عنوان مجله
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TWMS Journal of Applied and Engineering Mathematics
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چکیده
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An operator $T$ is said to be $k$-quasi class
$\mathcal{A}^{\ast}_{n}$ operator if \break
$T^{*k}\left( \vert T^{n+1} \vert ^{\frac{2}{n+1}}-\vert T^{*} \vert^{2}\right)
T^{k}\geq 0,$ for some positive integers $n$ and $k$.
In this paper, we prove that the
$k$-quasi class $\mathcal{A}^{\ast}_{n}$ operators have
Bishop$^{,}$s property $(\beta)$. Then, we give
a necessary and sufficient condition for $T\otimes S$ to be a
$k$-quasi class $\mathcal{A}^{\ast}_{n}$ operator, whenever $T$ and
$S$ are both non-zero operators. Moreover, it is shown that
the Riesz idempotent for a non-zero isolated point $\lambda_{0}$
of a $k$-quasi class $\mathcal{A}^{\ast}_{n}$ operator $T$ say
$\mathcal{R}_i$, is self-adjoint and
$ran(\mathcal{R}_i)=ker(T-\lambda_{0})=ker(T-\lambda_{0})^{*}$.
Finally, as an application in the last section, a necessary and
sufficient condition is given in such a way that the weighted
conditional type operators on $L^{2}(\Sigma)$, defined by
$T_{w,u}(f):= w E(uf)$, belong to $k$-quasi-
$\mathcal{A}^{\ast}_{n}$ class.
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