I ntenat. J. Math. & Math. Sci.
29
Vol 2. (1979) 29-34
SOME RESULTS ON COMPOSITION OPERATORS ON
R.K. SINGH
D.K. OUPTA
.
2
B.S. KOMAL
Mathematics Department
University of Jammu
Jammu IB000 i, India
(Received August 12, 1978)
ABSTRACT.
A necessary and sufficient condition for a bounded operator to be a
composition operator is investigated in this paper.
Normal, quasi-hyponormal,
paranormal composition operators are characterlsed.
KEY WORDS AND PHRASES. Invertible, normal, quasi-normal, hyponormal, quasi-hyppaanormal composition operators.
onorm,
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
Let
2
N
Primay 47B99, Secondary 47B99.
be the set of all non-zero positive integers and
be the Hilbert space of all square-summable sequences.
ping from
from
12
N
into itself.
Let
be a map-
Then we define a composition transformation
C
into the space of all complex valued sequences by
C f
In the case C
fo
is bounded and the range of
position operator.
The symbol
12
for every
C
is in
2
we call it a com-
B(l2) denotes the Banach algebra of all bounded
30
R. K. S INGH, D. K. GUPTA & B. S. KOMAL
linear operators on
2
In the first section of this paper, a criterion for a bounded operator to be
a composition operator is given.
In latter sections, Normal, Quasi-hyponormal,
Paranormal composition operators are characterized.
CRITERION FOR A BOUNDED OPERATOR TO BE A COMPOSITION OPERATOR.
2.
In this section we obtain a necessary and sufficient condition for a bounded
operator
A to be a composition operator.
THEOREM 2. I.
only if for every
(n)
A e B(
Let
N
n
Let
A
m N
e n (p)
np
is a composition operator if and
n e N
A * e (n)
Conversely suppose
m
This shows that
is unique.
A
C
C (m)
Thus
and hence
(n
A * e (n)
A
e
(m)
where
(the Kronecker delta).
2
Then
A
m
(n)
* (n)
C
e
for some
C$
* [6].
C
by definition of
Then define
e
A * e (n)
such that
A be a composition operator on
(n)
* e (n) ( (n)
Then A * e
Let
defined, since
Then
there is an
is the sequence defined by
PROOF.
3.
2)
is well
for every
N.
n
C
NORMAL COMPOSITION OPERATORS.
An operator A
B(H)
is normal if
A
commutes with its adjoint.
true in general that every invertible operator is normal.
It is not
It is true in case of
composition operators and is shown in the following theorem.
THEOREM 3.1.
Let
B(Z 2)
C
Then
C
is invertible if and only if
C
is normal.
PROOF.
Suppose that
C
is invertible.
Then by Theorem 2.3 of
[6J
is unitary and therefore, it is normal.
Conversely, if
C
is not invertible, then by theorem 2.2 of
is not invertible, and so either
is not one-to-one or
[6]
is not onto.
C
RESULTS ON COMPOSITION OPERATORS
If
some
C
N
n
for
Ic_
is not one-to-one, then
#
And if
N \ (N),
n
e
(n)
II
I
where
(N)
IC
and
IC
is not onto, then
31
e
(n)
II
e
1
(n)
II
and
>
for
IC
e
(n)
II
0
Thus in both the cases,
is the range of
This proves the sufficient part.
is not normal.
COROLLARY.
Let
n e N
Then
n
C
C_
is normal if and only if
is
normal.
4.
QUASI-HYPONORMAL COMPOSITION OPERATORS.
B(l 2)
Ce
Let
Then the measure
[4].
)t
respect to the measure
-i
is absolutely continuous with
It is clear that the measure
-I
absolutely continuous with respect to the measure
d
Let
f
d%
(the Radon-Nikodym derivative of the measure
o
-I
An operator
x e
12
x
vectors
A
in
go
I
A [I, p. 133]
Then by Theorem
A
B(Z 2)
-I
with
and
h
o
go
d% (@o@)
d
f
-I
h
o
o
IA*A xll < IAA xll
IA xJl 2 < IA2 xll for all
for
is quasi-hyponormal if
is called
Z2
is
.),
dl (@o@)
d%@
-1
#
-I
respect to the measure
all
)(o)
paranormal if
unit
It is shown in this section that the class of quasi-
hyponormal composition operators coincides with the class of paranormal composition
operators.
THEOREM 4.1.
if
C#
e
B(Z 2)
Then
C
is quasi-hyponormal if and only
fo < -o
PROOF.
where
or
Let
X_
Suppose
C$
is quasi-hyponormal.
is the characteristic function of the set
II MfoXEII 2
_<
CCCXEJ
Then
_<
CC+XEI 12
E
lXEO O l
by proof of Th. 3 [3],
32
R.K. SINGH, D. K. GUPTA & B. S. KOMAL
or
or
-
or
E
Since this is true for all
fo
<
-o
Conversely, if
f -<
O
go
N
d%
fE
(ho-fo2)
N
<
hod%
d% -> 0
therefore,
then
f2
_<
O
f
h
go
O
f2o
>
o
h
This shows that
O
" lfl2dfl2dX
j"
N
lf12d
()-i
N
lccfl 12
ICC
This completes the proof.
THEOREM 4.2.
if
C
C
C e B(2
C
Then
is quasi-hyponormal if and only
is paranormal.
PROOF.
if
Let
Necessity is true for any bounded operator
is paranormal and
N
n
A
then
for all
or
flX{n}Ol 2
or
fiX{ n} 12
or
fo(n)
or
(fo(n))
(:IX{n}l 2
d%-<
dX,- 1
_<
(fiX{n)]
d%
2
n e N
(o)-i) 1/2
dX (qoq)
-1 1/2
I
f o d% <
or
For the sufficiency,
h d%)
{n} o
(ho(n))
<
2
_<
ho(n)
go(n)" fo(n)
for all
n e N
33
RESULTS ON COMPOSITION OPERATORS
fo -< go
Thus
THEOREM 4.3.
Let
Hence
C
is quasi-hyponormal in view of the previous theorem.
C B(2)r.
N
and
/
N
be one-to-one.
Then the
following are equivalent.
PROOF.
is normal
(ii)
C
is hyponormal
(iii)
C#
is quasi-hyponormal
A
Here we show that (iii) implies (i).
be quasi-hyponormal.
taking
for
C
The implications (i) implies (ii), (ii) implies (iii)are true for
any bounded operator
C
(i)
n
to be
X{n)
N\ (N)
Then
is onto, because if
{(n)}
we have
which is a contradiction.
C
is not onto, then
the characteristic function of the singleton set
IC$CX{(n)}[[
1
I[ccx{(n)}II
and
0
is one-to-one by hypothesis, therefore
Since
By theorem 2.2 [6]
is invertible.
For this let
C
is invertible and so by theorem 3.1
is normal.
REMARK.
- - -
One has the following inclusion relation for classes of operators.
Normal
Quasi-normal
All the inclusions are proper [2].
Hyponormal
Quasi-hyponormal
We show with the help of examples that these
inclusions are also proper for composition operators.
EXAMPLE i.
Quasi-normal but not normal.
Let #
(n)
N
N
(n+l)/2
n/2
be defined by
if
n
is odd
if
n
is even
34
R.K. SINGH, D. K. GUPTA & B. S. KOMAL
Then from theorem 3 of [3] it follows that
C0C 0 Mf
21, where
I
C
identi’ty
is the
is quasi-normal since
0
C
But
operator.
O
is not normal
0
in view of theorem 3.1.
EXAMPLE 2.
Let
f (n)
o
But
N
0
0(3n+m)
n+l
3
Hyponormal but not quasi-normal.
for
if
(fo0)
N
n
EXAMPLE 3.
Let
equals I
0
and
0,1,2
m
# 1.
(2) #
and
llCx]l
3
n e N.
It is clear that
fo(2),
C
hence
Then
0(2)
and
f (n)
2
o
Hence
f o0 < f
o
o
1
equal
1
and
and
if
n
C
is hyponormal.
0
is not quasi-normal.
0
Quasi-hyponormal but not hyponormal.
N
+
N
0(3n+m)
x(1)
and
0(1)
be the mapping such that
n+2
for
hyponormal in view of theorem 4.1.
is such that
0(1)
be the mapping such that
2. x(3)
llC0xll
=
1,2,3
m
But
C
0
and
n
E
0(3) equal 2,
N
Then
C
0
0(2)
is quasi-
is not hyponormal because if
I and x(n) --0
Thus
and
llCxll
x
otherwise, then
llCxll
REFERENCES
i.
Halmos, P. R., Measure Theory, Springer- Verlag, Berlin- HeidelbergNew York, East West Press, New Delhi-Madras, 1974.
2.
Shah, N. C. and Sheth, I. H., Some results on quasi-hyponormal operators
J. of Indian Math. Soc. Vol. 39 (1975), 285-291.
3.
Singh, R. K., Compact and Quasi-mormal composition operators Proc. Amer.
Math. Soc. Vol. 45 (1974), 80-82.
4.
Singh, R. K., Composition operators induced by rational functions Proco
Amer. Math. Soc. Vol. 59 (1976), 329-333.
5.
Singh, R. K. and Komal, B. S., Composition operators, Bull. Austral. Math.
Soc. Vol. 18 (1978), 439-446.
6.
Singh, R. K. and Komal, B. S., Composition operator on
Proc. Amer. Math. Soc. Vol. 70 (1978) 21-25.
P
and its adjoint
E
2