• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.345-353
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• 2015

Subnormality of bounded weighted composition operators on $L^2({\Sigma})$ of the form $Wf=uf{\circ}T$, where T is a nonsingular measurable transformation on the underlying space X of a ${\sigma}$-finite measure space (X, ${\Sigma}$, ${\mu}$) and u is a weight function on X; is studied. The standard moment sequence characterizations of subnormality of weighted composition operators are given. It is shown that weighted composition operators are subnormal if and only if $\{J_n(x)\}^{+{\infty}}_{n=0}$ is a moment sequence for almost every $x{{\in}}X$, where $J_n=h_nE_n({\mid}u{\mid}^2){\circ}T^{-n}$, $h_n=d{\mu}{\circ}T^{-n}/d{\mu}$ and $E_n$ is the conditional expectation operator with respect to $T^{-n}{\Sigma}$.

• 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.

• Kyungpook Mathematical Journal
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• v.49 no.4
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• pp.683-689
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• 2009

We investigate the gaps among classes of weakly hyponormal composition operators induced by Embry characterization for the subnormality. The relationship between subnormality and weak hyponormality will be discussed in a version of composition operator induced by a non-singular measurable transformation.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.821-831
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• 2016

A case study is added to our recent work on Quillen phenomenon. Pointwise positivity of polynomials on generalized lemniscates of the complex plane is related to sums of hermitian squares of rational functions, and via operator quantization, to essential subnormality.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.2
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• pp.261-267
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• 2015

We study backward extensions of 2-variable weighted shifts with finite atomic Berger measure. We provide a necessary and sufficient condition for the subnormality of such extensions. As an application, we give a simple counterexample for the Curto-Muhly-Xia conjecture [10].

• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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• pp.347-352
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• 2000

In this note we answer to a question of Curto: Non-first two equal weights in the weighted shift force subnormality in the presence of quadratic hyponormality. Also it is shown that every hyponormal weighted shift with two equal weights cannot be polynomially hyponormal without being flat.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.2
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• pp.287-292
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• 2008

We introduce jointly weak subnormal operators. It is shown that if $T=(T_1,T_2)$ is subnormal then T is weakly subnormal and if f $T=(T_1,T_2)$ is weakly subnormal then T is hyponormal. We discuss the flatness of weak subnormal operators.

• Journal of the Korean Mathematical Society
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• v.56 no.6
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• pp.1489-1502
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• 2019

This paper describes when a partial isometry satisfies several weak normal properties. Topics treated include quasi-normality, subnormality, hyponormality, p-hyponormality (p > 0), w-hyponormality, paranormality, normaloidity, spectraloidity, the von Neumann property and Weyl's theorem.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.943-957
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• 2016

We introduce a 2-variable weighted shift, denoted by $S_2$(a, b, c, d), which arises naturally from analytic function space theory. We investigate when it is subnormal, and compute the Berger measure of it when it is subnormal. And we apply the results to investigate the relationship among 2-variable subnormal, hyponormal and 2-hyponormal weighted shifts.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.633-646
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• 2017

Recently, Curto, Lee and Yoon considered the properties (such as, hyponormality, subnormality, and flatness, etc.) for 2-variable weighted shifts and constructed several families of commuting pairs of subnormal operators such that each family can be used to answer a conjecture of Curto, Muhly and Xia negatively. In this paper, we consider the weak and quadratic hyponormality of 2-variable weighted shifts ($W_1,W_2$). In addition, we detect the weak and quadratic hyponormality with some interesting 2-variable weighted shifts.