• Title/Summary/Keyword: $Ces{\grave{a}}ro$ operator

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REMARKS CONCERNING SOME GENERALIZED CESÀRO OPERATORS ON ℓ2

  • Rhaly, Henry Crawford Jr.
    • Journal of the Chungcheong Mathematical Society
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    • v.23 no.3
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    • pp.425-434
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    • 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 $k{\geq}1$.

SUBNORMALITY OF THE WEIGHTED CESÀRO OPERATOR Ch∈l2(h)

  • Hechifa, Abderrazak;Mansour, Abdelouahab
    • Korean Journal of Mathematics
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    • v.25 no.1
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    • pp.117-126
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    • 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.

THE FINE SPECTRA OF THE RHALY OPERATORS ON c.

  • Yildirim, M.
    • East Asian mathematical journal
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    • v.23 no.2
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    • pp.135-149
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    • 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.

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CESÀRO OPERATORS IN THE BERGMAN SPACES WITH EXPONENTIAL WEIGHT ON THE UNIT BALL

  • Cho, Hong Rae;Park, Inyoung
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.705-714
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    • 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 $A^2_{{\alpha},{\beta}(\mathbb{B}_n)$.

POSINORMAL TERRACED MATRICES

  • Rhaly, H. Crawford, Jr.
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.1
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    • pp.117-123
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    • 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.