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| THE (U + K)-ORBIT OF ESSENTIALLY NORMAL OPERATORS AND COMPACT PERTURBATIONS OF STRONGLY IRREDUCIBLE OPERATORS |
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Citation： |
JI Youqing,JIANG Chunlan,WANG Zongyao.THE (U + K)-ORBIT OF ESSENTIALLY NORMAL OPERATORS AND COMPACT PERTURBATIONS OF STRONGLY IRREDUCIBLE OPERATORS[J].Chinese Annals of Mathematics B,2000,21(2):237~248 |
| Page view： 1062
Net amount： 1066 |
Authors： |
JI Youqing; JIANG Chunlan;WANG Zongyao |
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| Abstract： |
Let $\Cal H$ be a complex, separable, infinite dimensional Hilbert space,
$T\in\Cal L(\Cal H)$. $(\Cal U+\Cal K)(T)$ denotes the $(\Cal U+\Cal K)$-orbit of $T$, i.e.,
$(\Cal U+\Cal K)(T)=\{R^{-1}TR:\ R$ is invertible and of the form unitary plus
compact$\}$. Let $\Omega$ be an analytic and simply connected Cauchy domain
in $\Bbb C$ and $n\in \Bbb N$. $\Cal A(\Omega,n)$ denotes the class of operators,
each of which satisfies
(i) $T$ is essentially normal;
\quad
(ii) $\sigma(T)=\overline{\Omega}$, $\rho_F(T)\cap \sigma(T)=\Omega$;
(iii) $\ind(\lambda-T)=-n$, $\nul(\lambda-T)=0$ $(\lambda\in \Omega)$.
It is proved that given $T_1$, $T_2\in \Cal A(\Omega,n)$ and $\epsilon>0$,
there exists a compact operator $K$ with $\|K\|<\epsilon$ such that
$T_1+K\in(\Cal U+\Cal K)(T_2)$. This result generalizes a result of P. S.
Guinand and L. Marcoux$^{[6, 15]}$. Furthermore, the authors give a
character of the norm closure
of $(\Cal U+\Cal K)(T)$, and prove that for each $T\in \Cal A(\Omega,n)$,
there exists a compact (SI) perturbation of $T$ whose norm can be
arbitrarily small. |
Keywords： |
Essentially normal, (U + K)-orbit, Compact perturbation, Spectrum,
Strongly irreducible operator |
Classification： |
47A55, 47B, 47C |
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