B OLLETTINO
U NIONE M ATEMATICA I TALIANA
B. Yousefi, S. Jahedi
Composition operators on Banach spaces of
formal power series
Bollettino dell’Unione Matematica Italiana, Serie 8, Vol. 6-B (2003),
n.2, p. 481–487.
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Bollettino dell’Unione Matematica Italiana, Unione Matematica Italiana, 2003.
Bollettino U. M. I.
(8) 6-B (2003), 481-487
Composition Operators on Banach Spaces
of Formal Power Series.
B. YOUSEFI - S. JAHEDI
dedicated to the memory of Karim Seddighi
Sunto. – Supponiamo che ]b(n)(Q
n 4 0 sia una successione di numeri positivi e 1 G p E
Q
f×(n) z n , tali
Q . Consideriamo lo spazio H p (b) di tutte le serie di potenze f (z) 4
Q
che
!
n40
N f×(n) Np b(n)p E Q . Supponiamo che
1
p
1
1
q
Q
41 e
!
!
n qj n 4 0
q
n 4 1 b(n)
p
4 Q per un in-
tero non-negativo j . Dimostriamo che se Cf è compatto su H (b), allora il limite
non-tangenziale di f ( j 1 1 ) ha modulo maggiore di uno, in ogni punto della frontiera del disco unitario aperto. Dimostriamo anche che se Cf è di Fredholm su H p (b),
allora W deve essere un automorfismo del disco unitario aperto.
Summary. – Let ]b(n)(Q
n 4 0 be a sequence of positive numbers and 1 G p E Q . We conQ
f×(n) z n such that
sider the space H p (b) of all power series f (z) 4
!
Q
! N f×(n) N b(n) E Q . Suppose that
p
p
n40
1
p
1
1
q
Q
4 1 and
!
n40
n qj
q
n 4 1 b(n)
p
4 Q for some non-
negative integer j . We show that if CW is compact on H (b), then the non-tangential
limit of W ( j 1 1 ) has modulus greater than one at each boundary point of the open
unit disc. Also we show that if CW is Fredholm on H p (b), then W must be an automorphism of the open unit disc.
Introduction.
First in the following, we generalize the defintions coming in [5].
Let ]b(n)( be a sequence of positive numbers with b(0)41 and 1GpEQ.
We consider the space of sequences f 4 ] f×(n)(Q
n 4 0 such that
V f Vp 4 V f Vpb 4
Q
The notation f(z) 4
! f×(n) z
n
Q
! N f×(n) N b(n) E Q .
p
p
n40
shall be used whether or not the series con-
n40
verges for any value of z . These are called formal power series. Let H p (b) denotes the space of such formal power series. These are reflexive Banach
spaces with the norm V Q Vb ([4]) and the dual of H p (b) is H q (b p/q ) where
1
p
1
482
1
q
B. YOUSEFI
- S.
JAHEDI
4 1 and b p/q 4 ]b(n)p/q (n ([6]). Also if g(z) 4
V gVq 4
Q
Q
! g×(n) z
n
H q (b p/q ), then
n40
! N g×(n) N b(n) . The Hardy, Bergman and Dirichlet spaces can be
q
p
n40
viewed in this way when p 4 2 and respectively b(n) 4 1 , b(n) 4 (n 1 1 )21 /2
and b(n) 4 (n 1 1 )1 /2 . If lim
b(n 1 1 )
n
b(n)
4 1 or lim inf b(n)1 /n 4 1 , then H p (b) conn
sists of functions analytic on the open unit disc U . It is convenient and helpful
to introduce the notation a f , gb to stand for g( f ) where f H p (b) and g Q
H p (b)* . Note that af , gb 4
f×(n) g×(n) b(n)p . Let f×k (n) 4 d k (n). So fk (z) 4 z k
!
n40
and then ] fk (k is a basis such that V fk V 4 b(k). Clearly Mz , the operator of multiplication by z on H p (b) shifts the basis ] fk (k .
Remember that a complex number l is said to be a bounded point evaluation on H p (b) if the functional of point evaluation at l , el , is bounded. The
functional of evaluation of the j-th derivative at l is denoted by el( j ) .
The function W in H p (b) that maps the unit disc U into itself induces a composition operator CW on H p (b) defined by CW f 4 f i W . The operator CW is Fredholm, if it is invertible modulo the compact operators. If CW is a bounded invertible operator, then W must be an automorphism of U , that is a one to one
map of U onto U .
We say an analytic self-map W of U has an angular derivative at w ¯U , if
for some h ¯U the non-tangential limit of
W(z) 2 h
z2w
when z K w , exists and is
finite. We call this limit the angular derivative of W at w and denoted it by
W 8 (w).
Main results.
We suppose that H p (b) consists of functions analytic on the open unit disc
U . We study the Fredholm composition operator CW and investigate the compactness and essential norm of CW acting on the Banach space H p (b).
LEMMA 1. – Let X be a Banach space of analytic functions on a domain V in
C . If there exists a sequence of functions gk in the dual space X * such that
V gkV41 and gk K 0 weakly with VCW*(gk)VK0, then CW is not Fredholm on X .
PROOF. – Suppose S is any bounded operator on X * . Then by the hypothesis VSCW* ( gk ) VGVSV VCW* ( gk ) V K 0 as k K Q . Now let Q be an arbitrary compact
operator on X * . Since Q is necessarily completely continuous, then we have
VQ( gk ) V K 0 ([2, p. 177, Proposition 3.3]). Thus V(I 1 Q) gk V K 1 for every compact operator Q on X * . This implies that SCW* 2 I can not be compact, since
else it should be V(I 1 (SCW* 2 I) ) gk V K 1 that is a contradiction. Thus CW* , and
hence CW , is not Fredholm. r
COMPOSITION OPERATORS ON BANACH SPACES ETC.
483
Q
!
In the following we use the fact that ew H q (b p/q ) and Vew Vq 4
n40
Q for all w in U ([6]).
THEOREM 2. – Let
1
p
1
1
q
4 1 and
Q
n qj
n40
b(n)q
!
NwNnq
b(n)q
E
4 Q for some non-negative inte-
ger j . If CW is Fredholm on H p (b), then W is an automorphism of the
disc.
PROOF. – It is well known that if CW is Fredholm, then W is univalent since
else the kernel of CW* will contain an infinite linearly independent set whose
elements are differences of evaluation functionals. This is a contradiction,
since dim ker CW* E Q . So we need only show that W maps U onto U . If not,
there exists v ¯W(U) O U and zk U such that W(zk ) K v . By the Open Mapping Theorem it should be Nzk N K 1 .
n qj
Let j be the least non-negative integer such that the sum
4 Q . If
q
ez
n F 0 b(n)
j 4 0 , set ek 4 k . Then V ek V 4 1 . But
!
V ezk V
lim V ezk Vq 4 lim
k
k
!
nF0
Nzk Nnq
b(n)
q
4
!
nF0
1
b(n)q
4Q
and so if p is a polynomial in H p (b), then lim ap , ek b 4 lim
k
k
p(zk )
V ezk V
4 0 . But poly-
nomials are dense in H p (b), thus ek K 0 weakly as k K Q . Since v is in U and
W(zk ) K v , we have eW (zk ) K ev . Since we also have Vezk V K Q , we conclude that
VCW* ezk V 4 VeW(zk ) V/V ezk V tends to zero. So by Lemma 1, CW is not Fredholm that
is a contradiction.
ez( j)
If j D 0 , let ek 4 (kj) where ez(kj) is the functional of evaluation of the j-th
V ezk V
Q
derivative at zk . Note that ew (z) 4
!
1
p
n 4 0 b(n)
wn z n and ew( j) 4
Q
ez(kj) 4
Since Nzk NK 1 and
!
n40
!
nF0
n(n 2 1 )(n 2 2 ) R(n 2 j 1 1 )
n jq
b(n)p
k
k
!
n40
b(n)p
ew . Thus
z n.
4 Q , we have
Q
lim Vez(kj) Vq 4 lim
(zk )n 2 j
dj
dwj
(n(n 2 1 ) R(n 2 j 1 1 ) )q
Nzk N(n 2 j) q
b(n)q
4Q.
Since polynomials are dense in H p (b), by the same manner as in the previous
case, we can see that ek K 0 weakly as k K Q . Now we show that VCW* ek V K 0 as
484
B. YOUSEFI
- S.
JAHEDI
k K Q . A straightforward computation gives the following equalities:
(1)
CW* ez(k1 ) 4 W 8 (zk ) eW(z
k)
(2)
(1)
1 W 9 (zk ) eW(z
CW* ez(k2 ) 4 W 8 (zk )2 1 eW(z
k)
k)
(3)
(1)
(2)
(2)
1 WR(zk ) eW(z
1 2 W 9 (zk ) eW(z
1 W 8 (zk ) W 9 (zk ) eW(z
CW* ez(k3 ) 4 W 8 (zk )3 eW(z)
k)
k)
k)
QQQ
( j)
(1)
1 W ( j) (zk ) eW(z
1 lower order terms
CW* ez(kj) 4 W 8 (zk ) j eW(z
k)
k)
where the lower order terms involves functionals of evaluation of derivatives
of order less than j at W(zk ) with coefficients involving products of derivatives
of W at zk of order less than j . From this it follows that VCW* ek V K 0 as k K 0 . To
see this first suppose that j 4 1 . Thus we have
CW* ek 4
(1)
W 8 (zk ) eW(z
k)
V ez(k1 ) V
(1)
4 aW , ek b eW(z
.
k)
(1)
V K V en( 1 ) V E Q . Also since ek K 0 weakly,
But W(zk ) K n , where n U . So VeW(z
k)
aW , ek b K 0 as k K Q . Thus indeed VCW* ek V K 0 as k K Q .
If j D 1 , remark that for all i E j we have
(i)
4
eW(z
k)
Q
n!
(W(zk ))n 2 i
n41
(n 2 i) !
b(n)p
!
and so
Q
(i)
Vq 4
V eW(z
k)
!
n4i
(n(n 2 1 ) R(n 2 i 1 1 ) )q
Q
G
!
n4i
n iq
b(n)q
G
Q
n ( j21)q
n4i
b(n)q
!
since j is the least non-negative integer such that
Nv(zk ) Nn 2 i
b(n)q
EQ,
Q
n jq
n40
b(n)q
!
4 Q . Thus the limit
of the norms of the functionals of evaluation of derivatives at W(zk ) of order
less than j remain bounded in U . Also, by the Principle of Uniform Boundedness Theorem sup V ez(i)
V E Q for i E j and all derivatives of W at zk of order less
k
k
than j are bounded. Note that V ez(kj) V K Q and W(zk ) K n U . Thus we have
lim VCW* ek V 4 0 provided that
kKQ
lim
kKQ
1
V ez(kj) V
( j)
(1)
V(W 8 (zk ) ) j eW(z
1 W ( j) (zk ) eW(z
V40 .
k)
k)
COMPOSITION OPERATORS ON BANACH SPACES ETC.
485
Clearly
1
(˜)
Vez(kj) V
( j)
(1)
V(W 8 (zk ) ) j eW(z
1 W ( j) (zk ) eW(z
VG
k)
k)
NW 8 (zk ) Nj
V ez(kj) V
( j)
VeW(z
V1
k)
VWVjH p (b) V ez(k1 ) V j
V ez(kj) V
NW ( j) (zk ) N
Vez(kj) V
(1)
VeW(z
VG
k)
( j)
(1)
VeW(z
V 1 NaW , ek b N . VeW(z
V.
k)
k)
(i)
VK
Note that V ez(kj) V K Q and lim V ez(k1 ) V E Q , since 1 E j . Also VeW(z
k)
k
Ven(i) V E Q for i 4 1 , j and aW , ek b K 0 , since ek K 0 weakly. Thus indeed the
term in (˜) tends to zero as k K Q and so VCW* ek V K 0 which by the lemma implies that CW is not Fredholm that is a contradiction. r
Note that by the Julia Caratheodory Theorem ([3]), W has an angular
derivative at w ¯U if and only if W 8 has non-tangential limit at w , and W has
non-tangential limit of modulus one at w . Consider the open Euclidean disc,
Julia disc, J(j , a) 4 ]z U ; Nj 2 zN2 E a( 1 2 NzN2 )( of radius
at
j
11a
a
11a
and center
, whose boundary is tangant to ¯U at j . By the Julia’s Lemma ([1]), if
j ¯U and W is an analytic function such that BW 4 inf NW 8 (j) N E Q , then
j ¯U
W(J(j , a) ) ’ J(W(j), aBW ).
Recall that the essential norm of CW is denoted by VCW Ve and is the distance
in the operator norm from CW to the compact operators.
THEOREM 3. – Let
1
p
1
1
q
4 1 and
(i)
Q
n qj
n41
b(n)q
!
4 1Q for some non-negative
integer j . Also for 0 G i G j let W be an analytic self map of the unit disc U . If
CW is a bounded operator on H p (b) and NW ( j 1 1 ) (j) N G 1 for some j ¯U , then
VCW Ve F 1 and CW is not compact.
PROOF. – Let ]zk ( be any sequence in U with zk K j . Also let j be the least
Q
non-negative integer such that the sum
!
n41
n qj
b(n)q
4 1Q . Set ek 4
ez(kj)
V ez(kj) V
. Then
V ek V 4 1 and by the same method used in the proof of Theorem 2, ek K 0 weakly
as k K Q . If K is any compact operator, then K * is completely continuous and
since ek K 0 weakly, it should be VK * ek V K 0 . By definition VCW Ve 4 inf ]VCW 2
KV : K is compact( and
VCW 2 K V 4 V(CW 2 K)* V F V(CW 2 K)* ek V F VCW* ek V 2 VK * ek V .
486
B. YOUSEFI
- S.
JAHEDI
If k K Q , then since VK * ek V K 0 , we have VCW Ve F lim VCW* ek V . Now we show
k
that
lim VCW* ek V 4 lim
k
k
( j)
V eW(z
V
k)
Vez(kj) V
.
Note that since NW ( j 1 1 ) (j) N G 1 , by the Julia’s Caratheodory theorem the nontangential limit of W (i) (j) have modulus one for i 4 0 , 1 , R , j .
If j 4 0 , then ek 4
(1)
eW(z
k)
CW* ek 4 W 8 (zk )
V ez(k1 ) V
ezk
V ezk V
and CW* ek 4
eW(zk )
V ezk V
. If j 4 1 , then ek 4
ez(k1 )
V ez(k1 ) V
and
. But the non-tangential limit of W 8 (j) has modulus one
(1)
V/Vez(k1 ) V .
and so lim VCW* ek V 4 lim V eW(z
k)
k
k
If j D 1 , then ek 4 ez(kj) /V ez(kj) V and
CW* ek 4
1
V ez(kj) V
( j)
(W 8 (zk ) j eW(z
1 Lj , k )
k)
where Lj , k is the sum of lower order terms and involves derivatives of order
(i)
(i E j), with coefficients involvless than j at W(zk ), i.e., terms of the type eW(z
k)
ing product of derivatives of W at zk of order less than or equal to j . Remark
that since j is the least non-negative integer such that
we have
Q
(i)
VeW(z
Vq 4
k)
!
n41
(n(n 2 1 ) R(n 2 i 1 1 ) )q
NW(zk ) Nn 2 i
b(n)q
Q
n jq
n40
b(n)q
G
!
4 1Q , then
Q
n ( j21)q
n41
b(n)q
!
EQ
(i)
V remains bounded for all i less than j . Also since
for i E j . So lim V eW(z
k)
kKQ
Vez(kj) V K Q and W (i) (j) has the non-tangential limit of modulus one for all i G j ,
thus indeed lim
k
Therefore
VLj , k V
V ez(kj) V
40.
lim VCW* ek V 4 lim NW 8 (zk ) Nj
k
k
4 lim
k
( j)
V eW(z
V
k)
Vez(kj) V
( j)
VeW(z
V
k)
Vez(kj) V
.
Now to complete the proof it is sufficient to show that lim
g
this set zk 4 1 2
1
k
( j)
V eW(z
V
k)
V ez(kj) V
F 1 . For
h j . Then there exists a sequence ]r ( of non-negative
k
k
numbers such that zk is the point on ¯J(j , rk ) closest to 0. Therefore by the Ju-
COMPOSITION OPERATORS ON BANACH SPACES ETC.
487
lia’s Lemma
W(zk ) W(¯J(j , rk ) ) ’ ¯W(J(j , rk ) ) ’ ¯J(W(j), rk ) .
It follows that NW(zk ) N F Nzk N for all k . Now since Vez( j) Vq 4
Q
n4j
( j)
VeW(z
V/V ez(kj) V F 1
k)
V ez( j) V
n2j
! (n 2n! j) ! NzNb(n)
q
,
the norm
increases with NzN . Thus for all k ,
and indeed
VCW Ve F 1 . This implies that CW is not compact and so the proof is complete. r
COROLLARY 4. – Let
1
p
1
1
q
4 1 and
Q
n qj
n41
b(n)q
!
4 1Q for some non-negative
integer j . If CW is compact on H p (b), then NW ( j 1 1 ) (j) N D 1 for all j in ¯U such
that W ( j 1 1 ) (j) exists.
REFERENCES
[1] L. AHLFORS, Conformal Invariants, McGraw-Hill, New York, 1973.
[2] J. B. CONWAY, A Course in Functional Analysis, Springer-Verlag, New York,
1985.
[3] W. RUDIN, Function Theory in the Unit Ball of Cn , Grundlehren der Mathematischen Wissenschaften, 241, Springer-Verlag, Berlin, 1980.
[4] K. SEDDIGHI - K. HEDAYATIYAN - B. YOUSEFI, Operators acting on certain Banach
spaces of analytic functions, Internat. J. Math. & Math. Sci., 18, No. 1 (1995),
107-110.
[5] A. L. SHIELDS, Weighted shift operators and analytic function theory, Math. Survey, A.M.S. Providence, 13 (1974), 49-128.
[6] B. YOUSEFI, On the space l p (b), Rendiconti del Circolo Matematico di Palermo Serie
II, Tomo XLIX (2000), 115-120.
B. Yousefi: Dept. of Math., College of Science, Shiraz University, Shiraz 71454, Iran
E-mail: yousefiHmath.susc.ac.ir
S. Jahedi: Dept. of Math., College of Science, Shiraz University, Shiraz 71454, Iran
Pervenuta in Redazione
il 5 marzo 2002