Affiliation:
1. School of Mathematics and Statistics, Shaanxi Normal University , Xi’an , China
Abstract
Abstract
Let
ℋ
{\mathcal{ {\mathcal H} }}
be an infinite dimensional complex Hilbert space and
ℬ
(
ℋ
)
{\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})
the algebra of all bounded linear operators on
ℋ
{\mathcal{ {\mathcal H} }}
. For an operator
T
∈
ℬ
(
ℋ
)
T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})
, we say property
(
t
)
\left(t)
holds for
T
T
if
σ
(
T
)
⧹
σ
u
w
(
T
)
=
Π
00
(
T
)
\sigma \left(T)\hspace{-0.08em}\setminus \hspace{-0.08em}{\sigma }_{uw}\left(T)={\Pi }_{00}\left(T)
, where
σ
(
T
)
\sigma \left(T)
and
σ
u
w
(
T
)
{\sigma }_{uw}\left(T)
denote the spectrum and the Weyl essential approximate point spectrum of
T
T
, respectively, and
Π
00
(
T
)
=
{
λ
∈
iso
σ
(
T
)
:
0
<
n
(
T
−
λ
)
<
∞
}
{\Pi }_{00}\left(T)=\left\{\lambda \in {\rm{iso}}\sigma \left(T):0\lt n\left(T-\lambda )\lt \infty \right\}
. In this paper, we consider the stability of property
(
t
)
\left(t)
under (small) compact perturbations. Also, we explore the relations between the stability of property
(
t
)
\left(t)
and the stability of Weyl-type theorems. Moreover, we characterize those operators
T
T
satisfying that property
(
t
)
\left(t)
holds for
f
(
T
)
f\left(T)
for each function
f
f
analytic on some neighborhood of
σ
(
T
)
\sigma \left(T)
.
Reference20 articles.
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and
P. Peña
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2. L. A. Coburn
, Weyl’s theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285–288.
3. R. Harte
and
W. Y. Lee
, Another note on Weyl’s theorem, Trans. Amer. Math. Soc. 349 (1997), no. 5, 2115–2124.
4. V. Rakočević
, Operators obeying a-Weyl’s theorem, Rev. Roumaine Math. Pures Appl. 34 (1989), no. 10, 915–919.
5. J. L. Shen
and
A. Chen
, A note on property (
gaR
) and perturbations, Ukraïn. Mat. Zh. 69 (2017), no. 7, 974–983.
Cited by
1 articles.
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