Products of Hankel operators - Semantic Scholar
Integr. equ. oper. theory 29 (1997) 339 - 363
I Integral Equations and OperatorTheory
0378-620X/97/030339-25 $1.50+0.20/0 9 Birkh~user Verlag, Basel, 1997
Products of Hankel Operators* Daoxing Xia
Dechao Zheng
This paper studies products of Hankel operators on the Hardy space.
We show t h a t
H~,o)HI~I2~H~,,{a) = 0 for all p e r m u t a t i o n cr if and only if either HI1 or HI~ or H/a is zero. Using Douglas' localization theorem and Izuchi's theorem on Sarason's three functions problem, we show that
H~176 N H~
N H~176 C H ~176 +C
is a sufficient condition for H~HgH~, H ~ H / H ; , and H~HhH~ to be compact.
Introduction Let D be the open unit disk in the complex plane. Let L 2 denote the Lebesgue space of square integrable functions on the unit circle OD. The Hardy space H 2 is the subspace of L 2 of analytic functions on D. T h u s {z'~}~ forms an o r t h o n o r m a l basis for H 2, and { z ' ~ } : ~ is an o r t h o n o r m a l basis for (H2) • Let P be the orthogonal projection from L 2 onto H 2. Given a function f in L ~ , the multiplication operator M I is defined by M / 9 = f 9 for g C L 2. T h e Hankel operator H I is defined by
H I = (I - P ) M I P . If f is b o u n d e d on OD with Fourier coefficients an, t h e n for any n >_ 0 and m < 0 we h ave
< H / z '~, z m > = < f z '~, z TM > = < z n-m, / > = a - n + m . It follows t h a t the m a t r i x of the Hankel operator H / u n d e r the above mentioned o r t h o n o r m a l bases for H 2 and (H2) • is a-1
a-2
a-3
a-4
a-2
a-3
a-4
a-5
a-3 :
a-4 :
a-5 :
a-6 :
T h e main characteristic of the above m a t r i x is t h a t the entries on each skew-diagonal are constants. In fact, this property determines a Hankel operator on the Hardy space. In this paper we will s t u d y the product H~HgH~ of three Hankel operators. This work was p a r t l y s u p p o r t e d by NSF grants. T h e second a u t h o r was Mso p a r t l y s u p p o r t e d by the Research Council of Vanderbilt University.
340
Xia and Zheng
For the product .H~H~ of two Hankel operators, Brown and Halmos [2] proved t h a t H]Hg is zero if and only if either Hf or H~ is zero. We will give examples to show t h a t it is not true t h a t
H]HgH~ is zero if and only if one of H f , Hg, and Hh is zero. For three inner functions 01, 02, and 03, we will show t h a t H ~ H ~ H ~ = 0 if and only if 0103 divides 02 - X for some constant A. This result provides a negative answer to a question in [11] whether the r a n k of H]HgH~ equals min{rankH], rankHg, rankHh }. Furthermore we will show t h a t H~,o)HI,(2)H~(a ) = 0 for all p e r m u t a t i o n cr if and only if either
Hz~ o r
HZ~ or
ttz~
is z e r o .
Axler, C h a n g and Sarason [1], and Volberg [14] proved t h a t H]Hg is compact if and only
if
H~~
n
H~176 C H ~~+ C;
here H~176 denotes the Douglas algebra generated by H ~ and f in L eO and C denotes the algebra of continuous functions on the unit circle. Using Douglas' localization theorem [4] and Izuchi's theorem [8] on Sarason's three functions problem, we will show t h a t H~
ClHC~[g]C?H~[h] C H ~176 + C is
a sufficient condition for H~HgH~, H~HfH~, and H~HhH; to be compact. It is n a t u r a l to make a conjecture t h a t H~(t)HI~(2)H]~(a) is compact for ali p e r m u t a t i o n a if and only if 3
N H~~
C H ~ + C.
i=1
Now we are not able to prove the conjecture. But we will verify the conjecture in the special case that f l , fg, and f3 are complex cm[jugates of inner functions.
1
Preliminaries For f in L ~176the Toeplitz operator with symbol f is the operator Tf on H 2 defined by
Tih = P(fh). Define the dual-Toeplitz operator S I to be (1 - P ) M f ( I -
P). We have the following
useful observations.
Lemma 1.1 Let f a n d g be in L ~
The~
1 H)H 9 = Tfg - TfTg. 2 ~H;
= si~ - szsg.
3 SfHg = HhTI i f f is in H ~. The proof of the above lemma is easy and so it is omitted. As a m a t t e r of fact, some part of this is well-known. The following Douglas' Localization Theorem is useful in studying the Fredholm theorem of Toeplitz operators [41, [51, [6}. It will be used later to get a sufficient condition for the compactness of products of three Hankel operators.
Xia and Zheng
341
D o u g l a s L o c a l i z a t i o n T h e o r e m Ifld is a C * - algebra, Z is a C * - subalgebra contained
in the center of Lt having maximal ideal space M ( Z ) and for x in M ( Z ) , I~. is the closed ideal in b[ generaled by the maximal ideal {z E Z : ~(x) = 0}, then Mz~M(Z)Ix = O. In particular, if ~
is the *-homomorphism from b! to bl/I~, then Gzeg(z)~bz is a *-isomorphism from Lt into
~'~eM(Z) ~ tl/Ix. To s t a t e the Izuchi theorem we need introduce some notation. A Douglas algebra is, by definition, a closed subalgebra of L ~176 which contains H ~176Let H~176 denote the Douglas algebra generated by the function f in L ~176T h e Gelfand space (space of nonzero multiplicative linear functionals) of the Douglas algebra B will be denoted by M(B). If B is a Douglas algebra, then
M(B) can be identified with the set of nonzero linear functionals in M ( H ~176whose representing measures (on M(L~176are multiplicative on B, and we identify the function f with its Gelfand transform on M(B). In particular, M ( H ~176 + C) = M ( H ~176 - D, and a function f E H ~176 may be thought of as a continuous function on M ( H ~176 + C). Let QC be the C * - a l g e b r a of L ~176 defined by
QC = (H ~176 + C) N (H ~176 + C), here H c~ + C denotes the Douglas algebra H~176 The set E~ is called a QC level set if E~ = (y E M ( L ~ ) : f(y) = f(x) for all f E QC}. A subset of M(L ~176 is called a set of a n t i s y m m e t r y for H ~ + C if any function in H ~176 + C which is real valued on the set is constant on it. A subset of M ( L c~) is called a support set if it is the (closed) support of the representing measure for a functional in M ( H ~176 + C). Clearly, sets of a n t i s y m m e t r y are contained in QC level sets, and support sets are sets of antisymmetry. T h e Izuchi theorem is stated as follows. Izuchi Theorem.
Let f , g, and h be in L ~176The following are equivalent.
(a) For each support set S, either f l s E H~176 or gls E H~176 or his E H~176 (b) For each QC level set Q, either flQ E H~IQ, or gIo c H~176 or hlQ E H~176 T h e Izuchi theorem above was first conjectured by Sarason [13]. The proof of the following lemma is in [10] and [12]. To make this paper readable and complete we include it. One p a r t of the lemma is the Adamyan-Arov-Krein theorem. L e m m a 1.2 Let f be in L ~ . If Ilfll~ < 1, then there is a unimodular function u e f + H ~ such
that Tu is invertible, and dist(~u, H c~) = 1. PROOF:
First we prove t h a t f + H ~ contains a u n i m o d u l a r function u and dist(~u,H ~176= 1.
In fact, this is the Adamyan-Arov-Krein T h e o r e m (see page 204 [10]). Define F : C --* [0, oo) by
F(c) = dist(f + c.zg~176
342
Xia and Zheng
T h e n F(c) is continuous. Since
F(e)
-- IIg~f+~cll--* oc,
we have f ( 0 ) = IIH~sll _< Ilfll~r < l, a n d F(c) --~ eo as z --~ oc. By t h e i n t e r m e d i a t e value theorem, t h e r e is a n u m b e r c such t h a t F(c) = 1. Thus
d i s t ( f + c, z H ~ ) = 1. Let Ic = I + c. T h e n dist(fc, H ~ ) = dist(f, H ~~ < 1, and dist(gfc, H ~176= 1. T h e later condition implies t h a t the n o r m of the Hankel o p e r a t o r H~fc is 1. Also for such a c we have
dist(~fc, H ~ + C) (1 - I I H ~ l t 2 ) l M I 2. Hence Tu is left invertible. Since HF a t t a i n s its norm, t h e r e is a function g in H 2, with 119N = 1
such t h a t
IIHFgll =
Ilgll. Thus IIr~,~gll 2 -- I I r , gll 2 = IIgll 2 - I I H F g l l 2 = 0
This implies T~g is in t h e kernel of Ts. B u t T~ is left invertible, t h e n T~ 9 is n o t zero and so T~,g = A for some nonzero c o n s t a n t A. T h u s 1 = : / ~ , a n d t h e n T f H 2 contains 1. B u t
z=Tzl
= T ~T~]9 = T z T u ~ ]g = TzT~[T,,(z g )]
and then TuH 2 c o n t a i n s z. By induction, we conclude t h a t TuH 2 c o n t a i n s {zn}n. So ~ ible.
is invert9
Xia and Zheng
2
343
Examples In this section we will completely characterize when H ~ H ~ H ~ is zero for inner functions
01, 02, and 03. Then we will present examples to show that it is not true that H~HgH~ is zero if and only if one of H f , / / 9 and Hh is zero. Moreover those examples answer negatively the question [11] whether the rank of H ~ H ~ H ~ equals the minimum of the ranks of H ~ , H ~ , H ~ . We need some notations. For two functions 0 and ~ in H ~176 let 0[j3 denote that 0 divides fl, i.e., ~ is in H ~176Define an operator V on L 2 by
Vf(w) = ~f(w) for f E L 2. It is easy to check that V is anti-unitary.
The operator V enjoys the following
properties. I.emma 2.1 Let f be in L ~
Then
(1) V 2 = I; (2) V P V = ( I - P); (3) VH]V = g~. The proof of the lemma is easy and is left for the reader. For a fixed point z in D, let kz denote the normalized reproducing kernel at z, and let
~z(w) denote the Mhbius transform Z--W
~z(w)
-
1 - -~w"
For two vectors x and y in L 2, we use x | y to denote the operator of rank one given by
(x | y)(f) -- (f,y)x for f C L 2. The following lemma is simple but useful. Lemma 2.2 Let fl, f2, and f3 be in L ~ mid z in D. Then
Tr H>,Hf, H}3S~
-
H;,Hf, H;3
= -(H~,HI, H>3Vkz) | (Vkz) - (VHf, kz) | (VT~,,TCH~,Hf, kz)+
PROOF:
Let z be a point in D. Then it is easy to check that the operator T ~ T ~ is
the orthogonal projection onto the orthogonal complement of the normalized kernel function k~. Hence
1 - T ~ T L = k~|
344
Xia and Zheng
By Property (2) of the operator V in Lemma 2.1, VT~.V = 1 - s~s~.
S~.
Thus we obtain
= (v~.) | ( v ~ z ) ,
and so
T~..r,A nf~ nla o~. = T~ H~ HI,(T~ T~. + k. | k.)H~aS~. * H * z~H~J~)(TLH]~&, * * = ( o. An elementary estimate gives
IIHzkzll~llHgkzll~llHhk, ll~ < (1 -ly(z)12)(, -Ig(z)12)0 -Ih(=)l=). Then limm(1 - I f ( z ) [ 2 ) ( 1 -Ig(z)12)(1 -Ih(z)l 2) > 0. Thus none of f, g, and h is constant on the support set Sin, and so (2) fails. This completes the proof.
9
T h e o r e m 4.2 Let f, g, and h be in L ~176The operator H~HaH ~ is compact if
H~176 n tt~176 n H~~ PROOF:
c H ~176 + C.
By Lemma 4.1 and the Izuchi theorem, we have that for each Q C - l e v e l set
Q, there are functions fQ, 9Q, and hQ in H ~ such that either flQ = fo[Q, or gIQ = gQIQ or
hlo = hQ]Q.
354
Xia and Zheng
To prove T h e o r e m 4.2, let T = H]HgH~Ha. First we need to prove t h a t T is compact. By L e m m a 1.1 we have
H;Hg
:
T]g
-
T-fTg.
Thus
T : [TTg - TTTg][T~h]2 - T~Th], and so T is in the Toeplitz algebra ,7 where J is the C* algebra generated by bounded Toeplitz operators on H 2 . Now let us apply Douglas' Localization Theorem to the algebra b / :
J / K : where ]C is the
ideal of compact operators on H 2. T h e n the center Z of 5 / i s identified with QC. As shown above, T is in 57. By Douglas' Localization Theorem, in order to prove t h a t the operator T is compact we need only show t h a t ~ ( T )
= 0 for all x E M ( Q C ) . Let Ix denote the smallest closed two-side
ideal of 57/K: which contains {[Tf] E J / ] C : f E L ~ ,
f ( ~ - l ( x ) ) = 0}. Let Q denote the Q C - l e v e I
set ~r-l(x). We first consider the case t h a t f]Q = fQIQ. Thus
T : (TTg - TTTg)(T[h[2 -- T~Th) : (Tf--r& + f~. : (T~_~
- f > r ~ T . - T ~ T ~ ) ( T ~ < , - T~T~) - TT_~T~)(TIhj2 - T~Th).
The last equality in the above equation follows from t h a t
So [T] is I~. Hence Ox([T]) = 0. Similarily, in the ease t h a t gIQ = gQIQ or h]Q = hQIQ, we can also obtain t h a t 9 x([T]) = O.
Thus we obtain t h a t T = H~HgH~Hh is compact. Next we show t h a t H~HgH~ is compact. Note t h a t
[HTHgH;][H~HgH;]* = H I H g H ; H h H ; H f = T H ; H f . Thus [H~HgH~][I~fH~tI~]* is compact. So H~HgH~ is compact. This completes the proof.
9
T h e following is a consequence of thc above theorem. Corollary 4.3 Let fl, f2, and f3 be in L ~176Then H*fo(a)'tl zzf~(2)ul*, f~(a) is compact for any permutation
a if W c I / ] n H ~ [ A ] n H~[f3] C H ~ + C. We conjecture t h a t the converse of the above corollary is true.
HT.(~)HI~(z~H]~(z ) is compact for all p e r m u t a t i o n o if and only if n a = l H ~ [ k ] c [:r~~ + C.
T h a t is, the operator
Xia and Zheng
355
Although we are not able to prove the conjecture, we will verify the conjecture in the special case t h a t fi is the complex conjugate of a n inner function for i = 1, 2, 3. First we need the following lemmas. I.emma 4.4 Let K be a bounded operator from (H2) • to H 2. I l K is compact, then lim IlK - T~o KSw~ II = O.
z~OD
PROOF:
Since operators with finite rank are dense in the ideal of compact operators, to
prove L e m m a 4.4, it suffices to show lim IIF - T*~.FS~,~II -- 0
z---~aD
for any finite r a n k operators F. Also for each finite rank operator F , there are finite functions fi in H 2 and gi in (H2) • such t h a t Tt
F = ~f~ | i=1
To finish the proof we then need only to show lim Hf |
|
z~OD
= O.
for f C H 2 and g E (H2) • B u t this follows easily from t h a t lim [l~zzf - z--ofl]2 = 0
z---~z o
and
lim ling
z~zo
- ~ g l [ 2 = 0.
So this completes the proof.
L e m m a 4.5 Let f be in L ~176For m in M ( H ~176 + C), let Sm denote the support set for ra. Then limz~mllHsk~ll~ = 0
if and only if f[s~ is in H~176m. PROOF:
We may assume t h a t [lflloo < 1. By L e m m a 1.2, there exists a u n i m o d u l a r
function u C f + H ~176 such t h a t Tu is invertible. By L e m m a 3 in [15], we have C(1 - l u ( z ) l 2) < [IHr
(1 -lu(z)[ 2)
for some positive c o n s t a n t C > 0. Thus
limz._.mC(1 - l u ( z ) l 2) _< limz__.mllHfkzll 2
I Integral Equations and OperatorTheory
0378-620X/97/030339-25 $1.50+0.20/0 9 Birkh~user Verlag, Basel, 1997
Products of Hankel Operators* Daoxing Xia
Dechao Zheng
This paper studies products of Hankel operators on the Hardy space.
We show t h a t
H~,o)HI~I2~H~,,{a) = 0 for all p e r m u t a t i o n cr if and only if either HI1 or HI~ or H/a is zero. Using Douglas' localization theorem and Izuchi's theorem on Sarason's three functions problem, we show that
H~176 N H~
N H~176 C H ~176 +C
is a sufficient condition for H~HgH~, H ~ H / H ; , and H~HhH~ to be compact.
Introduction Let D be the open unit disk in the complex plane. Let L 2 denote the Lebesgue space of square integrable functions on the unit circle OD. The Hardy space H 2 is the subspace of L 2 of analytic functions on D. T h u s {z'~}~ forms an o r t h o n o r m a l basis for H 2, and { z ' ~ } : ~ is an o r t h o n o r m a l basis for (H2) • Let P be the orthogonal projection from L 2 onto H 2. Given a function f in L ~ , the multiplication operator M I is defined by M / 9 = f 9 for g C L 2. T h e Hankel operator H I is defined by
H I = (I - P ) M I P . If f is b o u n d e d on OD with Fourier coefficients an, t h e n for any n >_ 0 and m < 0 we h ave
< H / z '~, z m > = < f z '~, z TM > = < z n-m, / > = a - n + m . It follows t h a t the m a t r i x of the Hankel operator H / u n d e r the above mentioned o r t h o n o r m a l bases for H 2 and (H2) • is a-1
a-2
a-3
a-4
a-2
a-3
a-4
a-5
a-3 :
a-4 :
a-5 :
a-6 :
T h e main characteristic of the above m a t r i x is t h a t the entries on each skew-diagonal are constants. In fact, this property determines a Hankel operator on the Hardy space. In this paper we will s t u d y the product H~HgH~ of three Hankel operators. This work was p a r t l y s u p p o r t e d by NSF grants. T h e second a u t h o r was Mso p a r t l y s u p p o r t e d by the Research Council of Vanderbilt University.
340
Xia and Zheng
For the product .H~H~ of two Hankel operators, Brown and Halmos [2] proved t h a t H]Hg is zero if and only if either Hf or H~ is zero. We will give examples to show t h a t it is not true t h a t
H]HgH~ is zero if and only if one of H f , Hg, and Hh is zero. For three inner functions 01, 02, and 03, we will show t h a t H ~ H ~ H ~ = 0 if and only if 0103 divides 02 - X for some constant A. This result provides a negative answer to a question in [11] whether the r a n k of H]HgH~ equals min{rankH], rankHg, rankHh }. Furthermore we will show t h a t H~,o)HI,(2)H~(a ) = 0 for all p e r m u t a t i o n cr if and only if either
Hz~ o r
HZ~ or
ttz~
is z e r o .
Axler, C h a n g and Sarason [1], and Volberg [14] proved t h a t H]Hg is compact if and only
if
H~~
n
H~176 C H ~~+ C;
here H~176 denotes the Douglas algebra generated by H ~ and f in L eO and C denotes the algebra of continuous functions on the unit circle. Using Douglas' localization theorem [4] and Izuchi's theorem [8] on Sarason's three functions problem, we will show t h a t H~
ClHC~[g]C?H~[h] C H ~176 + C is
a sufficient condition for H~HgH~, H~HfH~, and H~HhH; to be compact. It is n a t u r a l to make a conjecture t h a t H~(t)HI~(2)H]~(a) is compact for ali p e r m u t a t i o n a if and only if 3
N H~~
C H ~ + C.
i=1
Now we are not able to prove the conjecture. But we will verify the conjecture in the special case that f l , fg, and f3 are complex cm[jugates of inner functions.
1
Preliminaries For f in L ~176the Toeplitz operator with symbol f is the operator Tf on H 2 defined by
Tih = P(fh). Define the dual-Toeplitz operator S I to be (1 - P ) M f ( I -
P). We have the following
useful observations.
Lemma 1.1 Let f a n d g be in L ~
The~
1 H)H 9 = Tfg - TfTg. 2 ~H;
= si~ - szsg.
3 SfHg = HhTI i f f is in H ~. The proof of the above lemma is easy and so it is omitted. As a m a t t e r of fact, some part of this is well-known. The following Douglas' Localization Theorem is useful in studying the Fredholm theorem of Toeplitz operators [41, [51, [6}. It will be used later to get a sufficient condition for the compactness of products of three Hankel operators.
Xia and Zheng
341
D o u g l a s L o c a l i z a t i o n T h e o r e m Ifld is a C * - algebra, Z is a C * - subalgebra contained
in the center of Lt having maximal ideal space M ( Z ) and for x in M ( Z ) , I~. is the closed ideal in b[ generaled by the maximal ideal {z E Z : ~(x) = 0}, then Mz~M(Z)Ix = O. In particular, if ~
is the *-homomorphism from b! to bl/I~, then Gzeg(z)~bz is a *-isomorphism from Lt into
~'~eM(Z) ~ tl/Ix. To s t a t e the Izuchi theorem we need introduce some notation. A Douglas algebra is, by definition, a closed subalgebra of L ~176 which contains H ~176Let H~176 denote the Douglas algebra generated by the function f in L ~176T h e Gelfand space (space of nonzero multiplicative linear functionals) of the Douglas algebra B will be denoted by M(B). If B is a Douglas algebra, then
M(B) can be identified with the set of nonzero linear functionals in M ( H ~176whose representing measures (on M(L~176are multiplicative on B, and we identify the function f with its Gelfand transform on M(B). In particular, M ( H ~176 + C) = M ( H ~176 - D, and a function f E H ~176 may be thought of as a continuous function on M ( H ~176 + C). Let QC be the C * - a l g e b r a of L ~176 defined by
QC = (H ~176 + C) N (H ~176 + C), here H c~ + C denotes the Douglas algebra H~176 The set E~ is called a QC level set if E~ = (y E M ( L ~ ) : f(y) = f(x) for all f E QC}. A subset of M(L ~176 is called a set of a n t i s y m m e t r y for H ~ + C if any function in H ~176 + C which is real valued on the set is constant on it. A subset of M ( L c~) is called a support set if it is the (closed) support of the representing measure for a functional in M ( H ~176 + C). Clearly, sets of a n t i s y m m e t r y are contained in QC level sets, and support sets are sets of antisymmetry. T h e Izuchi theorem is stated as follows. Izuchi Theorem.
Let f , g, and h be in L ~176The following are equivalent.
(a) For each support set S, either f l s E H~176 or gls E H~176 or his E H~176 (b) For each QC level set Q, either flQ E H~IQ, or gIo c H~176 or hlQ E H~176 T h e Izuchi theorem above was first conjectured by Sarason [13]. The proof of the following lemma is in [10] and [12]. To make this paper readable and complete we include it. One p a r t of the lemma is the Adamyan-Arov-Krein theorem. L e m m a 1.2 Let f be in L ~ . If Ilfll~ < 1, then there is a unimodular function u e f + H ~ such
that Tu is invertible, and dist(~u, H c~) = 1. PROOF:
First we prove t h a t f + H ~ contains a u n i m o d u l a r function u and dist(~u,H ~176= 1.
In fact, this is the Adamyan-Arov-Krein T h e o r e m (see page 204 [10]). Define F : C --* [0, oo) by
F(c) = dist(f + c.zg~176
342
Xia and Zheng
T h e n F(c) is continuous. Since
F(e)
-- IIg~f+~cll--* oc,
we have f ( 0 ) = IIH~sll _< Ilfll~r < l, a n d F(c) --~ eo as z --~ oc. By t h e i n t e r m e d i a t e value theorem, t h e r e is a n u m b e r c such t h a t F(c) = 1. Thus
d i s t ( f + c, z H ~ ) = 1. Let Ic = I + c. T h e n dist(fc, H ~ ) = dist(f, H ~~ < 1, and dist(gfc, H ~176= 1. T h e later condition implies t h a t the n o r m of the Hankel o p e r a t o r H~fc is 1. Also for such a c we have
dist(~fc, H ~ + C) (1 - I I H ~ l t 2 ) l M I 2. Hence Tu is left invertible. Since HF a t t a i n s its norm, t h e r e is a function g in H 2, with 119N = 1
such t h a t
IIHFgll =
Ilgll. Thus IIr~,~gll 2 -- I I r , gll 2 = IIgll 2 - I I H F g l l 2 = 0
This implies T~g is in t h e kernel of Ts. B u t T~ is left invertible, t h e n T~ 9 is n o t zero and so T~,g = A for some nonzero c o n s t a n t A. T h u s 1 = : / ~ , a n d t h e n T f H 2 contains 1. B u t
z=Tzl
= T ~T~]9 = T z T u ~ ]g = TzT~[T,,(z g )]
and then TuH 2 c o n t a i n s z. By induction, we conclude t h a t TuH 2 c o n t a i n s {zn}n. So ~ ible.
is invert9
Xia and Zheng
2
343
Examples In this section we will completely characterize when H ~ H ~ H ~ is zero for inner functions
01, 02, and 03. Then we will present examples to show that it is not true that H~HgH~ is zero if and only if one of H f , / / 9 and Hh is zero. Moreover those examples answer negatively the question [11] whether the rank of H ~ H ~ H ~ equals the minimum of the ranks of H ~ , H ~ , H ~ . We need some notations. For two functions 0 and ~ in H ~176 let 0[j3 denote that 0 divides fl, i.e., ~ is in H ~176Define an operator V on L 2 by
Vf(w) = ~f(w) for f E L 2. It is easy to check that V is anti-unitary.
The operator V enjoys the following
properties. I.emma 2.1 Let f be in L ~
Then
(1) V 2 = I; (2) V P V = ( I - P); (3) VH]V = g~. The proof of the lemma is easy and is left for the reader. For a fixed point z in D, let kz denote the normalized reproducing kernel at z, and let
~z(w) denote the Mhbius transform Z--W
~z(w)
-
1 - -~w"
For two vectors x and y in L 2, we use x | y to denote the operator of rank one given by
(x | y)(f) -- (f,y)x for f C L 2. The following lemma is simple but useful. Lemma 2.2 Let fl, f2, and f3 be in L ~ mid z in D. Then
Tr H>,Hf, H}3S~
-
H;,Hf, H;3
= -(H~,HI, H>3Vkz) | (Vkz) - (VHf, kz) | (VT~,,TCH~,Hf, kz)+
PROOF:
Let z be a point in D. Then it is easy to check that the operator T ~ T ~ is
the orthogonal projection onto the orthogonal complement of the normalized kernel function k~. Hence
1 - T ~ T L = k~|
344
Xia and Zheng
By Property (2) of the operator V in Lemma 2.1, VT~.V = 1 - s~s~.
S~.
Thus we obtain
= (v~.) | ( v ~ z ) ,
and so
T~..r,A nf~ nla o~. = T~ H~ HI,(T~ T~. + k. | k.)H~aS~. * H * z~H~J~)(TLH]~&, * * = ( o. An elementary estimate gives
IIHzkzll~llHgkzll~llHhk, ll~ < (1 -ly(z)12)(, -Ig(z)12)0 -Ih(=)l=). Then limm(1 - I f ( z ) [ 2 ) ( 1 -Ig(z)12)(1 -Ih(z)l 2) > 0. Thus none of f, g, and h is constant on the support set Sin, and so (2) fails. This completes the proof.
9
T h e o r e m 4.2 Let f, g, and h be in L ~176The operator H~HaH ~ is compact if
H~176 n tt~176 n H~~ PROOF:
c H ~176 + C.
By Lemma 4.1 and the Izuchi theorem, we have that for each Q C - l e v e l set
Q, there are functions fQ, 9Q, and hQ in H ~ such that either flQ = fo[Q, or gIQ = gQIQ or
hlo = hQ]Q.
354
Xia and Zheng
To prove T h e o r e m 4.2, let T = H]HgH~Ha. First we need to prove t h a t T is compact. By L e m m a 1.1 we have
H;Hg
:
T]g
-
T-fTg.
Thus
T : [TTg - TTTg][T~h]2 - T~Th], and so T is in the Toeplitz algebra ,7 where J is the C* algebra generated by bounded Toeplitz operators on H 2 . Now let us apply Douglas' Localization Theorem to the algebra b / :
J / K : where ]C is the
ideal of compact operators on H 2. T h e n the center Z of 5 / i s identified with QC. As shown above, T is in 57. By Douglas' Localization Theorem, in order to prove t h a t the operator T is compact we need only show t h a t ~ ( T )
= 0 for all x E M ( Q C ) . Let Ix denote the smallest closed two-side
ideal of 57/K: which contains {[Tf] E J / ] C : f E L ~ ,
f ( ~ - l ( x ) ) = 0}. Let Q denote the Q C - l e v e I
set ~r-l(x). We first consider the case t h a t f]Q = fQIQ. Thus
T : (TTg - TTTg)(T[h[2 -- T~Th) : (Tf--r& + f~. : (T~_~
- f > r ~ T . - T ~ T ~ ) ( T ~ < , - T~T~) - TT_~T~)(TIhj2 - T~Th).
The last equality in the above equation follows from t h a t
So [T] is I~. Hence Ox([T]) = 0. Similarily, in the ease t h a t gIQ = gQIQ or h]Q = hQIQ, we can also obtain t h a t 9 x([T]) = O.
Thus we obtain t h a t T = H~HgH~Hh is compact. Next we show t h a t H~HgH~ is compact. Note t h a t
[HTHgH;][H~HgH;]* = H I H g H ; H h H ; H f = T H ; H f . Thus [H~HgH~][I~fH~tI~]* is compact. So H~HgH~ is compact. This completes the proof.
9
T h e following is a consequence of thc above theorem. Corollary 4.3 Let fl, f2, and f3 be in L ~176Then H*fo(a)'tl zzf~(2)ul*, f~(a) is compact for any permutation
a if W c I / ] n H ~ [ A ] n H~[f3] C H ~ + C. We conjecture t h a t the converse of the above corollary is true.
HT.(~)HI~(z~H]~(z ) is compact for all p e r m u t a t i o n o if and only if n a = l H ~ [ k ] c [:r~~ + C.
T h a t is, the operator
Xia and Zheng
355
Although we are not able to prove the conjecture, we will verify the conjecture in the special case t h a t fi is the complex conjugate of a n inner function for i = 1, 2, 3. First we need the following lemmas. I.emma 4.4 Let K be a bounded operator from (H2) • to H 2. I l K is compact, then lim IlK - T~o KSw~ II = O.
z~OD
PROOF:
Since operators with finite rank are dense in the ideal of compact operators, to
prove L e m m a 4.4, it suffices to show lim IIF - T*~.FS~,~II -- 0
z---~aD
for any finite r a n k operators F. Also for each finite rank operator F , there are finite functions fi in H 2 and gi in (H2) • such t h a t Tt
F = ~f~ | i=1
To finish the proof we then need only to show lim Hf |
|
z~OD
= O.
for f C H 2 and g E (H2) • B u t this follows easily from t h a t lim [l~zzf - z--ofl]2 = 0
z---~z o
and
lim ling
z~zo
- ~ g l [ 2 = 0.
So this completes the proof.
L e m m a 4.5 Let f be in L ~176For m in M ( H ~176 + C), let Sm denote the support set for ra. Then limz~mllHsk~ll~ = 0
if and only if f[s~ is in H~176m. PROOF:
We may assume t h a t [lflloo < 1. By L e m m a 1.2, there exists a u n i m o d u l a r
function u C f + H ~176 such t h a t Tu is invertible. By L e m m a 3 in [15], we have C(1 - l u ( z ) l 2) < [IHr
(1 -lu(z)[ 2)
for some positive c o n s t a n t C > 0. Thus
limz._.mC(1 - l u ( z ) l 2) _< limz__.mllHfkzll 2
