Internat. J. Math. & Math. Sci.
VOL. 14 NO. 3 (1991) 439-442
439
FACTORIZATION OF k-QUASIHYPONORMAL OPERATORS
S.C. ARORA
Department of Mathematics
Untverslty of Delhi
||0007
Delhi
Ini.a
and
J. K. THUKRAL
Department of Mathematics
S.R.C.C.
Universlty of De
If0007
Delhi
India
(Received January l, 1987 and in revised form March 17, 1989)
ABSTRACT.
A
Let
be the class of all operators T on a liilbert space H such that
R(T*kT),
the range space of T*kT, is contained in R(T*k+I), for a positive integer k.
It has been shown that if T e A, there exists a unique operator C on H such that
T
IICTII 2
(li)
(ill)
(iv)
T*k+ICT;
T*kT
(i)
N(T*kT)
N(C T)
.
Inf{.:
> 0 and
(T*kT)(T*kT) *
<
.
T ,k+l T
k+l
};
and
R(C T) c_ R(Tk+l)
The main objective of this paper is to characterize k-quasihyponormal; normal, and
self-adjoint operators T in A in terms of C T.
Throughout the paper, unless stated
otherwise, H wlll denote a complex Hilbert space and T an operator on H, i.e., a
bounded linear transformation from H into H itself.
For an operator T, we write R(T)
and N(T) to denote the range space and the null space of T.
KEY
AND
WORDS
quaslhyponormal,
PHRASES.
Self-adJoint, normal, unitary, quasinormal, hyponormal,
Isometry, partial isometry, null space, range
k-quaslhyponormal,
space and the projection.
1980 AMS SUBJECT CLASSIFICATION CODES. 47B20; 47B15.
I.
INTRODUCTION
T is
said
equtvalently
II
to
be
quasinormal
’ I xll
if
.
T(T*T)
(T*T)T, hyponormal if T*T
)TT* or
k-quasihyponoral (Campbell and Gupta
440
S.C. ARORA AND J.K. THUKRAL
[I]) for a poslttve iateger k if
llT*’rkxll llTk+Ixll
T*k(T*T T’t’*)Tky
0 or equivalently
in H.
for each
A
The purpose of this paper is to consder the class
R(T*kT)
that
R(T
*k+l)
for
a
T
A
of those operators T such
More prec[selv,
k.
integer
positive
our
aim
is
normal
and
self-
identify
those
adjolnt.
The motivation is due to Embry [2] who conaldered the class of operators T
operators
in
which
are
k-quashyponormal,
satisfying R(T)R(T*) and Patel [5] who discssed the class of operators T sat[fykng
R(T*2).
R(T*T)
If T
A
then by Douglas’ theo:em [3,
eorem
I] there exists a
unique operator C T such that
2.
(i)
T,kT
(il)
[[CT [2
(Ill)
N(C T)
(iv)
R(CT)
T,k+l CT;
Inf{:
0 and
N(T*kT):
(T,kT) (T,kT),
<
T,k+ Tk+
}.
and
R(rk+l).
MAIN RESULTS.
By Douglas’ theorem [3, Theorem I], the class A contains all k-quaslhyponormal
operators.
r
THEOREM 2.1. An operator
PROOF.
If
IICTII
I,
in
A
is k-quas[hyponormal if and only if
llT*Tkxll IICT*
Tk+Ix II
II
T
k+l
____[[CT[
I.
xll
for all x in H and hence T is k-quaslhyponorma[.
Conversely, assume that T is k-quasihyponormal.
Since
To prove our next result, we need the following lemma.
LEMMA 2. I. Let T be a quasinormal operator. Then for any positive integer k
(a)
T,Tk
Tk-IT,T
I(*/xll
(c)
I xll
for all vectors x in H
N(T*kT) c_N(T*k).
PROOF.
(a)
We prove it by induction on k. For k
I, trivial. For k
2, again
Now assume that the result is true for any positive
m+l
m-I (T*T)T Tm-1
integer m > 2.
(T*Tm)T (TmThen T*T
T
TT*T
TmT*T.
Hence by induction the result follows. (b) It is an immediate consequence of the fact
it holds since T is quasinormal.
1T*T)T
that
if
T ts quasinormal,
then
(T’T) k
T*kTk
for any positive
integer k.
(c)
FACTORIZATION OF k-QUASIHYPONORMAL OPERATORS
441
N(T*kT).
0 by (a). Thus T*k-lx e N(T*T)
Then T*kTx
Let x e
0, i.e., T*T T*k-lx
0, i.e.,
N(T). But N(T) c_ N(T*) since T is quasinormal. Therefore T*kx
N(T*k).
x e
By using the lemma we obtain the following
Let T e A be a quasinormal operator.
Then CT is a quasinormal
THEOREM 2.2.
partial isometry with R(C
Isometry on
R(Tk+I).
R(Tk+I).
T)
k+l)
set of all those vectors x satisfying
R(Tk+l),
is an isometry on
k+l)
Hence R(T
+/-
N(C)
llSxll -llxll
therefore R(T
N(T*kT)
follows that N(C
CT is
a
T)
k+l)
e
[4, p. 63] and since
N(C) +/-,
the initial space of
By making use of Lemma 2.
N(T *k+l)
N(T *k)
C.
N(C)
since
again, we see
R(Tk+I).
R(CT)
From this it
reduces CT and since C T is a partial isometry, CT is of the form
0, where A is an isometry.
A
C hence
R(C T) as R(C T) is closed.
R(C T)
We now prove that CT is quaslnormal.
that N(C T)
Therefore
Further, since the initial space of a partial Isometry S equals the
partial isometry.
C
N(C) +/-.
R(C T)
But R(T
This gives that CT commutes with C* T C T and hence CT
is quasinormal.
Let T
LEMMA 2.2.
PROOF.
have R(C
if x
6
Since R(C
R(T
T)
N(T*kT),
x E N(T*T)
6
A
T)
be such that R(C
k+l)
=...- R(T).
then
T*kTx
N(T).
Then
N(T*kT)
N(T).
R-o we
N(T*k+I). Now,
R(T) and, by hypothesis, R(CT)
R(T
k+l)
R(T).
T)
N(T .2)
Thus N(T*)
0, i.e. Tx
E
N(T
*k)
N(T *k)
N(T*).
That means T*Tx
0 or
This completes the proof.
Our next result gives a characterization of normal operators in A
THEOREM 2.3. An operator T in A is normal if and only if CT is a normal partial
Isometry with R(C T)
PROOF.
Then by Theorem 2.2, CT is a partial isometry with
t
R(T). Thus by Lemma 2.2,
N(T*)
and hence R(C
Let T be normal.
k+l)
(z,kz)
R(C T)
C CT
Since
is
conclude that
N(T*k+I
T)
R(T
(CT
R(T).
(Z).
the
R(Cz)
on R(C)
projection
(*)+/-=
R(Z)
Zhereore
and
CTC
is
(Cz)+/-= R(C*z).
(Z)
the
projection
on
R(CT),
CCT.
CTC
Assume on the other hand that CT is a normal partial isometry with R(C T)
Since R(C
k)
R(T
T)
we
k+l) e R(Tk)
R(T
e R(T),
we have R(C
R(T) and consequently N(T*)- N(T .2)
-...
T)
R(T).
R(Tk+l)
N(C* T)
N(CT)
N(T*kT)
442
S.C. ARORA OF k-QUASIHYPONORMAL OPERATORS
i
is a
R(Tk+I),
partial i some try on R(C T)
each
g
H.
in
Thus
llTyll
lT*Tkxll
we have
llTyll
for each y
IC*T Tk+l x II II Tk+l x II
In R(Tk). He,lee llT*xll
llTxll
for
for
in H, i.e., T is normal.
each
COROLLARY 2.1.
Let T
unitary operator with
PROOF.
E
A.
Then T is normal and one-to-one if and only if C is a
T
R(T’)-.
R(CT)
Suppose T is normal and one-to-one.
partial Isometry with
R(CT)
R(T).
Since N(C
T)
Then by Theorem 2.3, C is a normal
T
i
N(T)
{0}, we have N(CT)
H and
thus C T is an Isometry and consequently C is a unitary operator.
T
Conversely, if C T is a unitary operator with
R(T), T is normal by Theorem
R(CT)
Also by Lemma 2.2, N(T)
N(T*kT) N(CT) {0}, therefore T is one-to-one.
The next corollary characterizes self-adJoint operators in A.
2.3.
COROLLARY 2.2
on R(T).
PROOF.
Suppose T
TkT T k+l
+l k+l.
as R(T)
T is self-adjoint if and only if C is the projection
T
self-adjoint.
is
Then by Theorem 2.3,
C T and T is self-adjoint, we have T k+l
Since
c
A.
Let T
c
R(CT)
hi
N(C).
z
o
(+l
().
N(CT)
R(C T)
k+l)
N(T).
i
CT Tk+l
therefore
Tx for all x
as C
CTTk+I
T
T
is
k+l,
Ao
C
R(Tk+I).
i.e.,
0 o
(----+/-
R--.
Then R(CT)-R(T)and
in
hence by
Also, as in the proof of Theorem 2.3, we have
R(T) and thus N(T*)
Therefore T*x
k
T*T
N(T,kT)
R(T
Tk+ICT
R(T)
Therefore C T is the projection on
Assume now that C T is the projection on R(T).
Lemma 2.2,
R(CT)
R(Tk).
N(T
Moreover
.2)
T*kT
N(C)
Tk+lcT,
N(CT)
implies
But CT is the projection on R(T)
self-adjolnt.
That means T*y
k).
Ty for all y in R(T
N(T)
Thus T*x
R(Tk+I),
Tx for
all x in H or T is self-adjoint.
ACKNOWLEDGEMENT.
The author gratefully acknowledges the financial support by the
India, vide Research Grant No. F-26-1(1629)/85(SR-IV).
The authors are
thankful to Professor B.S. Yadav for his kind guidance and encouragement during the
preparation of this paper.
Thanks are also due to the referee for suggesting an
improvement over the original version of the paper.
U.G.C.
REFERENCES
1.
2.
3.
4.
5.
CAMPBELL, S.L. and GUPTA, B.C., On k-Quaslhyponormal Operators, Math. Japonica
23 (1978), 185-189.
EMBRY, M.R., Factorlzation of Hyponormal Operators, J. Austral. Math. Soc. 13
(1972), 323-326.
DOUGLAS, R.G., On Majorlzatlon, Factorlzatlon, and Range Inclusion of Operators
on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-415.
HALMOS, P.R., A Hilbert Space Problem Book, Van Nostrand, Princeton, N.J., 1967.
PATEL, S.M., Factorlzatlon of Quasihyponormal Operators, Publ. de l’inst. Math.
24(38), (1978), 121-123.