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SUBNORMALITY OF S<sub>2</sub>(a, b, c, d) AND ITS BERGER MEASURE
Duan, Yongjiang,Ni, Jiaqi Korean Mathematical Society 2016 대한수학회보 Vol.53 No.3
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.
WEAKLY SUBNORMAL WEIGHTED SHIFTS NEED NOT BE 2-HYPONORMAL
Lee, Jun Ik The Kangwon-Kyungki Mathematical Society 2015 한국수학논문집 Vol.23 No.1
In this paper we give an example which is a weakly subnormal weighted shift but not 2-hyponormal. Also, we show that every partially normal extension of an isometry T needs not be 2-hyponormal even though p.n.e.(T) is weakly subnormal.
Subnormality of $S_{2}(a,b,c,d)$ and its Berger measure
Yongjiang Duan,Jiaqi Ni 대한수학회 2016 대한수학회보 Vol.53 No.3
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.
Weakly Hyponormal Composition Operators and Embry Condition
Lee, Mi-Ryeong,Park, Jung-Woi Department of Mathematics 2009 Kyungpook mathematical journal Vol.49 No.4
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.
Hermitian algebra on generalized lemniscates
Mihai Putinar 대한수학회 2016 대한수학회보 Vol.53 No.3
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.
HERMITIAN ALGEBRA ON GENERALIZED LEMNISCATES
Putinar, Mihai Korean Mathematical Society 2016 대한수학회보 Vol.53 No.3
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.
A new approach to the 2-variable Subnormal Completion Problem
Curto, R.E.,Lee, S.H.,Yoon, J. Academic Press 2010 Journal of mathematical analysis and applications Vol.370 No.1
We study the Subnormal Completion Problem (SCP) for 2-variable weighted shifts. We use tools and techniques from the theory of truncated moment problems to give a general strategy to solve SCP. We then show that when all quadratic moments are known (equivalently, when the initial segment of weights consists of five independent data points), the natural necessary conditions for the existence of a subnormal completion are also sufficient. To calculate explicitly the associated Berger measure, we compute the algebraic variety of the associated truncated moment problem; it turns out that this algebraic variety is precisely the support of the Berger measure of the subnormal completion.
Hyponormality and subnormality of block Toeplitz operators
Curto, Raú,l E.,Hwang, In Sung,Lee, Woo Young Elsevier 2012 Advances in mathematics Vol.230 No.4
<P><B>Abstract</B></P><P>In this paper, we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space H<SUP>Cn</SUP>2 of the unit circle.</P><P>First, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator.</P><P>Second, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos’s Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We show that if Φ is a matrix-valued rational function whose co-analytic part has a coprime factorization then every hyponormal Toeplitz operator <SUB>TΦ</SUB> whose square is also hyponormal must be either normal or analytic.</P><P>Third, using the subnormal theory of block Toeplitz operators, we give an answer to the following “Toeplitz completion” problem: find the unspecified Toeplitz entries of the partial block Toeplitz matrix A≔[<SUP>U∗</SUP>??<SUP>U∗</SUP>] so that A becomes subnormal, where U is the unilateral shift on <SUP>H2</SUP>.</P>
JOINT WEAK SUBNORMALITY OF OPERATORS
이전익 충청수학회 2008 충청수학회지 Vol.21 No.2
Abstract. We introduce jointly weak subnormal operators. It is shown that if T = (T1, T2) is subnormal then T is weakly subnormal and if f T = (T1, T2) is weakly subnormal then T is hyponormal. We discuss the flatness of weak subnormal operators.
A new characterization of subnormality for a class of 2-variable weighted shifts with 1-atomic core
Kim, Jaewoong,Yoon, Jasang Elsevier 2018 Linear algebra and its applications Vol.538 No.-
<P><B>Abstract</B></P> <P>Given a pair T ≡ ( <SUB> T 1 </SUB> , <SUB> T 2 </SUB> ) of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) calls for necessary and sufficient conditions for the existence of a commuting pair N ≡ ( <SUB> N 1 </SUB> , <SUB> N 2 </SUB> ) of normal extensions of <SUB> T 1 </SUB> and <SUB> T 2 </SUB> . This is an old problem in operator theory. The aim of this paper is to study LPCS. There are three well-known subnormal characterizations for operators: the Berger Theorem, the Bram–Halmos characterization, and Franks' result. In our paper, we study a new subnormal characterization which is related to these three well-known ones for a class of 2-variable weighted shifts. Thus, we can provide a large nontrivial class of 2-variable weighted shifts in which <I>k</I>-hyponormal (some k ≥ 1 ) and subnormal are equal and the class is invariant under the action ( h , ℓ ) ↦ <SUP> T ( h , ℓ ) </SUP> : = ( T 1 h , T 2 ℓ ) ( h , ℓ ≥ 1 ).</P>