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Spectral Synthesis of Jordan Operators
Journal of Mathematical Analysis and Applications 249, 652᎐667 Ž2000. doi:10.1006rjmaa.2000.6967, available online at http:rrwww.idealibrary.com on
Spectral Synthesis of Jordan Operators Steven M. Seubert Department of Mathematics and Statistics, Bowling Green State Uni¨ ersity, Bowling Green, Ohio 43403-0221 E-mail: [email protected] Submitted by Paul S. Muhly Received November 22, 1999
The purpose of this paper is to extend results of J. Wermer Ž1952, Proc. Amer. Math. Soc. 3, 270᎐277., L. Brown et al. Ž1960, Trans. Amer. Math. Soc. 96, 162᎐183., and D. Sarason Ž1972, J. Reine Agnew. Math. 252, 1᎐15; 1966, Pacific J. Math. 17, 511᎐517. on spectral subspaces of diagonalizable operators on separable complex Hilbert space to the class of so-called Jordan operators or infinite direct sums of Jordan cells. 䊚 2000 Academic Press
1. INTRODUCTION In this paper, we are concerned with a special case of the following general problem: Let X be a Banach space and let T : X ª X be a bounded linear operator which is complete, that is, whose root vectors have dense linear span in X . Under what conditions will every subspace invariant for T be the closed linear span of the root vectors for T that it contains? ŽRecall that a closed subspace M of a Banach space X is in¨ ariant for a bounded linear operator T : X ª X if T M : M and that a vector x in X is a root ¨ ector for T if there exists a complex number and a positive integer n such that ŽT y I . n x is the zero vector.. Invariant subspaces for a complete operator which are the closed linear span of the root vectors for the operator they contain are called spectral subspaces for the operator and complete operators all of whose invariant subspaces are spectral are said to admit spectral synthesis. In particular, we take X to be a separable complex Hilbert space and T to be the infinite direct sum of Jordan cells or a so-called Jordan operator. Recall that a bounded linear operator T : H ª H on a finite-dimensional Hilbert space H is a Jordan cell if there exists a complex number and an 652 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
653
SYNTHESIS OF JORDAN OPERATORS
orthonormal basis e i : 1 F i F m4 for H such that the matrix representation for T with respect to e i : 1 F i F m4 is
J Ž , m. s
1
0 1
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ .. .
0 0 0
0 0 0
1
0
.
That is, a bounded linear operator J: H ª H on a separable complex Hilbert space H is a Jordan operator if there exists a bounded sequence n4 of complex numbers, a sequence m n4 of positive integers, and a Hn and for each positive sequence Hn4 of Hilbert spaces such that H s [H integer n, the restriction J < Hn of J to Hn is the Jordan cell J Ž n , m n .. In this paper, we seek necessary and sufficient conditions for a Jordan operator to admit spectral synthesis. The special case of diagonalizable Jordan operators [J Ž n , 1. Žthat is, of complete normal operators. was studied by Wermer w14x and Brown et al. w2x, and was solved by Sarason w10, 11x in 1972. The following is a synopsis of the results relevant to our study. The most important feature of the result is the connection between condition Žv. and the rest. Also see Nikolskii w7, Wermer᎐Sarason theorem, p. 99x. for additional conditions. THEOREM 1. Let n4 be a sequence of distinct points in the open unit disc D ' z g C : < z < - 14 and let D s [J Ž n , 1. be a diagonalizable operator acting on a Hilbert space H s span e n4 . The following are equi¨ alent: Ži. D admits spectral synthesis, Žii. a ¨ ector x in H is cyclic for D if and only if ² x, e n : / 0 for all positi¨ e integers n, Žiii. there does not exist a sequence wn4 of complex numbers for which 0 - Ý < wn < - ⬁ and Ýwn in s 0 for all nonnegati¨ e integers i, Živ. the weakly closed algebra generated by D and the identity coincides with the commutant of D, and Žv. there does not exist a bounded complex domain ⍀ such that sup< f Ž z .< : z g ⍀ 4 s sup< f Ž z .< : z g ⍀ l n44 for all functions f bounded and analytic on ⍀. If, in addition, the points n accumulate only on the boundary of D, then these conditions are equi¨ alent to: Žvi. there does not exist a sequence wn4 of complex numbers for which 0 - Ý < wn < - ⬁ and Ýwn e n z s 0 for all complex numbers z,
654
STEVEN M. SEUBERT
Žvii. the map T : H ⬁ ª l⬁Ž . from the space H ⬁ of functions bounded and analytic on the unit disc to l⬁Ž . where ' Ý ␦ n is the measure consisting of point masses at the n defined by T : f ª f Ž n .4 is not an isometry, and Žviii. not almost e¨ ery point of the unit circle is in the nontangential cluster set of n4 . Throughout this paper, we let n4 denote a bounded sequence of distinct complex numbers, we let m n4 denote a bounded sequence of positive integers, and we let J s [J Ž n , m n . denote a Jordan operator ⬁ ⬁ acting on a Hilbert space H s [ns1 Hn s [ns1 span e n, i : 1 F i F m n4 . It is understood that for each positive integer n, e n, i : 1 F i F m n4 denotes the unique orthonormal basis for Hn for which the matrix representation for J < Hn with respect to e n, i : 1 F i F m n4 is the Jordan cell J Ž n , m n .. For distinct points n , the Jordan operator J ' [J Ž n , m n . has a dense set of cyclic vectors Žsee w4x. and moreover, the spectral subspaces of J are precisely those subspaces of the form [span e i, j : i F d j 4 where d n4 is any sequence of nonnegative integers with d n F m n . ŽIf d n s 0, then we interpret span e i, j : i F d j 4 as the zero subspace.. We have restricted our attention to the case of distinct points n in order to avoid uninteresting complications. The techniques of the paper apply in the more general case and yield similar results when suitably modified. The restriction that the block sizes m n4 be bounded, however, is imposed only so our proofs are valid. For a survey of the present state of the spectral synthesis problem, see Nikolskii w6x. Also see Markus w5x.
2. EQUIVALENT CONDITIONS FOR SPECTRAL SYNTHESIS In this section, we give several equivalent conditions for a Jordan operator to admit spectral synthesis. These conditions are analogues of conditions Ži., Žii., Žiii., and Žvi. of Theorem 1 on diagonalizable operators. THEOREM 2. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s ⬁ [J Ž n , m n . be a Jordan operator acting on a Hilbert space H s [ns1 Hn s ⬁ [ns 1 span e n, i : 1 F i F m n4 . The following are equi¨ alent: Ži.
J admits spectral synthesis,
Žii. a ¨ ector x s [x n in H is cyclic for J if and only if for each positi¨ e integer n, x n is a cyclic ¨ ector for J Ž n , m n . on Hn ,
SYNTHESIS OF JORDAN OPERATORS
655
Žiii. there does not exist a set wn, j 4 of complex numbers such that ⬁ minŽ iq1, m n . Ž i . iyjq1 n
⬁
min Ž iq1, m n .
Ý
Ý
ns1
js1
i iyjq1 jy1 n
ž /
m nyjq1
Ý
² x n , e n , kqjy1 :² yn , e n , k : . Ž 1 .
ks1
Define wn, j ' ² x n , e n, kqjy1 :² yn , e n , k : for all positive integers n and all j n
656
STEVEN M. SEUBERT
the term ␥n occurs in the Ž m n y jn q 1.st coordinate. Since < wn, j < 1r2 F ␥n and sup m n - ⬁, x ' [x n and y ' [yn are norm convergent. Moreover, the vectors x n and yn are chosen so that wn, j s n yjq1 ² x , e :² yn , e n , k : for all positive integers n and all j s Ým ks 1 n n, kqjy1 1, 2, . . . , m n . Hence for all nonnegative integers i, we have that ² J i x, y : s iq1, m n . Ž i . iyjq1 Ý⬁ns 1 ÝminŽ wn, j s 0. j y 1 n js1 For each positive integer n, ² x n , e n : is nonzero. Hence x n is cyclic for Ž J n , m n . on Hn and so, by hypothesis, x s [x n is cyclic for J on H . Since iq1, m n . Ž i . iyjq1 ² J i x, y : s Ý⬁ns 1 ÝminŽ wn, j s 0 for all nonnegative intej y 1 n js1 gers i and x is cyclic for J, we have that y s [yn is the zero vector. By n
3. SUFFICIENT CONDITIONS In this section, we show that a Jordan operator J s [J Ž n , m n . admits spectral synthesis whenever for each positive integer i, i is in the unbounded component of Ž k : k / i 4 . c. THEOREM 3. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s ⬁ [J Ž n , m n . be a Jordan operator acting on a Hilbert space H s [ns1 Hn . If for each positi¨ e integer i, the orthogonal projection PH i : H ª Hi is in the weakly closed algebra generated by J and the identity, then the Jordan operator J s [J Ž n , m n . admits spectral synthesis. Proof. Let M be an arbitrary invariant subspace for J and let x be an arbitrary element of M . Since PH i is in the weakly closed algebra generated by J and the identity, M is invariant for J and the identity, and x is in M , it follows that PH i x is in M . For each positive integer n, let d n denote the largest integer j s 1, 2, . . . , m n for which there exists a vector x in M with ² x, e n, j : nonzero Žif no such vector exists, take d n to be zero.. Clearly M : [span e n, j : j F d n4 Žif d n s 0, we interpret span e n, j : j F d n4 as 04.. We show equality. Let n be any positive integer for which d n is nonzero and let x n be any vector in M for which ² x n , e n, d n : is nonzero. By the first part of the proof, PH nx n is in M . Moreover, ² PH nx n , e n, d n : s ² x n , e n, d n : is nonzero and so PH nx n is a cyclic vector for J Ž n , m n . restricted to span e n, j : j F d n4 . Hence M = pŽ J . PH nx n : p is a polynomial4 s pŽ J Ž n , m n .. PH nx n : p is a polynomial4 s span e n, j : j F d n4 . So M = [span e n, j : j F d n4 . That is, M s [span e n, j : j F d n4 is a spectral subspace of J.
657
SYNTHESIS OF JORDAN OPERATORS
THEOREM 4. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s ⬁ [J Ž n , m n . be a Jordan operator acting on a Hilbert space H s [ns1 Hn . Let i be any positi¨ e integer and let p␣ 4 be a set of polynomials. Then p␣ Ž J .4 con¨ erges in the weak operator topology to the projection operator PH i if and only if 0 if k / i , 1 if k s i , Ž j. Ž Žii. lim ␣ ˆ . p␣ k s 0 for all j, k G 1, and Žiii. sup␣ , k < ˆ p␣Ž j. Ž k .< - ⬁ for all j G 0 where 0 if j G m k , ˆp␣Ž j. Ž k . ' Ž j. p␣ Ž k . if j - m k . Ži. lim ␣ p␣ Ž k . s
½
½
Proof. Suppose that the polynomials p␣ 4 satisfy properties Ži., Žii., and Žiii.. We show that p␣ Ž J .4 converges in the weak operator topology to the projection operator PH i . Let x s [Ž x n, 1 , x n, 2 , . . . , x n, m n . and y s Hn . We have [Ž yn, 1 , yn, 2 , . . . , yn, m n . denote arbitrary elements of H s [H that ² p␣ Ž J . x, y : ⬁
s
m kyjq1
mk
yk , j
Ý Ý ks1 js1
Ý yi , j js1
ž
k/i
p␣Ž ry1. Ž k . x k , rqjy1
1
Ý Ž r y 1. ! p␣Ž ry1. Ž i . x i , rqjy1 rs1 m kyjq1
q Ý yk , j
1
Ž r y 1. !
rs1
m iyjq1
mi
s
ž
Ý
ž
Ý rs1
1
Ž r y 1. !
/
/
p␣Ž ry1. Ž k . x k , rqjy1 .
/
Ž 2.
By Ži. and Žii., m iyjq1
mi
lim ␣
Ý
yi , j
js1
ž
1
Ý Ž r y 1. ! p␣Ž ry1. Ž i . x i , rqjy1 rs1
/
s ² PH i x, y : .
Ž 3.
By Žiii., M ' sup␣ , k< ˆ p␣Ž j. Ž k .< : j s 0, 1, . . . , m k y 14 - ⬁ from which it follows that m kyjq1
Ý yk , j k/i
ž
Ý rs1
1
Ž r y 1. !
p␣Ž ry1. Ž k . x k , rqjy1 F M Ž sup m n . 5 x 5 5 y 5 - ⬁.
/
Ž 4.
658
STEVEN M. SEUBERT
By means of contradiction, suppose that ² p␣ Ž J . x, y :4 does not converge to ² PH i x, y :. Then there exists a sequence ␣ n4 such that ² p␣ Ž J . x, y :4 does not converge to ² PH x, y :. Hence by the dominated n i convergence theorem, Eq. Ž4., and Žiii., m kyjq1
lim
Ý yk , j
nª⬁ k/i
ž
1
Ý
Ž r y 1. !
rs1
m kyjq1
s
Ý
ž
yk , j
k/i
Ý rs1
p␣Ž ry1. Ž k . x k , rqjy1 n
1
Ž r y 1. !
/
lim p␣Ž ry1. Ž k . x k , rqjy1 n
nª⬁
/
s 0, a contradiction as ² p␣ Ž J . x, y :4 does not converge to ² PH i x, y :. Hence m kyjq1
lim ␣
Ý
yk , j
k/i
ž
Ý rs1
1
Ž r y 1. !
m kyjq1
s
Ý yk , j k/i
ž
Ý rs1
p␣Ž ry1. Ž k . x k , rqjy1
1
Ž r y 1. !
/
lim p␣Ž ry1. Ž k . x k , rqjy1 ␣
s 0.
/ Ž 5.
So by Eqs. Ž2. ᎐ Ž5., ² p␣ Ž J . x, y : ª ² PH i x, y : for arbitrary elements x and y in H . That is, p␣ Ž J .4 converges in the weak operator topology to the projection operator PH i . Conversely, suppose that p␣ Ž J .4 converges in the weak operator topology to the projection operator PH i . We show that the polynomials p␣ 4 satisfy properties Ži., Žii., and Žiii.. Let x s [x n s [Ž x n, 1 , x n, 2 , . . . , x n, m n . and y s [yn s [Ž yn, 1 , yn, 2 , . . . , yn, m n . denote arbitrary elements of H s Hn . Since p␣ Ž J .4 converges in the weak operator topology to PH i , we [H have that ⬁
m kyjq1
mk
Ý Ý ks1 js1
yk , j
ž
Ý rs1
1
Ž r y 1. !
p␣Ž ry1. Ž k . x k , rqjy1
/
s ² p␣ Ž J . x, y : mi
ª ² PH i x, y : s ² x i , yi : s
Ý x i , j yi , j .
Ž 6.
js1
Judicious choices of coordinates for the vectors x and y in Eq. Ž6. show that the polynomials p␣ 4 satisfy properties Ži. and Žii.. Hence we need only show that the polynomials p␣ 4 satisfy property Žiii.. To this end, fix j G 0
659
SYNTHESIS OF JORDAN OPERATORS
and let wn4 be an arbitrary sequence in l 1. Define x s [x k where x k s Ž0, . . . , 0, < w k < , 0, . . . , 0. with the term < w k < occuring in the jth coordinate Žtake x k s 0 g Hk if j ) m k .. Similarly, define y k s ŽŽ j y 1.!e i arg w k < w k < , 0, . . . , 0.. We have that ² p␣ Ž J . x, y : s Ý⬁ks1 ˆ p␣Ž j. Ž k . w k . By ² Ž . :4 ² : hypothesis, p␣ J x, y converges to PH i x, y . Hence Ý⬁ks1 ˆ p␣Ž j. Ž k . w k 4 converges for each w k in l 1. Moreover,
'
'
'
⬁
lim ␣
² p␣ Ž J . x, y : s ² PH x, y : . Ý ˆp␣Ž j. Ž k . wk s lim ␣ i
Ž 7.
ks1
For each w k 4 in l 1 , we define a functional L␣ : l 1 ª C by L␣ Ž w k 4. s ˆp␣Ž j. Ž k . wk . Since Ý⬁ks1 ˆp␣Ž j. Ž k . wk converges for all wk 4 in l1 , it follows from the Banach᎐Steinhaus theorem that ˆ p␣Ž j. Ž k .4⬁ks1 is in l⬁ . Hence L␣ is a bounded linear functional on l 1 , and, moreover, 5 L␣ 5 lU1 s 5 ˆ p␣Ž j. Ž k . w k is bounded in ␣ for p␣Ž j. Ž k .45 l⬁ . By Eq. Ž7., L␣ Ž w k 4. s Ý⬁ks1 ˆ all w k 4 in l 1. Hence the sequence L␣ 4 of functionals on l 1 is pointwise bounded and so uniformly bounded by the principle of uniform boundedness. That is, sup␣ 5 L␣ 5 lU1 s sup␣ 5 ˆ p␣Ž j. Ž k .45 l⬁ is finite for all j G 0. The result follows. Ý⬁ks 1
In general, the weak closure and the weak sequential closure of the set pŽ J . : p is a polynomial4 of polynomials in J do not coincide, even if J is a diagonalizable operator J s [J Ž n , 1. Žsee Wermer w14x, the Corollary to Theorem 1 of Sarason w11, p. 511x, and the remarks following Lemma 7.. In the following theorem, we give sufficient conditions for J to be the weak sequential limit of polynomials in J by applying Mergelyan’s theorem on polynomial approximation. THEOREM 5. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s ⬁ [J Ž n , m n . be a Jordan operator acting on a Hilbert space H s [ns1 Hn . Suppose that for each positi¨ e integer n, n is in the unbounded component of Ž m : m / n4 . c. Then there exists a sequence pn, i 4 of polynomials such that pn, i Ž J .4 con¨ erges in the weak operator topology to the projection operator PH n for each positi¨ e integer n. In particular, J s [J Ž n , m n . admits spectral synthesis. Proof. Let n be a fixed positive integer. By Theorem 4, it suffices to show that there exist polynomials pn, i 4 such that Ži. lim iª⬁ pn, i Ž k . s 0 1 Ž j. Ž Žii. lim iª⬁ ˆ . pn, s 0 i k Ž j. Ž Žiii. sup i, k < ˆ .< pn, i k -⬁
½
if k / n, if k s n, for all j, k G 1, and for all j G 0
660
STEVEN M. SEUBERT
where
ˆpnŽ j., i Ž k . '
½
if j G m k ,
0 pnŽ j., i
Ž k .
if j - m k .
We apply Mergelyan’s theorem Žsee Rudin w9, p. 390x.. Define K n s m : m / n4 . By hypothesis n is not in K n and so there exists ⑀ ) 0 such that B Ž n , ⑀ . l K n s ⭋ where here B Ž n , ⑀ . denotes the open ball of radius ⑀ with center n . Let Jn be any Jordan curve whose interior int Jn contains K n and whose exterior ext Jn contains B Ž n , ⑀ . . Define hŽ z . ' 0 on Ž Jn j int Jn . and hŽ z . ' 1 on B Ž n , ⑀ . . Since h is continuous on the compact set Cn s B Ž n , ⑀ . j Jn j int Jn4 and analytic on the interior Cn0 s B Ž n , ⑀ . j int Jn of Cn , by Mergelyan’s theorem there exist polynomials ˜ pn, i 4 converging uniformly to h on Cn . We show that the polynomials pn, i ' ˜ pn,N i satisfy properties Ži., Žii., and Ž j. 4 Žiii. where here N ' sup m n . It suffices to show that for each j G 0, pn, i Ž j. converges uniformly to h on an open set containing B Ž n , ⑀r2. j K n . We map to the open unit disc D and apply Cauchy’s integral formula. Since K n : int Jn and K n and Jn are compact, there exists an open set n for which K n : n : n: int Jn . So n l Jn s ⭋. By the Riemann mapping theorem Žsee Burckel w1, Theorem 9.7, pp. 299 and 303x., there exists a conformal map from int Jn onto D. Moreover, extends to a continuous map : Ž int Jn . ª D which is one-to-one on Ž int Jn . Žsee Burckel w1, Lemma 9.13, p.307x.. Since is one-to-one, it follows that Ž n . : D. Hence there exists r in Ž0, 1. for which Ž n . : B Ž0, r .. For any r ⬘ g Ž r, 1., we have that K n : n : y1 Ž B Ž0, r ⬘... Since ˜ pn, i 4 converges uniformly to h on Ž int Jn . , pn, i 4 converges uniformly to h N s h on Ž int Jn . , and so pn, i ( y1 4 converges uniformly to h( y1 on B Ž0, r ⬘.. Hence, by Cauchy’s integral formula, dzd Ž pn, i ( y1 . s Ž pXn, i ( y1 . ⭈ Ž y1 .⬘ converges uniformly to dzd Ž h( y1 . s Ž h⬘( y1 . ⭈ Ž y1 .⬘ on B Ž0, r ⬘. for all r ⬘ g Ž r, 1.. Since y1 is a conformal map from D to Ž int Jn . and continuous on B Ž0, r ⬘. for all r ⬘ g Ž r, 1., we have that inf<Ž y1 .⬘ < : z g B Ž 0, r ⬘ . 4 ) 0. Hence pXn, i ( y1 4 converges uniformly to h⬘( y1 on B Ž0, r ⬘. for all r ⬘ g Ž r, 1.. So by Cauchy’s integral formula, dzd Ž pXn, i ( y1 . s Ž pYn, i ( y1 . ⭈ Ž y1 .⬘ converges uniformly to h⬘( y1 s Ž h⬙ .(Ž y1 .⬘ on y1 ŽB Ž 0, r ⬘ . . for all r ⬘ g Ž r, 1.. Since inf<Ž y1 .⬘ < : z g B Ž 0, r ⬘ . 4 ) 0 for all r ⬘ g Ž r, 1., we have that pYn, i 4 converges uniformly to h⬙ on y1 ŽB Ž 0, r ⬘ . . for all Ž j. 4 Ž j. r ⬘ g Ž r, 1.. Induction yields that pn, on i converges uniformly to h y1 Ž y1 B Ž 0, r ⬘ . . for all r ⬘ g Ž r, 1. and hence on ŽB Ž 0, r ⬘ . . = n. Ž j. 4 Ž j. Similarly, pn, on B Ž n , 3 ⑀r4.. Hence i converges uniformly to h Ž j. Ž j. pn, i 4 converges uniformly to h on an open set containing B Ž n , ⑀r2. j K n for each j G 0. The result follows.
661
SYNTHESIS OF JORDAN OPERATORS
4. A NECESSARY CONDITION In this section we give a necessary condition for a Jordan operator to admit spectral synthesis. We begin with the following notation. Let n4 be a bounded sequence of distinct complex numbers and let m n4 be a bounded sequence of positive integers. Define R ' sup m n y 1. We denote by A the subalgebra A ' f g H ⬁ : fˆŽ i. Ž n . 4 g l⬁ for all i s 0, 1, 2, . . . , R
½
5
of H ⬁ where for each positive integer n, fˆŽ i. Ž n . '
½
if i G m n
0 f
Ž i.
Ž n .
if i - m n
R 5 ˆŽ i. Ž and norm A by defining 5 f 5 A ' 5 f 5 H ⬁ q Ý is1 f n .45 l⬁ . We denote by B the subalgebra
B ' Ž a n , 0 4 , a n , 1 4 , . . . , a n , R 4 : a n , i 4 g l⬁ for all i s 0, 1, 2, . . . , R and a n , i s 0 for all i ) m n 4 R R 5 of [is0 l⬁ and norm B by defining 5Ž a n, 0 4 , . . . , a n, R 4.5 B ' Ý is0 a n, i 45 l⬁ .
LEMMA 1.
If the bounded linear operator T : A ª B defined by T: fª
ž f Ž . 4 , fˆ
Ž1.
n
Ž n . 4 , . . . , fˆŽ R. Ž n . 4 /
is an isometry, then the Jordan operator J s [Ž n , m n . fails to admit spectral synthesis. Proof. We first show that T : A ª B is not onto. By means of contradiction, suppose that T maps A onto B. So for each sequence a n4 in l⬁ there exists a function f in A : H ⬁ for which T Ž f . s Ž a n4 , a n, 14 , a n, 2 4 , . . . , a n, R 4. where a n, i ' 0 for all n G 1 and all i s 1, 2, . . . , R. Hence f Ž n . s a n for all positive integers n, and so n4 is an interpolating sequence for H ⬁. Let B denote any interpolating Blaschke product having simple zeros n4 . Then B R Ž z . is in A and since T is an isometry, we have that 1 s 5 B R 5 A s 5 T Ž B R .5 B s 5Ž04 , . . . , 04.5 B s 0, a contradiction. Hence T : A ª B is not onto. Let C denote the subalgebra C ' Ž a n , 0 4 , a n , 1 4 , . . . , a n , R 4 . : a n , i 4 g l 1 for all i s 0, 1, 2, . . . , R and a n , i s 0 for all i ) m n 4
662
STEVEN M. SEUBERT
R R 5 of [is0 l⬁ and norm C by defining 5Ž a n, 0 4 , . . . , a n, R 4.5 C ' Ý is0 a n, i 45 l 1. Then B s C *. We show that the range of T is weak-star closed in B. By the Krein᎐Smulian theorem Žsee Conway w3, Corollary 12.7, p. 165x., it suffices to show that the range of T is weak-star sequentially closed. To this end, let f k 4 be any sequence of functions in A for which T Ž f k .4 converges weak-star to some vector Ž a n, 0 4 , a n, 1 4 , . . . , a n, R 4. in B. Since T Ž f k .4 converges weak-star, 5 T Ž f k .5 B 4 is bounded. But T is an isometry, and so 5 f k 5 A 4 is bounded. Since 5 f 5 H ⬁ F 5 f 5 A , 5 f k 5 H ⬁4 is bounded and so by Montel’s theorem, there exists a subsequence f k r 4 of f k 4 which converges uniformly to some function g on every compact subset of the unit disc. Since T Ž f k .4 converges weak-star to Ž a n, 0 4 , a n, 14 , . . . , a n, R 4. in B, it follows that a n, i s lim r ª⬁ fˆkŽ ri. Ž n . s ˆ g Ž i. Ž n . for all positive integers n and i. Hence Ž a n, 0 4 , a n, 1 4 , . . . , a n, R 4. s Tg is in the range of T and so the range of T is weak-star closed. Since T : A ª B is not onto and the range of T is weak-star closed in B, by the Hahn᎐Banach theorem there exists a nonzero weak-star continuous functional on B annihilating the range of T. That is, there exists a nonzero vector Ž u n, 0 4 , u n, 14 , . . . , u n, R 4. in C such that
⬁
R
Ý Ý u n , k fˆŽ k . Ž n .
0s
Ž 8.
ns1 ks0
for all f in A. Define wn, j ' u n, jy1Ž j y 1.! for all positive integers n and all j s 1, 2, . . . , R. Hence ⬁
⬁
mn
R
Ý Ý < wn , j < s Ý Ý < u n , k < k!F R!5 Ž u n , 0 4 , . . . , u n , R 4 . 5 C - ⬁. ns1 js1
ns1 ks0
For each nonnegative integer i, the function f i Ž z . ' z i is in A. Moreover,
ˆ
f iŽ k .
¡ ¢f
i!
Ž . s~ Ž i y k . ! n
Ž i.
if 0 F i y k and k - m n
iyk n
Ž n .
otherwise.
Hence for each nonnegative integer i, we have by Eq. Ž8. that ⬁
R
⬁
0 s Ý Ý u n , k fˆiŽ k . Ž n . s Ý ns1 ks0
ns1
min Ž iq1, m n .
Ý js1
wn , j
i iyjq1 , jy1 n
ž /
and so J fails to admit spectral synthesis by Theorem 2. It is not known if the converse of Lemma 1 is true.
663
SYNTHESIS OF JORDAN OPERATORS
5. SUNDBERG’S EXAMPLE In this section, we outline an example due to Sundberg w13x of a bounded sequence n4 of distinct complex numbers for which the diagonalizable operator D s [J Ž n , 1. admits spectral synthesis but the corresponding Jordan operator J s [J Ž n , 2. consisting of two-by-two Jordan cells does not. LEMMA 2. Let n G 2 and d in w1r2, 1x be gi¨ en, and let k and ␦ be positi¨ e numbers for which ln ␦ F lnŽ2 n. y ln 2 ⭈ ln d ln n q lnŽ2 n.4 and k G 16'2 nr␦ . Define Sk s Ž a q ib . rk : a, b g Z; < Ž a q ib . rk < F 1r2 4 . If f is any function in H ⬁ for which 5 f 5 ⬁ F n and < f ⬘Ž z .< F ␦ for all z g Sk , then < f Ž z . y f Ž0.< F 1rn for < z < F d. Proof. By Cauchy’s integral formula, we have that < f ⬙ Ž z .< F 16 n for < z < F 1r2. If < z < F 1r2, then there exists a point a in Sk such that < z y a < F 1rŽ'2 k .. It follows from Cauchy’s integral formula that < f ⬘Ž z .< - 2 ␦ for < z < F 1r2, and so < f Ž z . y f Ž0.< F ␦ for < z < F 1r2. The function g Ž z . ' lnŽ2 n. q lnŽ2 nr␦ .ln < z
2 2 n
½ ž Ž1 q r .Ž1 y d . y Ž1 y r . / r Ž2 r Ž1 y d . . 5 4 n
2 n
2 n
2 n
Ž 9.
is an integer. Define z j, k s r j e 2 i k r m j for k s 0, 1, . . . , m j y 1 and define jy1 E Sk n as in Lemma 2. Let n4 be an enumeration of E ' D ⬁ns2 D mjs0 n, j where En, j ' Ž z q z n, j .rŽ1 q z n, j z . : z g Sk n 4 . LEMMA 3. If f is any function in H ⬁ for which 5 f 5 ⬁ F n, and < f Ž z .< F 1 ny1 E < Ž .< < < and < f ⬘Ž z .< F n for all z in D mjs0 n, j , then f z F 1 q 1rn for z s r n .
664
STEVEN M. SEUBERT
Proof. We first show that for all n G 2 and all j s 0, 1, . . . , m n y 1, the hyperbolic distance n ' <Ž z n, j y z n, jq1 .rŽ1 y z n, j z n, jq1 .< between z n, j and z n, jq1 is d n . One readily checks that cosŽ2rm n . s Ž1 q rn4 .Ž1 y n2 . y Ž1 y rn2 . 2 4 r 2 rn2 Ž1 y n2 .4 . Moreover, by Eq. Ž9., cosŽ2rm n . s Ž1 q rn4 .Ž1 y d n2 . y Ž1 y rn2 . 2 4 r 2 rn2 Ž1 y d n2 .4 . Hence Ž1 q rn4 .Ž1 y n2 . y Ž1 y rn2 . 2 4Ž1 y d n2 . s Ž1 q rn4 .Ž1 y d n2 . y Ž1 y rn2 . 2 4Ž1 y n2 . from which it follows that d n s n as asserted. Let f be any function in H ⬁ for which 5 f 5 ⬁ F n, and < f Ž z .< F 1 and n y1 E < f ⬘Ž z .< F n for all z in D mjs0 Ž . ŽŽ . Ž .. n, j . Define g z ' f z q z n, j r 1 q z n, j z so that f Ž z . s g ŽŽ z y z n, j .rŽ1 y z n, j z ... If z is in Sk n, then Ž z q z n, j .rŽ1 q z n, j z . is in En, j and so < g ⬘Ž z . < s f ⬘
ž
z q zn , j 1 q zn , j z
/
⭈
1 y < zn , j < 2 <1 q z n , j z < 2
Fn
1 y rn2
Ž 1 y < z <.
2
F ␦n
by choice of rn . So by Lemma 2, < g Ž z . y g Ž0.< - 1rn for < z < F d n . Hence for all z between z n, j and z n, jq1 with < z < s d n , we have that < f Ž z . < F < f Ž zn , j . < q < f Ž z . y f Ž zn , j . < F1q g
ž
z y zn , j 1 y zn , j z
/
y g Ž 0 . F 1 q 1rn
since <Ž z y z n, j .rŽ1 y z n, j z .< F n s d n . Hence < f Ž z .< F 1 q 1rn for < z < s rn . LEMMA 4. If f is any function in H ⬁ and < f ⬘Ž z .< is bounded on E s jj En, j , then 5 f 5 ⬁ s sup E < f Ž z .<. In particular, the Jordan operator J s [J Ž n , 2. ha¨ ing eigen¨ alues n4 s E fails to admit spectral synthesis. Proof. Let f be any function in H ⬁ for which sup E < f ⬘ < - ⬁. The function g Ž z . ' f Ž z .rsup E < f < is in H ⬁ with sup E < g < s 1. For any n ) maxŽ5 f 5 ⬁ rsup E < f <; sup E < f ⬘
SYNTHESIS OF JORDAN OPERATORS
665
cally within 1r2 of z n, j , it follows that the nontangential cluster set of E has measure zero on the unit circle. Hence the diagonalizable operator D s [J Ž n , 1. admits spectral synthesis by w2, Theorem 3, p. 167x. 6. ALGEBRAS ASSOCIATED WITH JORDAN OPERATORS In this section, we study some algebras of operators associated with a Jordan operator J. In particular, we identify the commutant and double commutant of J and the weakly closed C *-algebra W * Ž J . ' p Ž J , J * . : p is a polynomial4 generated by J and the identity. We also give sufficient conditions for the weakly closed algebra W Ž J . ' p Ž J . : p is a polynomial4 generated by J and the identity to coincide with the commutant of J. The proof of the following result identifying the commutant of a Jordan operator, being routine, is omitted. LEMMA 6. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s [J Ž n , m n . ⬁ be a Jordan operator acting on a Hilbert space H s [ns1 Hn s ⬁ 4 [ns 1 span e n, i : 1 F i F m n . A bounded linear operator T : H ª H on H commutes with J if and only if T s [Tn where for each positi¨ e integer n, the matrix representation for Tn ' T < Hn with respect to e n, i : 1 F i F m n4 is upper triangular and constant on diagonals. As a consequence of Lemma 6, we have that the double commutant J 4 ⬙ of J consists of those operators S ' [Sn on H for which the matrix representation for each Sn s S < H n with respect to e n, i : 1 F i F m n4 is upper triangular and constant on diagonals. Also as a consequence of Lemma 6, we have that the commutant of the adjoint J * s [J *Ž n , m n . of the Jordan operator J consists of those operators S ' [Sn on H for which the matrix representation for each S n ' S < H n with respect to e n, i : 1 F i F m n4 is lower triangular and constant on diagonals. Hence a bounded linear operator Tn on Hn commutes with both J Ž n , m n . and J *Ž n , m n . if and only if Tn is a multiple ␣ n I of the identity I on Hn . By Lemma 6, the double commutant J, J *4 ⬙ of J, J *4 is J, J *4 ⬙ s [␣ n I : ␣ n g C4 ⬘ s [Tn : Tn is a bounded linear operator on Hn4 . Hence by the double commutant theorem Žsee Radjavi and Rosenthal w8, Theorem 7.5, p. 119x., the weakly closed C*-algebra W *Ž J . generated by J and the identity is W * Ž J . s J , J * 4 ⬙ s [Tn : Tn is a bounded linear operator on Hn 4 . A related problem is to identify the weakly closed algebra W Ž J . generated by the Jordan operator J and the identity. Certainly, W Ž J . : J 4 ⬘.
666
STEVEN M. SEUBERT
Since each subspace Hi reduces J, each orthogonal projection PH i : H ª Hi commutes with J. The converse is also true. LEMMA 7. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s [J Ž n , m n . ⬁ 4 4 be a Jordan operator acting on a Hilbert space H s [ns 1 Hn . Then J ⬘ s W J if and only if for each positi¨ e integer i, the orthogonal projection PH i : H ª Hi is in W Ž J .. Proof. Suppose that J 4 ⬘ s W Ž J .. For each positive integer i, the subspace Hi reduces J and so PH i is in J 4 ⬘ s W Ž J .. Conversely, if PH i is in J 4 ⬘ s W Ž J . for each positive integer i, then J Ž i , m i . s JPH i is in W Ž J . for each positive integer i. Hence J Ž i , m i .4 ⬘ s pŽ J Ž i , m i .. : p is a polynomial4 : W Ž J . for each positive integer i and so J 4 ⬘ s [Ti : Ti g J Ž i , m i .4 ⬘4 : W Ž J .. By Theorem 3 and Lemma 7, a necessary condition for J 4 ⬘ s W Ž J . is that J admit spectral synthesis. The converse is true for any diagonalizable Jordan operator D ' [J Ž n , 1.. ŽBy the remarks following Lemma 6, D4 ⬘ s W *Ž D ., and by Sarason w11, corollary to Theorem 1, p. 511x, the weakly closed algebra generated by a normal operator is a star-algebra if and only if every invariant subspace for the normal operator is reducing. Hence W Ž D . s D4 ⬘ if and only if D admits spectral synthesis.. So, in general, W Ž J . / J 4 ⬘. Sufficient conditions for W Ž J . s J 4 ⬘ are given in Theorem 5. An open question is under what additional conditions on n4 and m n4 , if any, does the spectral synthesis of J imply J 4 ⬘ s W Ž J .?
REFERENCES 1. R. B. Burckel, ‘‘An Introduction to Classical Complex Analysis: Volume 1,’’ Academic Press, New York, 1979. 2. L. Brown, A. Shields, and K. Zeller, On absolutely convergent exponential sums, Trans. Amer. Math. Soc. 96 Ž1960., 162᎐183. 3. J. B. Conway, ‘‘A Course in Functional Analysis,’’ Graduate Texts in Mathematics, No. 96, Springer-Verlag, New York, 1985. 4. J. P. Lesko and S. M. Seubert, Cyclicity results for Jordan and compressed Toeplitz operators, Integral Equations Operator Theory 31 Ž1998., 338᎐352. 5. A. S. Markus, The problem of spectral synthesis for operators with point spectrum, Math. USSR Iz¨ . 4 Ž1970., 670᎐696. 6. N. K. Nikolskii, The present state of the analysis-synthesis problem. I, in ‘‘Fifteen Papers on Functional Analysis,’’ Amer. Math. Soc. Trans., Ser. 2, Vol. 124, pp. 97᎐130, Am. Math. Soc., Providence, 1984. 7. N. K. Nikolskii, ‘‘Treatise on the Shift Operator,’’ A Series of Comprehensive Studies in Mathematics, No. 273, Springer-Verlag, BerlinrHeidelberg, 1986. 8. H. Radjavi and P. Rosenthal, ‘‘Invariant Subspaces,’’ Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, BerlinrHeidelberg, 1973.
SYNTHESIS OF JORDAN OPERATORS
667
9. W. Rudin, ‘‘Real and Complex Analysis,’’ 3rd ed., McGraw-Hill, New York, 1987. 10. D. Sarason, Weak-star density of polynomials, J. Reine Angew. Math. 252 Ž1972., 1᎐15. 11. D. Sarason, Invariant subspaces and unstarred operators algebras, Pacific J. Math. 17 Ž1966., 511᎐517. 12. J. E. Scroggs, Invariant subspaces of normal operators, Duke Math. J. 26 Ž1959., 95᎐112. 13. C. Sundberg, private communication, University of Tennessee, 1995. 14. J. Wermer, On invariant subspaces of normal operators, Proc. Amer. Math. Soc. 3 Ž1952., 270᎐277.
Spectral Synthesis of Jordan Operators Steven M. Seubert Department of Mathematics and Statistics, Bowling Green State Uni¨ ersity, Bowling Green, Ohio 43403-0221 E-mail: [email protected] Submitted by Paul S. Muhly Received November 22, 1999
The purpose of this paper is to extend results of J. Wermer Ž1952, Proc. Amer. Math. Soc. 3, 270᎐277., L. Brown et al. Ž1960, Trans. Amer. Math. Soc. 96, 162᎐183., and D. Sarason Ž1972, J. Reine Agnew. Math. 252, 1᎐15; 1966, Pacific J. Math. 17, 511᎐517. on spectral subspaces of diagonalizable operators on separable complex Hilbert space to the class of so-called Jordan operators or infinite direct sums of Jordan cells. 䊚 2000 Academic Press
1. INTRODUCTION In this paper, we are concerned with a special case of the following general problem: Let X be a Banach space and let T : X ª X be a bounded linear operator which is complete, that is, whose root vectors have dense linear span in X . Under what conditions will every subspace invariant for T be the closed linear span of the root vectors for T that it contains? ŽRecall that a closed subspace M of a Banach space X is in¨ ariant for a bounded linear operator T : X ª X if T M : M and that a vector x in X is a root ¨ ector for T if there exists a complex number and a positive integer n such that ŽT y I . n x is the zero vector.. Invariant subspaces for a complete operator which are the closed linear span of the root vectors for the operator they contain are called spectral subspaces for the operator and complete operators all of whose invariant subspaces are spectral are said to admit spectral synthesis. In particular, we take X to be a separable complex Hilbert space and T to be the infinite direct sum of Jordan cells or a so-called Jordan operator. Recall that a bounded linear operator T : H ª H on a finite-dimensional Hilbert space H is a Jordan cell if there exists a complex number and an 652 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
653
SYNTHESIS OF JORDAN OPERATORS
orthonormal basis e i : 1 F i F m4 for H such that the matrix representation for T with respect to e i : 1 F i F m4 is
J Ž , m. s
1
0 1
⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ .. .
0 0 0
0 0 0
1
0
.
That is, a bounded linear operator J: H ª H on a separable complex Hilbert space H is a Jordan operator if there exists a bounded sequence n4 of complex numbers, a sequence m n4 of positive integers, and a Hn and for each positive sequence Hn4 of Hilbert spaces such that H s [H integer n, the restriction J < Hn of J to Hn is the Jordan cell J Ž n , m n .. In this paper, we seek necessary and sufficient conditions for a Jordan operator to admit spectral synthesis. The special case of diagonalizable Jordan operators [J Ž n , 1. Žthat is, of complete normal operators. was studied by Wermer w14x and Brown et al. w2x, and was solved by Sarason w10, 11x in 1972. The following is a synopsis of the results relevant to our study. The most important feature of the result is the connection between condition Žv. and the rest. Also see Nikolskii w7, Wermer᎐Sarason theorem, p. 99x. for additional conditions. THEOREM 1. Let n4 be a sequence of distinct points in the open unit disc D ' z g C : < z < - 14 and let D s [J Ž n , 1. be a diagonalizable operator acting on a Hilbert space H s span e n4 . The following are equi¨ alent: Ži. D admits spectral synthesis, Žii. a ¨ ector x in H is cyclic for D if and only if ² x, e n : / 0 for all positi¨ e integers n, Žiii. there does not exist a sequence wn4 of complex numbers for which 0 - Ý < wn < - ⬁ and Ýwn in s 0 for all nonnegati¨ e integers i, Živ. the weakly closed algebra generated by D and the identity coincides with the commutant of D, and Žv. there does not exist a bounded complex domain ⍀ such that sup< f Ž z .< : z g ⍀ 4 s sup< f Ž z .< : z g ⍀ l n44 for all functions f bounded and analytic on ⍀. If, in addition, the points n accumulate only on the boundary of D, then these conditions are equi¨ alent to: Žvi. there does not exist a sequence wn4 of complex numbers for which 0 - Ý < wn < - ⬁ and Ýwn e n z s 0 for all complex numbers z,
654
STEVEN M. SEUBERT
Žvii. the map T : H ⬁ ª l⬁Ž . from the space H ⬁ of functions bounded and analytic on the unit disc to l⬁Ž . where ' Ý ␦ n is the measure consisting of point masses at the n defined by T : f ª f Ž n .4 is not an isometry, and Žviii. not almost e¨ ery point of the unit circle is in the nontangential cluster set of n4 . Throughout this paper, we let n4 denote a bounded sequence of distinct complex numbers, we let m n4 denote a bounded sequence of positive integers, and we let J s [J Ž n , m n . denote a Jordan operator ⬁ ⬁ acting on a Hilbert space H s [ns1 Hn s [ns1 span e n, i : 1 F i F m n4 . It is understood that for each positive integer n, e n, i : 1 F i F m n4 denotes the unique orthonormal basis for Hn for which the matrix representation for J < Hn with respect to e n, i : 1 F i F m n4 is the Jordan cell J Ž n , m n .. For distinct points n , the Jordan operator J ' [J Ž n , m n . has a dense set of cyclic vectors Žsee w4x. and moreover, the spectral subspaces of J are precisely those subspaces of the form [span e i, j : i F d j 4 where d n4 is any sequence of nonnegative integers with d n F m n . ŽIf d n s 0, then we interpret span e i, j : i F d j 4 as the zero subspace.. We have restricted our attention to the case of distinct points n in order to avoid uninteresting complications. The techniques of the paper apply in the more general case and yield similar results when suitably modified. The restriction that the block sizes m n4 be bounded, however, is imposed only so our proofs are valid. For a survey of the present state of the spectral synthesis problem, see Nikolskii w6x. Also see Markus w5x.
2. EQUIVALENT CONDITIONS FOR SPECTRAL SYNTHESIS In this section, we give several equivalent conditions for a Jordan operator to admit spectral synthesis. These conditions are analogues of conditions Ži., Žii., Žiii., and Žvi. of Theorem 1 on diagonalizable operators. THEOREM 2. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s ⬁ [J Ž n , m n . be a Jordan operator acting on a Hilbert space H s [ns1 Hn s ⬁ [ns 1 span e n, i : 1 F i F m n4 . The following are equi¨ alent: Ži.
J admits spectral synthesis,
Žii. a ¨ ector x s [x n in H is cyclic for J if and only if for each positi¨ e integer n, x n is a cyclic ¨ ector for J Ž n , m n . on Hn ,
SYNTHESIS OF JORDAN OPERATORS
655
Žiii. there does not exist a set wn, j 4 of complex numbers such that ⬁ minŽ iq1, m n . Ž i . iyjq1 n
⬁
min Ž iq1, m n .
Ý
Ý
ns1
js1
i iyjq1 jy1 n
ž /
m nyjq1
Ý
² x n , e n , kqjy1 :² yn , e n , k : . Ž 1 .
ks1
Define wn, j ' ² x n , e n, kqjy1 :² yn , e n , k : for all positive integers n and all j n
656
STEVEN M. SEUBERT
the term ␥n occurs in the Ž m n y jn q 1.st coordinate. Since < wn, j < 1r2 F ␥n and sup m n - ⬁, x ' [x n and y ' [yn are norm convergent. Moreover, the vectors x n and yn are chosen so that wn, j s n yjq1 ² x , e :² yn , e n , k : for all positive integers n and all j s Ým ks 1 n n, kqjy1 1, 2, . . . , m n . Hence for all nonnegative integers i, we have that ² J i x, y : s iq1, m n . Ž i . iyjq1 Ý⬁ns 1 ÝminŽ wn, j s 0. j y 1 n js1 For each positive integer n, ² x n , e n : is nonzero. Hence x n is cyclic for Ž J n , m n . on Hn and so, by hypothesis, x s [x n is cyclic for J on H . Since iq1, m n . Ž i . iyjq1 ² J i x, y : s Ý⬁ns 1 ÝminŽ wn, j s 0 for all nonnegative intej y 1 n js1 gers i and x is cyclic for J, we have that y s [yn is the zero vector. By n
3. SUFFICIENT CONDITIONS In this section, we show that a Jordan operator J s [J Ž n , m n . admits spectral synthesis whenever for each positive integer i, i is in the unbounded component of Ž k : k / i 4 . c. THEOREM 3. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s ⬁ [J Ž n , m n . be a Jordan operator acting on a Hilbert space H s [ns1 Hn . If for each positi¨ e integer i, the orthogonal projection PH i : H ª Hi is in the weakly closed algebra generated by J and the identity, then the Jordan operator J s [J Ž n , m n . admits spectral synthesis. Proof. Let M be an arbitrary invariant subspace for J and let x be an arbitrary element of M . Since PH i is in the weakly closed algebra generated by J and the identity, M is invariant for J and the identity, and x is in M , it follows that PH i x is in M . For each positive integer n, let d n denote the largest integer j s 1, 2, . . . , m n for which there exists a vector x in M with ² x, e n, j : nonzero Žif no such vector exists, take d n to be zero.. Clearly M : [span e n, j : j F d n4 Žif d n s 0, we interpret span e n, j : j F d n4 as 04.. We show equality. Let n be any positive integer for which d n is nonzero and let x n be any vector in M for which ² x n , e n, d n : is nonzero. By the first part of the proof, PH nx n is in M . Moreover, ² PH nx n , e n, d n : s ² x n , e n, d n : is nonzero and so PH nx n is a cyclic vector for J Ž n , m n . restricted to span e n, j : j F d n4 . Hence M = pŽ J . PH nx n : p is a polynomial4 s pŽ J Ž n , m n .. PH nx n : p is a polynomial4 s span e n, j : j F d n4 . So M = [span e n, j : j F d n4 . That is, M s [span e n, j : j F d n4 is a spectral subspace of J.
657
SYNTHESIS OF JORDAN OPERATORS
THEOREM 4. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s ⬁ [J Ž n , m n . be a Jordan operator acting on a Hilbert space H s [ns1 Hn . Let i be any positi¨ e integer and let p␣ 4 be a set of polynomials. Then p␣ Ž J .4 con¨ erges in the weak operator topology to the projection operator PH i if and only if 0 if k / i , 1 if k s i , Ž j. Ž Žii. lim ␣ ˆ . p␣ k s 0 for all j, k G 1, and Žiii. sup␣ , k < ˆ p␣Ž j. Ž k .< - ⬁ for all j G 0 where 0 if j G m k , ˆp␣Ž j. Ž k . ' Ž j. p␣ Ž k . if j - m k . Ži. lim ␣ p␣ Ž k . s
½
½
Proof. Suppose that the polynomials p␣ 4 satisfy properties Ži., Žii., and Žiii.. We show that p␣ Ž J .4 converges in the weak operator topology to the projection operator PH i . Let x s [Ž x n, 1 , x n, 2 , . . . , x n, m n . and y s Hn . We have [Ž yn, 1 , yn, 2 , . . . , yn, m n . denote arbitrary elements of H s [H that ² p␣ Ž J . x, y : ⬁
s
m kyjq1
mk
yk , j
Ý Ý ks1 js1
Ý yi , j js1
ž
k/i
p␣Ž ry1. Ž k . x k , rqjy1
1
Ý Ž r y 1. ! p␣Ž ry1. Ž i . x i , rqjy1 rs1 m kyjq1
q Ý yk , j
1
Ž r y 1. !
rs1
m iyjq1
mi
s
ž
Ý
ž
Ý rs1
1
Ž r y 1. !
/
/
p␣Ž ry1. Ž k . x k , rqjy1 .
/
Ž 2.
By Ži. and Žii., m iyjq1
mi
lim ␣
Ý
yi , j
js1
ž
1
Ý Ž r y 1. ! p␣Ž ry1. Ž i . x i , rqjy1 rs1
/
s ² PH i x, y : .
Ž 3.
By Žiii., M ' sup␣ , k< ˆ p␣Ž j. Ž k .< : j s 0, 1, . . . , m k y 14 - ⬁ from which it follows that m kyjq1
Ý yk , j k/i
ž
Ý rs1
1
Ž r y 1. !
p␣Ž ry1. Ž k . x k , rqjy1 F M Ž sup m n . 5 x 5 5 y 5 - ⬁.
/
Ž 4.
658
STEVEN M. SEUBERT
By means of contradiction, suppose that ² p␣ Ž J . x, y :4 does not converge to ² PH i x, y :. Then there exists a sequence ␣ n4 such that ² p␣ Ž J . x, y :4 does not converge to ² PH x, y :. Hence by the dominated n i convergence theorem, Eq. Ž4., and Žiii., m kyjq1
lim
Ý yk , j
nª⬁ k/i
ž
1
Ý
Ž r y 1. !
rs1
m kyjq1
s
Ý
ž
yk , j
k/i
Ý rs1
p␣Ž ry1. Ž k . x k , rqjy1 n
1
Ž r y 1. !
/
lim p␣Ž ry1. Ž k . x k , rqjy1 n
nª⬁
/
s 0, a contradiction as ² p␣ Ž J . x, y :4 does not converge to ² PH i x, y :. Hence m kyjq1
lim ␣
Ý
yk , j
k/i
ž
Ý rs1
1
Ž r y 1. !
m kyjq1
s
Ý yk , j k/i
ž
Ý rs1
p␣Ž ry1. Ž k . x k , rqjy1
1
Ž r y 1. !
/
lim p␣Ž ry1. Ž k . x k , rqjy1 ␣
s 0.
/ Ž 5.
So by Eqs. Ž2. ᎐ Ž5., ² p␣ Ž J . x, y : ª ² PH i x, y : for arbitrary elements x and y in H . That is, p␣ Ž J .4 converges in the weak operator topology to the projection operator PH i . Conversely, suppose that p␣ Ž J .4 converges in the weak operator topology to the projection operator PH i . We show that the polynomials p␣ 4 satisfy properties Ži., Žii., and Žiii.. Let x s [x n s [Ž x n, 1 , x n, 2 , . . . , x n, m n . and y s [yn s [Ž yn, 1 , yn, 2 , . . . , yn, m n . denote arbitrary elements of H s Hn . Since p␣ Ž J .4 converges in the weak operator topology to PH i , we [H have that ⬁
m kyjq1
mk
Ý Ý ks1 js1
yk , j
ž
Ý rs1
1
Ž r y 1. !
p␣Ž ry1. Ž k . x k , rqjy1
/
s ² p␣ Ž J . x, y : mi
ª ² PH i x, y : s ² x i , yi : s
Ý x i , j yi , j .
Ž 6.
js1
Judicious choices of coordinates for the vectors x and y in Eq. Ž6. show that the polynomials p␣ 4 satisfy properties Ži. and Žii.. Hence we need only show that the polynomials p␣ 4 satisfy property Žiii.. To this end, fix j G 0
659
SYNTHESIS OF JORDAN OPERATORS
and let wn4 be an arbitrary sequence in l 1. Define x s [x k where x k s Ž0, . . . , 0, < w k < , 0, . . . , 0. with the term < w k < occuring in the jth coordinate Žtake x k s 0 g Hk if j ) m k .. Similarly, define y k s ŽŽ j y 1.!e i arg w k < w k < , 0, . . . , 0.. We have that ² p␣ Ž J . x, y : s Ý⬁ks1 ˆ p␣Ž j. Ž k . w k . By ² Ž . :4 ² : hypothesis, p␣ J x, y converges to PH i x, y . Hence Ý⬁ks1 ˆ p␣Ž j. Ž k . w k 4 converges for each w k in l 1. Moreover,
'
'
'
⬁
lim ␣
² p␣ Ž J . x, y : s ² PH x, y : . Ý ˆp␣Ž j. Ž k . wk s lim ␣ i
Ž 7.
ks1
For each w k 4 in l 1 , we define a functional L␣ : l 1 ª C by L␣ Ž w k 4. s ˆp␣Ž j. Ž k . wk . Since Ý⬁ks1 ˆp␣Ž j. Ž k . wk converges for all wk 4 in l1 , it follows from the Banach᎐Steinhaus theorem that ˆ p␣Ž j. Ž k .4⬁ks1 is in l⬁ . Hence L␣ is a bounded linear functional on l 1 , and, moreover, 5 L␣ 5 lU1 s 5 ˆ p␣Ž j. Ž k . w k is bounded in ␣ for p␣Ž j. Ž k .45 l⬁ . By Eq. Ž7., L␣ Ž w k 4. s Ý⬁ks1 ˆ all w k 4 in l 1. Hence the sequence L␣ 4 of functionals on l 1 is pointwise bounded and so uniformly bounded by the principle of uniform boundedness. That is, sup␣ 5 L␣ 5 lU1 s sup␣ 5 ˆ p␣Ž j. Ž k .45 l⬁ is finite for all j G 0. The result follows. Ý⬁ks 1
In general, the weak closure and the weak sequential closure of the set pŽ J . : p is a polynomial4 of polynomials in J do not coincide, even if J is a diagonalizable operator J s [J Ž n , 1. Žsee Wermer w14x, the Corollary to Theorem 1 of Sarason w11, p. 511x, and the remarks following Lemma 7.. In the following theorem, we give sufficient conditions for J to be the weak sequential limit of polynomials in J by applying Mergelyan’s theorem on polynomial approximation. THEOREM 5. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s ⬁ [J Ž n , m n . be a Jordan operator acting on a Hilbert space H s [ns1 Hn . Suppose that for each positi¨ e integer n, n is in the unbounded component of Ž m : m / n4 . c. Then there exists a sequence pn, i 4 of polynomials such that pn, i Ž J .4 con¨ erges in the weak operator topology to the projection operator PH n for each positi¨ e integer n. In particular, J s [J Ž n , m n . admits spectral synthesis. Proof. Let n be a fixed positive integer. By Theorem 4, it suffices to show that there exist polynomials pn, i 4 such that Ži. lim iª⬁ pn, i Ž k . s 0 1 Ž j. Ž Žii. lim iª⬁ ˆ . pn, s 0 i k Ž j. Ž Žiii. sup i, k < ˆ .< pn, i k -⬁
½
if k / n, if k s n, for all j, k G 1, and for all j G 0
660
STEVEN M. SEUBERT
where
ˆpnŽ j., i Ž k . '
½
if j G m k ,
0 pnŽ j., i
Ž k .
if j - m k .
We apply Mergelyan’s theorem Žsee Rudin w9, p. 390x.. Define K n s m : m / n4 . By hypothesis n is not in K n and so there exists ⑀ ) 0 such that B Ž n , ⑀ . l K n s ⭋ where here B Ž n , ⑀ . denotes the open ball of radius ⑀ with center n . Let Jn be any Jordan curve whose interior int Jn contains K n and whose exterior ext Jn contains B Ž n , ⑀ . . Define hŽ z . ' 0 on Ž Jn j int Jn . and hŽ z . ' 1 on B Ž n , ⑀ . . Since h is continuous on the compact set Cn s B Ž n , ⑀ . j Jn j int Jn4 and analytic on the interior Cn0 s B Ž n , ⑀ . j int Jn of Cn , by Mergelyan’s theorem there exist polynomials ˜ pn, i 4 converging uniformly to h on Cn . We show that the polynomials pn, i ' ˜ pn,N i satisfy properties Ži., Žii., and Ž j. 4 Žiii. where here N ' sup m n . It suffices to show that for each j G 0, pn, i Ž j. converges uniformly to h on an open set containing B Ž n , ⑀r2. j K n . We map to the open unit disc D and apply Cauchy’s integral formula. Since K n : int Jn and K n and Jn are compact, there exists an open set n for which K n : n : n: int Jn . So n l Jn s ⭋. By the Riemann mapping theorem Žsee Burckel w1, Theorem 9.7, pp. 299 and 303x., there exists a conformal map from int Jn onto D. Moreover, extends to a continuous map : Ž int Jn . ª D which is one-to-one on Ž int Jn . Žsee Burckel w1, Lemma 9.13, p.307x.. Since is one-to-one, it follows that Ž n . : D. Hence there exists r in Ž0, 1. for which Ž n . : B Ž0, r .. For any r ⬘ g Ž r, 1., we have that K n : n : y1 Ž B Ž0, r ⬘... Since ˜ pn, i 4 converges uniformly to h on Ž int Jn . , pn, i 4 converges uniformly to h N s h on Ž int Jn . , and so pn, i ( y1 4 converges uniformly to h( y1 on B Ž0, r ⬘.. Hence, by Cauchy’s integral formula, dzd Ž pn, i ( y1 . s Ž pXn, i ( y1 . ⭈ Ž y1 .⬘ converges uniformly to dzd Ž h( y1 . s Ž h⬘( y1 . ⭈ Ž y1 .⬘ on B Ž0, r ⬘. for all r ⬘ g Ž r, 1.. Since y1 is a conformal map from D to Ž int Jn . and continuous on B Ž0, r ⬘. for all r ⬘ g Ž r, 1., we have that inf<Ž y1 .⬘ < : z g B Ž 0, r ⬘ . 4 ) 0. Hence pXn, i ( y1 4 converges uniformly to h⬘( y1 on B Ž0, r ⬘. for all r ⬘ g Ž r, 1.. So by Cauchy’s integral formula, dzd Ž pXn, i ( y1 . s Ž pYn, i ( y1 . ⭈ Ž y1 .⬘ converges uniformly to h⬘( y1 s Ž h⬙ .(Ž y1 .⬘ on y1 ŽB Ž 0, r ⬘ . . for all r ⬘ g Ž r, 1.. Since inf<Ž y1 .⬘ < : z g B Ž 0, r ⬘ . 4 ) 0 for all r ⬘ g Ž r, 1., we have that pYn, i 4 converges uniformly to h⬙ on y1 ŽB Ž 0, r ⬘ . . for all Ž j. 4 Ž j. r ⬘ g Ž r, 1.. Induction yields that pn, on i converges uniformly to h y1 Ž y1 B Ž 0, r ⬘ . . for all r ⬘ g Ž r, 1. and hence on ŽB Ž 0, r ⬘ . . = n. Ž j. 4 Ž j. Similarly, pn, on B Ž n , 3 ⑀r4.. Hence i converges uniformly to h Ž j. Ž j. pn, i 4 converges uniformly to h on an open set containing B Ž n , ⑀r2. j K n for each j G 0. The result follows.
661
SYNTHESIS OF JORDAN OPERATORS
4. A NECESSARY CONDITION In this section we give a necessary condition for a Jordan operator to admit spectral synthesis. We begin with the following notation. Let n4 be a bounded sequence of distinct complex numbers and let m n4 be a bounded sequence of positive integers. Define R ' sup m n y 1. We denote by A the subalgebra A ' f g H ⬁ : fˆŽ i. Ž n . 4 g l⬁ for all i s 0, 1, 2, . . . , R
½
5
of H ⬁ where for each positive integer n, fˆŽ i. Ž n . '
½
if i G m n
0 f
Ž i.
Ž n .
if i - m n
R 5 ˆŽ i. Ž and norm A by defining 5 f 5 A ' 5 f 5 H ⬁ q Ý is1 f n .45 l⬁ . We denote by B the subalgebra
B ' Ž a n , 0 4 , a n , 1 4 , . . . , a n , R 4 : a n , i 4 g l⬁ for all i s 0, 1, 2, . . . , R and a n , i s 0 for all i ) m n 4 R R 5 of [is0 l⬁ and norm B by defining 5Ž a n, 0 4 , . . . , a n, R 4.5 B ' Ý is0 a n, i 45 l⬁ .
LEMMA 1.
If the bounded linear operator T : A ª B defined by T: fª
ž f Ž . 4 , fˆ
Ž1.
n
Ž n . 4 , . . . , fˆŽ R. Ž n . 4 /
is an isometry, then the Jordan operator J s [Ž n , m n . fails to admit spectral synthesis. Proof. We first show that T : A ª B is not onto. By means of contradiction, suppose that T maps A onto B. So for each sequence a n4 in l⬁ there exists a function f in A : H ⬁ for which T Ž f . s Ž a n4 , a n, 14 , a n, 2 4 , . . . , a n, R 4. where a n, i ' 0 for all n G 1 and all i s 1, 2, . . . , R. Hence f Ž n . s a n for all positive integers n, and so n4 is an interpolating sequence for H ⬁. Let B denote any interpolating Blaschke product having simple zeros n4 . Then B R Ž z . is in A and since T is an isometry, we have that 1 s 5 B R 5 A s 5 T Ž B R .5 B s 5Ž04 , . . . , 04.5 B s 0, a contradiction. Hence T : A ª B is not onto. Let C denote the subalgebra C ' Ž a n , 0 4 , a n , 1 4 , . . . , a n , R 4 . : a n , i 4 g l 1 for all i s 0, 1, 2, . . . , R and a n , i s 0 for all i ) m n 4
662
STEVEN M. SEUBERT
R R 5 of [is0 l⬁ and norm C by defining 5Ž a n, 0 4 , . . . , a n, R 4.5 C ' Ý is0 a n, i 45 l 1. Then B s C *. We show that the range of T is weak-star closed in B. By the Krein᎐Smulian theorem Žsee Conway w3, Corollary 12.7, p. 165x., it suffices to show that the range of T is weak-star sequentially closed. To this end, let f k 4 be any sequence of functions in A for which T Ž f k .4 converges weak-star to some vector Ž a n, 0 4 , a n, 1 4 , . . . , a n, R 4. in B. Since T Ž f k .4 converges weak-star, 5 T Ž f k .5 B 4 is bounded. But T is an isometry, and so 5 f k 5 A 4 is bounded. Since 5 f 5 H ⬁ F 5 f 5 A , 5 f k 5 H ⬁4 is bounded and so by Montel’s theorem, there exists a subsequence f k r 4 of f k 4 which converges uniformly to some function g on every compact subset of the unit disc. Since T Ž f k .4 converges weak-star to Ž a n, 0 4 , a n, 14 , . . . , a n, R 4. in B, it follows that a n, i s lim r ª⬁ fˆkŽ ri. Ž n . s ˆ g Ž i. Ž n . for all positive integers n and i. Hence Ž a n, 0 4 , a n, 1 4 , . . . , a n, R 4. s Tg is in the range of T and so the range of T is weak-star closed. Since T : A ª B is not onto and the range of T is weak-star closed in B, by the Hahn᎐Banach theorem there exists a nonzero weak-star continuous functional on B annihilating the range of T. That is, there exists a nonzero vector Ž u n, 0 4 , u n, 14 , . . . , u n, R 4. in C such that
⬁
R
Ý Ý u n , k fˆŽ k . Ž n .
0s
Ž 8.
ns1 ks0
for all f in A. Define wn, j ' u n, jy1Ž j y 1.! for all positive integers n and all j s 1, 2, . . . , R. Hence ⬁
⬁
mn
R
Ý Ý < wn , j < s Ý Ý < u n , k < k!F R!5 Ž u n , 0 4 , . . . , u n , R 4 . 5 C - ⬁. ns1 js1
ns1 ks0
For each nonnegative integer i, the function f i Ž z . ' z i is in A. Moreover,
ˆ
f iŽ k .
¡ ¢f
i!
Ž . s~ Ž i y k . ! n
Ž i.
if 0 F i y k and k - m n
iyk n
Ž n .
otherwise.
Hence for each nonnegative integer i, we have by Eq. Ž8. that ⬁
R
⬁
0 s Ý Ý u n , k fˆiŽ k . Ž n . s Ý ns1 ks0
ns1
min Ž iq1, m n .
Ý js1
wn , j
i iyjq1 , jy1 n
ž /
and so J fails to admit spectral synthesis by Theorem 2. It is not known if the converse of Lemma 1 is true.
663
SYNTHESIS OF JORDAN OPERATORS
5. SUNDBERG’S EXAMPLE In this section, we outline an example due to Sundberg w13x of a bounded sequence n4 of distinct complex numbers for which the diagonalizable operator D s [J Ž n , 1. admits spectral synthesis but the corresponding Jordan operator J s [J Ž n , 2. consisting of two-by-two Jordan cells does not. LEMMA 2. Let n G 2 and d in w1r2, 1x be gi¨ en, and let k and ␦ be positi¨ e numbers for which ln ␦ F lnŽ2 n. y ln 2 ⭈ ln d ln n q lnŽ2 n.4 and k G 16'2 nr␦ . Define Sk s Ž a q ib . rk : a, b g Z; < Ž a q ib . rk < F 1r2 4 . If f is any function in H ⬁ for which 5 f 5 ⬁ F n and < f ⬘Ž z .< F ␦ for all z g Sk , then < f Ž z . y f Ž0.< F 1rn for < z < F d. Proof. By Cauchy’s integral formula, we have that < f ⬙ Ž z .< F 16 n for < z < F 1r2. If < z < F 1r2, then there exists a point a in Sk such that < z y a < F 1rŽ'2 k .. It follows from Cauchy’s integral formula that < f ⬘Ž z .< - 2 ␦ for < z < F 1r2, and so < f Ž z . y f Ž0.< F ␦ for < z < F 1r2. The function g Ž z . ' lnŽ2 n. q lnŽ2 nr␦ .ln < z
2 2 n
½ ž Ž1 q r .Ž1 y d . y Ž1 y r . / r Ž2 r Ž1 y d . . 5 4 n
2 n
2 n
2 n
Ž 9.
is an integer. Define z j, k s r j e 2 i k r m j for k s 0, 1, . . . , m j y 1 and define jy1 E Sk n as in Lemma 2. Let n4 be an enumeration of E ' D ⬁ns2 D mjs0 n, j where En, j ' Ž z q z n, j .rŽ1 q z n, j z . : z g Sk n 4 . LEMMA 3. If f is any function in H ⬁ for which 5 f 5 ⬁ F n, and < f Ž z .< F 1 ny1 E < Ž .< < < and < f ⬘Ž z .< F n for all z in D mjs0 n, j , then f z F 1 q 1rn for z s r n .
664
STEVEN M. SEUBERT
Proof. We first show that for all n G 2 and all j s 0, 1, . . . , m n y 1, the hyperbolic distance n ' <Ž z n, j y z n, jq1 .rŽ1 y z n, j z n, jq1 .< between z n, j and z n, jq1 is d n . One readily checks that cosŽ2rm n . s Ž1 q rn4 .Ž1 y n2 . y Ž1 y rn2 . 2 4 r 2 rn2 Ž1 y n2 .4 . Moreover, by Eq. Ž9., cosŽ2rm n . s Ž1 q rn4 .Ž1 y d n2 . y Ž1 y rn2 . 2 4 r 2 rn2 Ž1 y d n2 .4 . Hence Ž1 q rn4 .Ž1 y n2 . y Ž1 y rn2 . 2 4Ž1 y d n2 . s Ž1 q rn4 .Ž1 y d n2 . y Ž1 y rn2 . 2 4Ž1 y n2 . from which it follows that d n s n as asserted. Let f be any function in H ⬁ for which 5 f 5 ⬁ F n, and < f Ž z .< F 1 and n y1 E < f ⬘Ž z .< F n for all z in D mjs0 Ž . ŽŽ . Ž .. n, j . Define g z ' f z q z n, j r 1 q z n, j z so that f Ž z . s g ŽŽ z y z n, j .rŽ1 y z n, j z ... If z is in Sk n, then Ž z q z n, j .rŽ1 q z n, j z . is in En, j and so < g ⬘Ž z . < s f ⬘
ž
z q zn , j 1 q zn , j z
/
⭈
1 y < zn , j < 2 <1 q z n , j z < 2
Fn
1 y rn2
Ž 1 y < z <.
2
F ␦n
by choice of rn . So by Lemma 2, < g Ž z . y g Ž0.< - 1rn for < z < F d n . Hence for all z between z n, j and z n, jq1 with < z < s d n , we have that < f Ž z . < F < f Ž zn , j . < q < f Ž z . y f Ž zn , j . < F1q g
ž
z y zn , j 1 y zn , j z
/
y g Ž 0 . F 1 q 1rn
since <Ž z y z n, j .rŽ1 y z n, j z .< F n s d n . Hence < f Ž z .< F 1 q 1rn for < z < s rn . LEMMA 4. If f is any function in H ⬁ and < f ⬘Ž z .< is bounded on E s jj En, j , then 5 f 5 ⬁ s sup E < f Ž z .<. In particular, the Jordan operator J s [J Ž n , 2. ha¨ ing eigen¨ alues n4 s E fails to admit spectral synthesis. Proof. Let f be any function in H ⬁ for which sup E < f ⬘ < - ⬁. The function g Ž z . ' f Ž z .rsup E < f < is in H ⬁ with sup E < g < s 1. For any n ) maxŽ5 f 5 ⬁ rsup E < f <; sup E < f ⬘
SYNTHESIS OF JORDAN OPERATORS
665
cally within 1r2 of z n, j , it follows that the nontangential cluster set of E has measure zero on the unit circle. Hence the diagonalizable operator D s [J Ž n , 1. admits spectral synthesis by w2, Theorem 3, p. 167x. 6. ALGEBRAS ASSOCIATED WITH JORDAN OPERATORS In this section, we study some algebras of operators associated with a Jordan operator J. In particular, we identify the commutant and double commutant of J and the weakly closed C *-algebra W * Ž J . ' p Ž J , J * . : p is a polynomial4 generated by J and the identity. We also give sufficient conditions for the weakly closed algebra W Ž J . ' p Ž J . : p is a polynomial4 generated by J and the identity to coincide with the commutant of J. The proof of the following result identifying the commutant of a Jordan operator, being routine, is omitted. LEMMA 6. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s [J Ž n , m n . ⬁ be a Jordan operator acting on a Hilbert space H s [ns1 Hn s ⬁ 4 [ns 1 span e n, i : 1 F i F m n . A bounded linear operator T : H ª H on H commutes with J if and only if T s [Tn where for each positi¨ e integer n, the matrix representation for Tn ' T < Hn with respect to e n, i : 1 F i F m n4 is upper triangular and constant on diagonals. As a consequence of Lemma 6, we have that the double commutant J 4 ⬙ of J consists of those operators S ' [Sn on H for which the matrix representation for each Sn s S < H n with respect to e n, i : 1 F i F m n4 is upper triangular and constant on diagonals. Also as a consequence of Lemma 6, we have that the commutant of the adjoint J * s [J *Ž n , m n . of the Jordan operator J consists of those operators S ' [Sn on H for which the matrix representation for each S n ' S < H n with respect to e n, i : 1 F i F m n4 is lower triangular and constant on diagonals. Hence a bounded linear operator Tn on Hn commutes with both J Ž n , m n . and J *Ž n , m n . if and only if Tn is a multiple ␣ n I of the identity I on Hn . By Lemma 6, the double commutant J, J *4 ⬙ of J, J *4 is J, J *4 ⬙ s [␣ n I : ␣ n g C4 ⬘ s [Tn : Tn is a bounded linear operator on Hn4 . Hence by the double commutant theorem Žsee Radjavi and Rosenthal w8, Theorem 7.5, p. 119x., the weakly closed C*-algebra W *Ž J . generated by J and the identity is W * Ž J . s J , J * 4 ⬙ s [Tn : Tn is a bounded linear operator on Hn 4 . A related problem is to identify the weakly closed algebra W Ž J . generated by the Jordan operator J and the identity. Certainly, W Ž J . : J 4 ⬘.
666
STEVEN M. SEUBERT
Since each subspace Hi reduces J, each orthogonal projection PH i : H ª Hi commutes with J. The converse is also true. LEMMA 7. Let n4 be a bounded sequence of distinct complex numbers, let m n4 be a bounded sequence of positi¨ e integers, and let J s [J Ž n , m n . ⬁ 4 4 be a Jordan operator acting on a Hilbert space H s [ns 1 Hn . Then J ⬘ s W J if and only if for each positi¨ e integer i, the orthogonal projection PH i : H ª Hi is in W Ž J .. Proof. Suppose that J 4 ⬘ s W Ž J .. For each positive integer i, the subspace Hi reduces J and so PH i is in J 4 ⬘ s W Ž J .. Conversely, if PH i is in J 4 ⬘ s W Ž J . for each positive integer i, then J Ž i , m i . s JPH i is in W Ž J . for each positive integer i. Hence J Ž i , m i .4 ⬘ s pŽ J Ž i , m i .. : p is a polynomial4 : W Ž J . for each positive integer i and so J 4 ⬘ s [Ti : Ti g J Ž i , m i .4 ⬘4 : W Ž J .. By Theorem 3 and Lemma 7, a necessary condition for J 4 ⬘ s W Ž J . is that J admit spectral synthesis. The converse is true for any diagonalizable Jordan operator D ' [J Ž n , 1.. ŽBy the remarks following Lemma 6, D4 ⬘ s W *Ž D ., and by Sarason w11, corollary to Theorem 1, p. 511x, the weakly closed algebra generated by a normal operator is a star-algebra if and only if every invariant subspace for the normal operator is reducing. Hence W Ž D . s D4 ⬘ if and only if D admits spectral synthesis.. So, in general, W Ž J . / J 4 ⬘. Sufficient conditions for W Ž J . s J 4 ⬘ are given in Theorem 5. An open question is under what additional conditions on n4 and m n4 , if any, does the spectral synthesis of J imply J 4 ⬘ s W Ž J .?
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