplications on cer- tain H? or L? spaces in several variables We concentrate our attention on the unila- teral case where we will develop the basic analytic function theory We will follow the outline of [12] as a model for our theory, and we omit the details of proofs which are obvious extensions of the single operator case We assume that the reader is familiar with the basic theory of operators in Hilbert space We present several applications of our theory to the theory of general commuting contractions, commut- ing subnormal operators, Toeplitz operators in several complex variables, and several variable analytic function theory It is these applications which, to a great extent, motivate this presentation of the general theory We also note that [12] con- tains a number of open problems most of which have natural extensions to the multivariable case
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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
&(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
Arg = Ais 2 AW,
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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
_ _ 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
suppose
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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
= 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
From Corollary 3 we see that the spectrum of each T,,j = 1, ., N as well
as the various parts of the spectrum have circular symmetry about the origin, and the joint spectrum is invariant under ‘‘torus’’ rotations, i.e., (4), -; Hw) € (joint spectrum of 7) implies that (u,e", , wye?®%) € (joint spectrum of 7) for all 6,¢
€ (0, 2x], j = 1, , N Also, Corollary 2 shows that, for shifts with non-zero weights,
we may always assume that {w,, ;} is a set of positive real numbers
PROPOSITION 4
{74 | = sup [Wy wWytey, Noe + Wet (in—Dew, NMI+iney,N-1 +
>
WJ+iwEy+ (NT1—=UEN sa, NH1 + * * WJ+yEN+ÍN -1EN sa + + Íz°z+ Cu—1)8y,1Ì*
PROPOSITION 5
Tre, — | Wrenstime,
WV ww | =
(If T is a bilateral shift, then
Tye; => Wr—e,, jÊJ—e» J= 1,, , NV.)
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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,
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
w= + DEM + Ð$:
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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,ñ=n
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
w7,; =0 otherwise
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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
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
T! eg = B, e,, 1 > 0.
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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
From now on we will assume that each M,, is bounded and we note that we can now interchange freely between either viewing the operators as weighted shifts
or as multiplication operators By examining the conditions for similarity of shifts
we note, amongst other facts, that
H*T*) # H*S®M) (N> 1)
EXAMPLES (1) In Example ! of Section 2, Ø; = l1 for all 7 and
H*(B) = H°*Œ*)
(2) In Example 2 of Section 2,
8; =[H/(H| + N — Ii? and H?(B) = XS)
(3) In Example 3 of Section 2,
By = LENA ?
(4) In Example 4 of Section 2,
By = 1 if i #0, Bp = 2 if i = 0.
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Note that {8,} is uribounded, but My Mẹ are éach bounded operators “This is
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° ¢ H®(B) we have oz° = g € H*(f) for any pe H(6), ie., H°* (8) 6 H*) So ° has the representation
oz) =.) 972"
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)
g(z) 7 = ys @¡ ztÍ = — ers? 1
_ 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™())
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)
5 ~ c 1056
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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
For any operator A let o(A) denote its spectrum and r(A) its spectral radius For a system of N commuting operators T = (7j, , Ty), let jo(T) denote the joint spectrum of (Tj, ., Ty)