• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.3
• /
• pp.425-434
• /
• 2010

Here we see that the $p-Ces{\grave{a}}ro$ operators, the generalized $Ces{\grave{a}}ro$ operators of order one, the discrete generalized $Ces{\grave{a}}ro$ operators, and their adjoints are all posinormal operators on ${\ell}^2$, but many of these operators are not dominant, not normaloid, and not spectraloid. The question of dominance for $C_k$, the generalized $Ces{\grave{a}}ro$ operators of order one, remains unsettled when ${\frac{1}{2}}{\leq}k<1$, and that points to some general questions regarding terraced matrices. Sufficient conditions are given for a terraced matrix to be normaloid. Necessary conditions are given for terraced matrices to be dominant, spectraloid, and normaloid. A very brief new proof is given of the well-known result that $C_k$ is hyponormal when $k{\geq}1$.

• Korean Journal of Mathematics
• /
• v.25 no.1
• /
• pp.117-126
• /
• 2017

The subnormality of some classes of operators is a very interesting property. In this paper, we prove that the weighted $Ces{\grave{a}}ro$ operator $C_h{\in}{\ell}^2(h)$ is subnormal and we described completely the set of the extended eigenvalues for the weighted $Ces{\grave{a}}ro$ operator, some other important results are also given.

• Communications of the Korean Mathematical Society
• /
• v.24 no.1
• /
• pp.57-66
• /
• 2009

In this paper, we characterize the boundedness and compactness of the extended $Ces{\grave{a}}ro$ operators from general function spaces F(p, q, s) to Bloch-type spaces ${\mathcal{B}}_{\mu}$ where $\mu$ is normal function on [0,1).

• East Asian mathematical journal
• /
• v.23 no.2
• /
• pp.135-149
• /
• 2007

In 1975, Wenger [4] determined the fine spectra of $Ces{\grave{a}}ro$ operator $C_1$ on c, the space of convergent sequences. In [7], the spectrum of the Rhaly operators on $c_0$ and c, under the assumption that ${lim}\limits_{n{\rightarrow}{\infty}}(n+1)a_n\;=\;L\;{\neq}\;0$, has been determined. In this paper the author determine the fine spectra of the Rhaly matrix $R_a$ as an operator on the space c, with the same assumption.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.705-714
• /
• 2017

Let $A^2_{{\alpha},{\beta}}(\mathbb{B}_n)$ denote the space of holomorphic functions that are $L^2$ with respect to a weight of form ${\omega}_{{\alpha},{\beta}}(z)=(1-{\mid}z{\mid}^{\alpha}e^{-{\frac{\beta}{1-{\mid}z{\mid}}}}$, where ${\alpha}{\in}\mathbb{R}$ and ${\beta}$ > 0 on the unit ball $\mathbb{B}_n$. We obtain some results for the boundedness and compactness of $Ces{\grave{a}}ro$ operator on $A^2_{{\alpha},{\beta}(\mathbb{B}_n)$.

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.1
• /
• pp.117-123
• /
• 2009

This paper is a study of some properties of a collection of bounded linear operators resulting from terraced matrices M acting through multiplication on ${\ell}^2$; the term terraced matrix refers to a lower triangular infinite matrix with constant row segments. Sufficient conditions are found for M to be posinormal, meaning that $MM^*=M^*PM$ for some positive operator P on ${\ell}^2$; these conditions lead to new sufficient conditions for the hyponormality of M. Sufficient conditions are also found for the adjoint $M^*$ to be posinormal, and it is observed that, unless M is essentially trivial, $M^*$ cannot be hyponormal. A few examples are considered that exhibit special behavior.