WEYL'S THEOREM HOLDS FOR p-HYPONORMAL
OPERATORS*
by MUNEO CHO, MASUO ITOH and SATORU OSHIRO
(Received 3 January, 1996)
1. Introduction. Let 5if be a complex Hilbert space and B(ffl) the algebra of all
bounded linear operators on #f. Let 3if(2if) be the algebra of all compact operators of
fl(2if). For an operator TeB(%), let a(T), crp(T), (rn(T) and noo(T) denote the
spectrum, the point spectrum, the approximate point spectrum and the set of all isolated
eigenvalues of finite multiplicity of T, respectively. We denote the kernel and the range of
an operator 7 by ker(7) and R(T), respectively. For a subset % of %C, the norm closure of
<3/ is denoted by <§. The Weyl spectrum <o(T) of T e B(%) is defined as the set
u>(T)= n
<r(T + K).
We say that Weyl's theorem holds for T if the following equality holds;
An operator T e B{^€) is said to be p-hyponormal if (T*Tf > (TT*y. Especially,
when p = 1 and p = \, T is called hyponormal and semi-hyponormal, respectively. It is
well known that a p-hyponormal operator is g-hyponormal for q <p by Lowner's
Theorem. In [8], Coburn showed that Weyl's theorem holds for hyponormal operators. In
this paper, we shall prove the following results.
THEOREM 0. Let T be a p-hyponormal operator on $? where 0<p < 1 . Then Weyl's
theorem holds for T.
2. Proof of Theorem 0. Throughout this section, let p satisfy 0 <p < 1. First in [2]
Baxley proved the following result.
THEOREM A (Lemma 3 of [2]). Let T e B(ffl). Suppose that T satisfies the following
condition C-l.
C-l. / / {An} is a infinite sequence of distinct points of the set of eigenvalues of finite
multiplicity of T and {xn} is any sequence of corresponding normalized eigenvectors, then
the sequence {xn} does not converge.
Then
*(T)-^(T)
<zto(T).
Cho and Huruya proved the following result.
THEOREM B (Corollary 5 of [5]). Let T be p-hyponormal. Let a,/3 G ap{T) where
a ¥> /3. If x and y are eigenvectors of a and /3, respectively, then (x,y) = 0.
"This research was partially supported by a grant from the National Science Foundation.
Glasgow Math. J. 39 (1997) 217-220.
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218
M. CHO, M. ITOH AND S. OSHIRO
By Theorem B, it follows that if T is p-hyponormal, then T satisfies C-l. Hence it is
clear that if T is p-hyponormal, then
For the proof of the converse inclusion relation, we shall prove the following result.
THEOREM
1. Let T be p-hyponormal. If\ is an isolated point ofa(T), then A e crp(T).
Since the theorem holds for A T^O, by Theorem 1 of [7], we need only prove the case
A = 0.
For this proof, we need the Aluthge transform (cf. [1]). Let U\T\ be the polar
decomposition of T e B{VC). Then Aluthge introduced the transform
T^T
= \T\mU\T\m,
and proved the following result.
THEOREM
C (Theorems
1 and 2 of
f = \T\m U \T\m is (p + ^-hyponormal.
[1]). Let
T be p-hyponormal.
Then
Though the operator U in Aluthge's paper is unitary, it is easy to check that Theorem
C holds for any p-hyponormal operator.
We need some further results.
LEMMA
\T\mU\T\m.
1. Let T=U\T\
be p-hyponormal.
Then a(T) = a(T),
where T =
Proof. To see this write T = (U \T\m) \T\m and consider separately A = 0 and A * 0.
LEMMA
0SCTP(T).
2. Let T be semi-hyponormal. If 0 is an isolated point of a(T), then
Proof. Let T = U \T\ be the polar decomposition of T and T = \T\m U \T\m. Since 0
belongs to the boundary of o-(T), by Lemma 1 it follows 0 e <r(T) = <T(7). Therefore, 0 is
an isolated point of cr(7"). Since, by Theorem C, T is hyponormal, from a Stampfli result
(Theorem 2 of [10]) it follows that 0 is an eigenvalue of t. Hence there exists a nonzero
x0 e 2i?such that tx0 = 0. Since \T\m U \t\mx0 = 0, we have U \T\ll2x0 e kei(\T\m). Since,
by Lemma 1 of [5], ker(7) c ker(7*), It follows that
Hence \T\xo = 0. Therefore we have 0 e ap(T).
Proof of Theorem 1 for A = 0 and 0 <p < \. Let T = U \T\ be the polar decomposition of T and T = \T\m U \T\m. By Lemma 1, it follows that 0 E <J{T) and 0 is an isolated
point of cr(T). Since, by Theorem C, t is semi-hyponormal, by Lemma 2 it follows that
0 e crp(t). Hence also it follows that 0 e (TP(T) on the analogy of the proof of Lemma 2.
Proof of the inclusion relation. w(7) c a(T) - noo(T).
Let A e noo(T). By Theorem 4 of [5] or Theorem 2 of [9], we have
ker(7 - A) c ker((7 - A)*) = (R(T - A))1.
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p-HYPONORMAL OPERATORS
219
Hence we have the following decomposition of T — A:
lo s) on
Since
0
)
5 + A/'
5 + A is a p-hyponormal operator on R((T - A)*). If A e a(S + A), by Theorem 1 we have
A E crp(5 + A) because A is an isolated point of a(5 + A). This is a contradiction. Hence
A * a(S + A). Therefore 0 £ <r(S). Let
K
~\o o)-
Then K e 3iT(#f) and
T+K x
~ ~\o
s)
is an invertible operator. Therefore A g (o(T). Hence we have
and the proof of the theorem is complete.
3. Application.
COROLLARY
1. Let T be p-hyponormal. If noo(T) = 0 , then for every K E
Proof. By Corollary 10 of [5], we have that r(T) =\\T\\, where r(T) is the spectral
radius of T. Hence from Theorem 1 it follows that ||7|| < \\T + K\\ for every K
COROLLARY 2. Let T be p-hyponormal. Then there exist orthogonal reducing
subspaces M and M for T such that 3€= JK&Jf, T\M is a normal operator on M and
Proof. For A e o-p(T), let
Mk = {x I Tx = AJC}.
Then, by Theorem 4 of [5], MK is a reducing subspace for T. Let
M=
©
A
and Jf = M±.
Then M reduces T and T\M is normal. Let 5 = T\jf. Then 5 is a p-hyponormal
operator on Jf. By Theorem 0, Weyl's theorem holds for 5. Since noo(S) = 0 , it follows
that w(S) = <r(S).
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220
M. CHO, M. ITOH AND S. OSHIRO
COROLLARY 3. Let T be p-hyponormal. Then
\\{T*Ty-{TT*y\\<- ff r^
Proof. Let /x be planar Lebesgue measure. Then we have fi(noo(T)) = 0. Hence the
result follows from Theorem 5 of [6].
COROLLARY 4. Let T be p-hyponormal. Then, for every K e
Tf -{TT*Y || < - ff r^-'
<r(T+K)
Proof. Since a>(T) a a(T + K) for every Ke%(%),
Corollary 3.
the result follows from
REFERENCES
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13 (1990), 307-315.
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(1972), 447-452.
4. M. Cho, Spectral properties of p-hyponormal operators, Glasgow Math. J. 36 (1994),
117-122.
5. M. Cho and T. Huruya, p-hyponormal operators for 0 < p < \, Comment. Math. 33 (1993),
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6. M. Cho and M. Itoh, Putnam's Inequality for p-hyponormal operators, Proc. Amer. Math.
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8. L. A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1965),
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10. J. G. Stampfli, Hyponormal operators, Pacific J. Math. 12 (1962), 1453-1458.
11. D. Xia, Spectral theory of hyponormal operators (Birkhauser Verlag, Boston, 1983).
MUNEO CHO
MASUO ITOH
DEPARTMENT OF MATHEMATICS
TOKYO PUBLIC MITA SENIOR HIGH SCHOOL
1-4-46
FACULTY OF ENGINEERING
MITA
KANAGAWA UNIVERSITY
TOKYO 108
YOKOHAMA 221
JAPAN
JAPAN
SATORU OSHIRO
DEPARTMENT OF MATHEMATICS
JOETSU UNIVERSITY OF EDUCATION
JOETSU 943
JAPAN
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