Internat. J. Math. & Math. Sci.
Vol. I0, No.3 (1987) 621-623
621
M-QUASI-HYPONORMAL COMPOSITION OPERATORS
PUSHPA R. SURI and N. SINGH
Department of Mathematics
Kurukshetra University
132 119, India
Kurukshetra
(Received November 12, 1985)
A necessary and sufficient condition is obtained for M-quasi-hyponormal
composition operators. It has also been poved that the class of M-quasi-hyponormal
ABSTRACT.
composition operators coincides with the class of M-paranormal composition operators.
Existence of M-hyponormal composition operators which are not hyponormal; and M-quasi-
hyponormal composition operators which are not M-hyponormal and quasi-hyponormal are
also shown.
KEY WORDS AND PHRASES.
M-hyponormal, M-quasi-hyponormal, M-paranormal, normal
composition operators.
1980 AMS SUBJECT CLASSIFICATION CODE. 47
PRELIMINARIES.
1.
Let (X,S,m) be a sigma-finite measure space and T a measurable transformation
from X into itself (that is one
0 whenever m(E)
0 for E e S). Then
mT-I(E)
the equation
C
T
fo T for every f in
CTf
is bounded with range in
L2(m),
L2(m)
defines a linear transformation.
then it is called composition operator.
If
If X
the set of all non-zero positive integers and m is counting measure on the family
of all subsets of N, then
sequences).
Let f
o
dmT
dm
-i
L2(m) 2
(the Hilbert space of all square surmnable
be the Radon-Nikodym derivative of the measure mT
-I with
respect to the measure m,
dm(ToT)
-I
dmT
Then
dm(ToT)
o’
dm
h
o
ho fo" o"
Let B(H) denote the Banach algebra of all bounded linear operators on the Hilbert
space H. An operator T e B(H) is called M-quasi-hypornormal if there exists M > 0
such that
M2T *
2
T 2-
(T’T)2
>_
0
N
622
P.R. SURI AND N. SINGH
IT*Txll
or equivalently
112xll
M
for all x in H [I].
T is said to be
M-paranormal [2] if for all unit vectors x in H
l[Txll 2 _-< M
IlT2xl[.
T
is said to be
M-hyponormal [2] if
llTxll
llTxll
_-< M
for all x in H.
The purpose of this paper is to generalize the results on quasi-hyponormal
composition operators in [3] for M-quasi-hyponormal composition operators.
2.
M-QUASI-HYPONORMAL COMPOSITION OPERATORS.
In this section we obtain a necessary and sufficient condition for M-quasihyponormal composition operators and then show that the class of M-quasi-hyponormal
2
coincides with the class of M-paranormal composition
composition operators on
operators.
We also show the existence of M-hyponormal composition operators which
are not hyponormal, and M-quasi-hyponormal composition operators which are not
M-hyponormal and quasi-hyponormal.
Let C
THEOREM 2.1.
f2o M2ho
PROOF.
e
T
B(L2).
Since for any f in L
(C
,2
T
Then C
T
is M-quasi-hyponormal if and only if
2,
(Cf,Cf)
2
CTf,f)
(Mh
h
lel z
d,
f,f),
o
,2
where M
h
is the multiplication operator induced by h
o
CTCT Mf o
Similarly it can be seen that
if
,2
2
M C
C
T
(C T* C T) 2
2
T
C
T
o’ therefore
2
CT CT Mho
is M-quasi-hyponormal if and only
O.
This implies that
M2 M
that is
f2o
M
2
h
2
h
o
Mf o
0,
o
Hence the result.
f
o
COROLLARY.
2
M go"
PROOF.
Let C
Since
T
e
B(2).
ho fo’go
and
Then C
fo
T
is M-quasi-hyponumal if and
only if
is positive, therefore, by above theorem we
get the result.
if and only if
THEOREM 2.2. Let C T e B(2). Then C T is M-quasi-hyponormal
C is M-paranormal.
T
the sufficiency,
PROOF. Necessity is true for any bounded operator A. For
let C be M-paranormal, then
T
M-QUASI-HYPONORMAL COMPOSITION OPERATORS
2
2
IICTX {n}ll MIICT X {n}l for all n N
I/2
J" IX {n} TlZdm -<- M( I IX{n oT212dm)
M(I IX {n}12dm(ToT)-l) I/2
/ IX
{n} 12dmT-I
or
or
am
f} fo
or
{n
f2
or
O
(n)
for all n in N.
is M-quasi-hyponormal.
Let C
THEOREM 2.3.
e
M({n} ho din)I/2
_-<
M2h
(n)
O
f2o M2ho; CT
Hence
623
e
T
B(-) and T:N
N be one-to-one.
Then the following
are equivalent.
(i)
(ii)
(iii)
PROOF.
operator A.
ICT
then
CTfll
Normal
M-hyponormal
M-quasi-hyponormal.
(i) implies (li), (ii) implies (iii) are always true for any bounded
We show that (iii) implies (i). Let C be M-quasi-hyponormal. Then
T
<_- M
Now T is onto because if T is not onto
fll for all f in
NIT(N)
2.
IC
NIT(N)
is non-empty and for n e
ICTCT X{n}l
There exists no M>0 such that C
T
Since T is one-to-one, therefore,
invertible and C
T
ICTCT X{n}l
and
O.
is M-quasi-hyponormal which is a contradiction.
T is invertible, by Theorem 2.2 [4] C is
T
[3].
is normal by Theorem 2.1
Here we give an example of a composition operator on
2
which is M-hyponormal
but not hyponormal.
EXAMPLE i.
Let T:N
T(1)
2,
N be the mapping such that
T(2)
T(3n+m)= n+2,
Then C
T
EXAMPLE 2.
Let T:N
0,1,2 and n e N.
m
-hyponormal.
C
T
1,2,3
m
is not nyponormal as
T(3)
i,
--
foT
$
2 and
and n
fo
for n
N be defined by T(1)
Then C
T
is
N.
i.
C
T
is M-hyponormal for M
2, T(2)
T(3n+m)
I,
quasi-hyponormal but C
T
/.
n+l,
is not
is not quasi-hyponormal also.
REFERENCES
M-Quasi-Hyponormal Operators, Bull. Austral. Math. Soc.
I.
SURI, P.R. and SINGH, N.
(Communicated).
2.
ARORA, S.S. and KUMAR, R. M-Paranormal Operators, Publications De L’Institut
Mathematique Nouvelle Serie 2__9(1981), 5-13.
SINGH, R.K., GUPTA, D.K. and KOMAL, B.S. Some Results On Composition Operators
on 2, Internat. J. Math. and Math. Sci. 2(1979), 29-34.
3.
4.
SINGH, R.K. and KOMAL, B.S. Composition Operator on
Proc. Amer. Math. Soc. 70(1978), 21-25.
P
and its Adjoint,