• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.721-727
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• 2008

Let $W_{\alpha}$ be a weighted shift with positive weight sequence ${\alpha}=\{\alpha_i\}_{i=0}^{\infty}$. The semi-cubical hyponormality of $W_{\alpha}$ is introduced and some flatness properties of $W_{\alpha}$ are discussed in this note. In particular, it is proved that if ${\alpha}_n={\alpha}_{n+1}$ for some $n{\geq}1$, ${{\alpha}_{n+k}}={\alpha}_n$ for all $k{\geq}1$.

• Bulletin of the Korean Mathematical Society
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• v.61 no.4
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• pp.1019-1031
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• 2024

In this paper we consider the sufficient condition for hyponormal Toeplitz operators T_𝛗 with non-harmonic symbols $${\varphi}(z)=\sum_{\ell=1}^{k}{\alpha}_{\ell}z^{{m_{\ell}}{\bar{z}}n_{\ell}}$$ with m_ℓ-n_ℓ = δ > 0 for all 1 ≤ ℓ ≤ k, and α_ℓ ∈ ℂ on the Bergman spaces. In particular, we will observe the characteristics of the hyponormality of the Toeplitz operators T_𝛗 according to the positional relationship of the coefficients α_ℓ's.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.579-590
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• 2020

The flatness property of a unilateral weighted shifts is important to study the gaps between subnormality and hyponormality. In this paper, we first summerize the results on the flatness for some special kinds of a weighted shifts. And then, we consider the flatness property for a local-cubically hyponormal weighted shifts, which was introduced in [2]. Let α : ${\sqrt{\frac{2}{3}}}$, ${\sqrt{\frac{2}{3}}}$, $\{{\sqrt{\frac{n+1}{n+2}}}\}^{\infty}_{n=2}$ and let W_α be the associated weighted shift. We prove that W_α is a local-cubically hyponormal weighted shift W_α of order ${\theta}={\frac{\pi}{4}}$ by numerical calculation.

• 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.

• Korean Journal of Mathematics
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• v.31 no.1
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• pp.109-120
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• 2023

The Berezin symbol à of an operator A on the reproducing kernel Hilbert space 𝓗 (Ω) over some set Ω with the reproducing kernel k_λ is defined by $${\tilde{A}}(\lambda)=\,\;{\lambda}{\in}{\Omega}$$. The Berezin number of an operator A is defined by $$ber(A):=\sup_{{\lambda}{\in}{\Omega}}{\mid}{\tilde{A}}({\lambda}){\mid}$$. We study some problems of operator theory by using this bounded function Ã, including estimates for Berezin numbers of some operators, including truncated Toeplitz operators. We also prove an operator analog of some Young inequality and use it in proving of some inequalities for Berezin number of operators including the inequality ber (AB) ≤ ber (A) ber (B), for some operators A and B on 𝓗 (Ω). Moreover, we give in terms of the Berezin number a necessary condition for hyponormality of some operators.