Internat. J. Math. & Math. Sci.
Vol. I0, No. 3 (1987) 473-482
473
A NEW CLASS OF COMPOSITION OPERATORS
S. PATTANAYAK, C.K. MOHAPATRA and A.K. MISHRA
School of Mathematical Sciences
Sambalpur University
768019, INDIA
Jyoti Vihar
(Received May 30, 1984 and in revised form June 14, 1985)
introduced.
obtained.
P: H2(T) H2(T),
A new class of composition operators
ABSTRACT.
Sufficient conditions on
P
Properties of
a
parameters
and
b
on the compactness of
for
(e it)
with
Pit
:
T/
is
to be bounded and Hilbert-Schmidt are
+
ae
have been investigated.
with
be
-it
for different values of the
This paper concludes with a discussion
P
KEY WORDS AND PHRASES.
pact operator.
Space, Composition operator, Hilbert Schmidt operator, Com-
1980 AMS SUBJECT CLASSIFICATION CODES.
47BJ7, 47B99, 46J15.
PRELIMINARIES.
i.
For a complex valued function
f
analytic in
zl
{z:
D
<
I}
and for
I
p <
set
MpCr,f)
(I 2
M (r,f)
Sup
0<2
ifCreie)ip
0
,de
-c) p
l<p <
and
The function
f
is said to be in
If
HP(D)
lim
if
r/l--
HP(T),
{z:
T
Izl
I}, be the class of
(e 18)
12f
0
[1,2] that for
It is known
8
and
f,
belongs to
is in
HP(D).
then,
f,
Also, if
f
HP(T).
f
in
in
e
in8
M (r,f) <
P
LP(T)
functions in
d8
HP(D)
0
lira
n
f(re
Similarly, let
such that
1,2,3
i8)
f,(e i8)
exists for almost all
Conversely, the Poisson integral of a function in
HP(D)
has the sequence
{a
n
as its Taylor coefficients
has the same sequence as its Fourier coefficients and vice versa.
spondence establishes an isometrical isomorphism between
HP(D)
HP(T)
and
HP(T).
This corre-
Thus, these
S. PATTANAYAK, C.K. MOHAPATRA AND A.K. MISHRA
474
interchangeably used and are usually referred to as the Hardy Space
two spaces are
HP[I,2].
In the sequel we came across another space familiarly known as the weighted Hardy
Let 0(n) be a sequence of positive numbers. An analytic function f:D+C,
space [3]
given by
Z
f(z)
n
anZ
{e
normal basis
n
H.
Then,
in
H
H
Let
Also we need the following definition.
linear operator on
Ha(0)
is said to be in the class
e (e
n
2.
it)
e
such that
{e
We note that
be a bounded
T
be a Hilbert space and
p(n)<.
l[Ten II 2
Z
<
0,1,2,..., the function
n
H 2.
forms an orthonormal basis for
n
A NEW CLASS OF CONPOSlTION OPERATORS.
Let
:
f((z)), z
D
D
D.
in
C:
be analytic and let
C
The operator
[4].
study a new class of composition operators
analytic’ also.
That is
:
Let
DEFINITION.
T
Hp (D)
(Cf)(z)
be defined by
HP(D)
P
and
In the present paper we introduce and
on
H2(T)
where
:
many be ’non-
T
nonvanishing negative Fourier coefficients.
D
E
for every set
P
H (D)
is known as a composition operator on
is extensively studied in the literature
(a)
Elan 12
is said to be Hilbert Schmidt if there exists an ortho-
T
Throughout in the present paper we denote by e n
int
lfl Ip
if
satisfy the following properties:
T, of linear measure zero,
-I(E)
{z
(z)
T:
w, w
E
T} is
also a set of linear measure zero and
(b)
for every
H2(T), fo is
H2(T) H2(T) by
f
P:
Then, define
L2(T).
in
in
P(fo)
Pf
P
where
is the projection of
L2(T)
H2(T).
into
Here some explanations are in order.
be extended analytically into
fo
D
is defined almost everywhere on
series
n
0
ane
in@
We observe that a function
as described in Section I.
T
Futher, let
Then by the Weierstrass theorem,
in
H2(T)
can
be represented by the Fourier
f
n
f
So with the condition (a),
0
an( (ei) )n
converges point-
f((eiS)) for all 8 such that (e i8) D and by a result of Carleson [5]
such that
n0 an( (ei0))n converges pointwise to f((ei@)) for almost all
(eie) T. Hence n 0 an((eie))n converges pointwise almost everywhere on T to
f((eie)). Thus throughout in this paper we write 0 an((ei@)) n in place of f (#(elSe.
wise to
n
We note theat if
Graph Theorem,
tions on
,
satisfies the conditions of the definition, then by the Closed
is a bounded operator.
P#
fo$
#
is in
L2(T)
for all
f
So a natural question is:
in
H2(T).
under what condi-
The present paper primarily deals
with this question.
.
In the following sections we first obtain bounds for the norm of
conditions on
Then we consider
defined by
(e
it
ae
it
+
be
P
-i t
under suitable
and study
475
NEW CLASS OF COMPOSITION OPERATORS
foe e L2(T)
such that
b
and
a
conditions on
PC
section we have discussed the compactness of
3.
for all
f
H2(T).
in
In the last
with the help of some examples.
P.
NORM OF
We have the following results.
THEOREM I.
:
Let
D
T
be such that
f2
o
P
Then,
dt
Let
f
H2(T)
in
(3.1)
<
I1%11 --< (M()12) I/2
is Hilbert Schmidt and
PROOF.
M()
-I( it)
f(z)
be given by
Jf((elO)
2
Ea ((e
n
Y.an z n
le) n 12
<
z
T.
in
Then,
(Elan 12) (El(el8)
2n)
ie)
so,
Next, with the orthonormal basis
n
rO lP, CCr) 12 s
P
Thus,
<
M()
ilfll 2
2
(M(*)/2) I/2"
IIP, II
and we get
ilfoll 22
2 <
iiPfll
e
n
n
we have
$2l(elt)
12n
0
leno*l 12 nO
0
H2(T),
of
0,1,2
dt
(M()/2)<
.
is Hilbert Schmidt.
COROLLARY 2.
:
If
T+D
is continuous then
PC
is Hilbert Schmidt.
The condition (3.1) is trivially satisfied if # is continuous.
By an example in the next section we will show that (3.1) is only a sufficient conto be Hilbert Schmidt. We need the following lemma due to Gabriel [6]
dition for
PROOF.
P
for the proof of our next theorem.
LEMMA.
Let
f
H2
every
in
THEOREM 2.
(i)
Let
F
:
be a rectifiable convex curve in the closed unit disc.
T+D
be such that
describes a closed rectifiable convex curve in
(il)
inf[’ (eli)
m
llPll
Then,
PROOF.
Then, for
> 0,
0
D
and
2
t
(2/m) I/2
By lemma and the condition (ll) we have
2
2
lifll
If( (eit ))1
i (=)1
dt > m
12
I(fo) (eit )ldt
0
so that
lP,fl
<
o
,2r
2
Cfo,) ( it,I 2 dt <
lfl
S. PATTANAYAK, C.K. MOHAPATRA AND A.K. MISHRA
476
P
The conditions (i) and (ii) in the above theorem are not necessary for
to be
As an example consider
bounded.
(eit
(t)
e
0 < t <
0
_< t_< 2
does not satisfy any of the conditions (i) or (ii) of the theorem.
so that
Now,
I 02 If(
P
This shows that
4.
If(eit) 12
=i f0
)(e it))12 dt
II PII
is bounded with
2
at +
lfl 122
If() Imdt -<
</2.
A FAMILY OF COMPOSITION OPERATORS.
P
In this section we study the properties of
T
tions
D
given by
(z)
where
lal
+
Ibl
for the particular family of func-
! I.
+
az
lal
We note that if
#
Ibl
ellipse containing the orgin in its interior.
P
Hence by Theorem 2,
(4.1)
z e T
bz
then the curve traced by
Also
inflae -it
m
P
It turns out that
is bounded.
perties for different values of the parameters
and
a
bl
is an
>
has many interesting pro-
We need the following tech-
b
nical lemma.
LEMMA I.
For all
n
k
2k>
k
PROOF.
Z+
in
+
n
<
2n+2k
(4.2)
We shall prove (4.2) by method of induction on
Let
n
0
n
so that we have
to show
2k
<
(k)
22k
for
k
(4.3)
1,2,3
We establish (4.3), also by the process of induction on
is trivially true.
I,
k
(i)
Next, assume that
(2kk
To complete induction on
k
<
22k,
(2k+2)!
(k+l) (k+l)
k
2n+2k
n+l+2k.
k
Thus (4.3) is true for all
k-- 1,2
2k
k
<
22k
(2k+l) <
(k+l)
2
k
2
(2k+l)
(k+l)
2
Next, let
1,2,3
2k+l
k
.n+2k. <
k
22 k
<
we consider
k+l
Thus, (4.3) is true for all
(2k)
(k!) (k!)
i.e.
2(k+l)
Now, assume that
For
k
2
<
22(k+I)
Then,
n
2k+l
To complete the induction we consider
(n+2k)
(k!) (n+k)
and
(n+l+2k)
(n+l+k)
n
0,I,2
<
2n+2k
2<
2n+l+2k
2
2 < 2
NEW CLASS OF COMPOSITION OPERATORS
(1)
:
Let
THEOREM 3.
T
D
I,
[a]
[b],
then
fo need
[a + [b]
If
lal
(ll) If
+
Ibl
la[
The inequality
PROOF.
0
b
<
z
<
e
_’n+2k"
[k
tmn
0,1,2,
n
Z+
k
a
n--E0 (fan Imn + k-El
+
lal
.n+2k. 2
k
For
a
<
n-ZO k--El
lal 2n labl 2k
which also implies
(il)
Schmldt.
in
H2
P
so that
n
H2
(ab)k
With respect
if
n < m
if
n
m -2k
if
n
m
2k
+
2n
2k
.n+2k. 2
k
lal 2n labl 2k)
.n+2k. 2
k
al 2n abl 2k
to show that the second sum in the right hand side is convergent.
kl n0
since
f
Now
.n+2k. 2
We use Lemma
is Hilbert Schmldt.
for
0
and
n
m
e
for all
0
where
T.
in
then
L2
not be in
n
has a matrix representation
P
la[
and
bz
in (1) is best possible.
Consider the orthonormal basis
to this basis
+
az
H2
is not defined on the whole of
(ill)
(z)
be given by
477
ZZ
22n+4k lal 2n lab
1-12al 2
labl <
(eie)
cos
f(eie))
e
ZZ
2k
12al 2n
1-14abl 2
P
This proves that
H2
For the proof of (ll) consider the
b
2n
function
f(z)
is Hilbert
(l-z)
-a
0<a<1/2
Thus,
f(cos O)
(1-cos e)
a
(2
a
s
in2a
e
and
f2
I(f((e ie) )12
0
if
a >
for
(iii)
/ 2 22-2a
4a
de
0
sin
de
e
>
f/2 22-2a
4a
0
In the above we have made use of the well known inequality
0 <
e
de
e
2e <
sin
<-
2
In view of (ii), for the proof of (iii), it is sufficient to show that
not Hilbert Schmidt if
condition
e<e
mE Itm, ml 2--
Observe that
a
/
b
and
a > b.
P@
is
In fact, we show that under the above
S. PATTANAYAK, C.K. MOHAPATRA AND A.K. MISHRA
478
(aeiO
en((eiO)
(a + b)
Since
n
+
kOn (k)n
be-iO)n
n
n
k
k=Z0 (k) an-k b
we have
a
n-k
b
k e i(n-k)O
e-ikO
Hence considering this sum as the
I.
inner product of two vectors
We see
()an b 0 ()an-I
that, since II [,
n
n
kffiZ0 [(k
a
n-k
(nn)
b
b
k
]II
[2 >
a > b
Further, we observe that if
I
us to
n
It n,n [2
land
..(.n.+.l!..t.e.r.s,
I,I
by Cauchy Schwarz inequality
(n+l)
()
a
n-k _> k
n-r
b
r >
n
(n_r)
a
r
bn-r
[[P en[ [z _> il2(n+l)
and so
(z)
Also, with the help of the same function
az
+ bz
P
is not a necessary condition for
b
a
f2
0
ab > 0
R
in
and
f2
0
dt
l_l(e it )12
al
bl
+
I.
we show that the condlto be a Hilbert Schmidt.
Then,
(1-(a-b) 2) sin 2 t
P
is Hilbert Schmidt.
In the following theorem we present a sufficient condition on
THEOREM 4.
be given by
PROOF.
fo
:
n an
0
First let
z
Further if
if((eiO))12
az
an
(e io)
so that
b
a
(z)
be given by
T/D
f(z)
f
in
H2
to
L2(T).
is in
Let
leading
dt
However, in view of Theorem 3, it follows that
ensure that
so that over half
This completes the proof of the theorem.
tion (3. I) of Theorem
For this take
n
b
(n+l)
then
of the above sum is from terms where
0
a
+
lal
bz
0
with
Ibl
>
and
f
then
fo
in
H2
E
L 2.
n
cos O.
Now,
n
_<
into cos= o
(4.4)
n+B
(4.5)
We know that
n-E0-
(I-z) B+
n
z
n
and [7]
Taking
B
-a
n+B
n
n
r(B+l)
(4.6)
in (4.5) and (4.6), we get
cos
nffi0
n
0
n
r(-a+l)
(1-cosO) (l-a)
Thus, to complete the proof, it is sufficient to show that
f2
0
(1-cosO)
(2-2)d0
sin(4a-4)
f2
0
However, this is ture because of the condition
a >
_.0
2
3
dO <
To dispose of the general case
NEW CLASS OF COMPOSITION OPERATORS
2a
we observe that if
ia
e
eiV
25
and
a-+28
ei (a+V)/2 cos()
ei8)
$(
then
479
and
this leads to similar calculations as above.
fo@ e L 2
Taking cue from the above theorem we next show that
p(n).
for a suitable choice of the sequence
Let
THEOREM 5.
o(n)
(i)
8.
n
fo#
T
L2
for all
(e tO)
be given by
aeiO + be-iO
is in
8 <
f
H 2 (0(n))
in
there is a function
PROOF.
As in the previous theorem we assume
Let
in
H2(0)
f(z)
be given by
If(*(e io) )i 2
ow
1
ib
and
8 > _I2
if
f8
in
a
b
L2
f
la
Then,
for each
(ii)
:
H2(p(n))
f e
for
an z
n__E0
I0 %
cos
n
12
-
H2(D(n))
f8o@
such that
is not in
f(@(eiO))
so that
f(cos e).
We have
(-o ai
<
p(n))
(-o
2n
cos
p(n)
o)
using (4,5) and (,6) as in the previous theorem it can be shom that
sin
fo
Thus
28-2 8
L2
is in
n=E0
n-I
r(-8+I)
cos
2n 8.
8 >
if
For the proof of (ll) consider the function
a
f(z)
(l-z)
An z n
a+1
By (4.6),
ZlAIn 2 nS,,,
Z n 2a+8
The sum on the right hand converges if
in
H2(O),
n8
p(n)
for
if
a a-
but
fo
REMARK.
Thus, for given
is not in
<
<
I/2
n
2
(i)
fo#
Let
:
T
is in
L2
for all
(il) for each
L2
(8+1)/2.
a <
Thus, f
is
if we chose
is in
H2(D)
dO
(3+28+2)/8
a
remains open in the above theorem.
but with slower rate than
THEOREM 6.
i.e.
sl-n4a4-
theorem we prove the same result for a sequence
n
-1
However,
22a/I f
0
dO
8
+ 8
f
L 2.
0(n)
The case
(8+I)/2.
a <-
27
/0 ]f((ei8))[2
2a
n
I/2+
0
(n) having faster
in
H2 (p(n))
8 < 0, there is a function
f8
rate of growth than
0
for any
be as in the previous theorem and
f
However, in the next
if
in
D(n)
nl/2(logn)8.
Then,
8 > I,
H2(p(n))
such that
f8o
is not in
S. PATTANAYAK, C.K. MOHAPATRA AND A.K. MISHRA
480
Let
PROOF (i)
f(z)
f, given by
anzn
nffiZ0
be in
H2(0(n))
so that
: = 1/2(log n) 8 fan 12<
II I
By Cauchy Schwarz inequality,
If((eiS))l 2
8
n.0 nl/2 (log n) lan 12)(n0=
<_.
cos
2n
8
(4.7)
1/2 (log n) B.))
n
It is known [7, p. 192] that if
(l_z)a+l
A
8
(a,8)
(4.8)
R
a, 8
(log n) 8
n
r(a+1)
n
n
A (a 8) z
n
Z=
n 0
# -I,-2,-3,
a > 2, a
then for
a
-L)
(log
so that
RI
(117
/ n I/2(log n) 8
n
ieee
!
/
(cos 2 O)
n
n__Z0 nl/2(logn) 8
(log
0)I/2
(l-cos 2
-8
a
l-cos 2 6)
2
-8
b
sln-----
(log Kin
O
-8
(4.9)
In view of (4.7) and (4.9), in order to show that
fo@
is in
L2
it is suffi-
cient to show that
f2
0
(log sin
sin 8
8)-8 dO
<
(28/) < sin 8 < 8
(0,6), of the function
it is sufficient to show the
Further, because of the inequality
integrability, in an interval
=.,
(log
Making the substitution
log
h(O)
_61
()
(log )
lim/0
-8
-8
u, we get
(log
dO
lira
(ll)
Let
p(n)
n
112(
f(z)
(l-z)
where
-I/2
parison of
> y <
f
(8-I)12.
I-8
8 > I.
1/4 (log
_I.) I-8
This completes the proof of (i).
Now consider the function
8 > 0.
a
(log
8
/0
Thus, the above integral converges if
logn) -8
)
Tz
y
n
7.AYz
n
We first observe that
f
is in
H2(p(n)).
with (4.8) shows that
A7
n
n
-3/4
r’(I14)
(log n) Y
Thus,
ZlAVnl
2 p(n)
rn -I (log
n) 2Y-
and the right hand side series converges because
Y < (8-I)/2.
Now,
In fact, a com-
CLASS OF CONPOSITIO OPERATORS
If(*(e iO) )12
$20
where
a
b
2y-1/2
dO
because of the condition
REMARK.
dO
O
de
>
Thus, we prove that although
n2(logn) 8,
p(n)
The case
0 _< 8 <
H2(p(n))
f
fo,
is
remains unsettled.
We conclude this section by showing that
(e it +
*(e it)
H2
e- it )/2
cos t
gn(t)
g
f(z)
and
f (,(e
n
L
is in
k=El
knf iZ
It))
--=
cskt
g
f2w ign(t)
S(t)
n+m 0
L2.
gn
and
log
Z
for
f
n
Let
so that
())
f_,_elt__
g(t)
and
k
but is not in
Fourier coefficients of
lim
H2
k
fn(Z) =kE=l.
Observe that
is an unbounded operator on
This we do by exhibiting a sequence of function
k
Let
P,
lP,fnl
for which lim
n+oo
k
0(18 ’sin
L2
not in
th
sin
0
The above integral diverges with the integral
f/2 (log ) 2
0
in
f/2
4
481
2y
b
a
k
and
a
(n)
k=gl
cskt
k
respectively be the
n
Since
0
dt
we get
a
IP, 2
1tm
5.
(n)
a..
ltm
n+oo
n
k_n
Now,
lal(n) 12
k.z_.
/Ig(e it) 12
lak 12
at
COHPACTNESS OF
In this section we discuss some examples illustrating cases when
and when it is not.
(1)
(ll)
’I (elt
ae
[eL
,2(e It)
Let
T
’1 ’2 ’3
-it
/
D
P,
is compact
be defined by
Jaj
0 < t < 2
it
0 _< t <
0
e
(ill)
(i)
f(z)
*3 (eli)
P,I
n__Z0
It
Lae it +
0 _< t _<
be
-it
_< t _< 2
a+b
is a finite rank, hence a compact, operator.
a z
n
n
then
(p,
f)
(elt
P
(n=0
a a
n
n
e
-int
a
# b,
For, if
a
o
f
a,b
_> 0
in
H2
is given by
S. PATTANAYAK, C.K.
482
MOHAPATRA AND A.K.
For composition operators with analytic
HP(D)
C:
limit of
HP(D)
(z).
is compact then
Schwartz [8] has shown that if
I(e )1
<
a.e. where
PI
We observe in this example that
(e it)
is the radial
deviates in behaviour from
lP211
We have shown at the end of Section 3 that
(ii)
MISHRA
/2
<
C
We show here that
P2
is not compact.
By Riemann-Lebesgue Lemma the sequence
in
H2
However,
P2 (en)
the Fourier series of
o0 2
e
P(en2)’
e
n
n
0,I,2,... converges to zero weakly
does not converge strongly to zero.
is given by
(enOO2)(e it)
Z
a
e
imt
For, if
then by direct
computation it can be seen that
i
a
m
and
(n-m)
if
n-m
is odd
0
if
n-m
is even
if
nffim
[IPqb 2 (en)[[ 22 mEffi0 [am 12
>
Thus,
P2
is not compact.
By a similar argument as in (ii) it can be shown that
compact operator.
ACKNOWLEDGEMENT.
definition of
P3
is bounded but not a
The authors thank the referee for his useful suggestions regarding the
P@
REFERENCES
I.
2.
3.
DUREN, P.L.,
HP.SPace
s, Academic Press, New York, 1970.
HOFFMAN, K., Banach Space of......alvtlc Fnctions, Prentic-Hall, Englewood Cliffs,
New Jersey, 1962.
HALMOS, P.R., A Hilbert Space..rblem Book, Sprlnger-Verlag, New York, Inc., Gra
duate texts in Mathematics, Vol. 19.
4.
5.
6.
7.
8.
NORDGREN, E.A., Compo.s#io.Oerators iR Hilbert Spaces, Lecture notes in Mathematics, 639, Springer-Verlag, Berlin, 1978.
CARLESON, L., On Convergence Growth of Partial Sums of Fourier Series. Acta Math,
116 (1966), 135-157.
GABRIEL, R.M., Some Results Concerning the Moduli of Regular Functions Along Curves
of Certain Types, Proc. London Math. Soc. 28(1928), 121-127.
ZYGMOND, A., Trigonometric Sers., 2nd Ed. CambrldKe University Press, 1959.
SCHWARTZ, H. omp.osit%on ODe.ators oK H Dissertation, University of Toledo, 196
,