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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>
Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols
Curto, R.E.,Hwang, I.S.,Kang, D.O.,Lee, W.Y. Academic Press ; Elsevier Science B.V. Amsterdam 2014 Advances in mathematics Vol.255 No.-
In this paper we deal with the subnormality and the quasinormality of Toeplitz operators with matrix-valued rational symbols. In particular, in view of Halmos's Problem 5, we focus on the question: Which subnormal Toeplitz operators are normal or analytic? We first prove: Let Φ@?L<SUB>M'n</SUB><SUP>~</SUP> be a matrix-valued rational function having a ''matrix pole'', i.e., there exists α@?D for which kerH<SUB>Φ</SUB>@?(z-α)H<SUB>C^n</SUB><SUP>2</SUP>, where H<SUB>Φ</SUB> denotes the Hankel operator with symbol Φ. If(i)T<SUB>Φ</SUB> is hyponormal; (ii)ker[T<SUB>Φ</SUB><SUP>@?</SUP>,T<SUB>Φ</SUB>] is invariant for T<SUB>Φ</SUB>, then T<SUB>Φ</SUB> is normal. Hence in particular, if T<SUB>Φ</SUB> is subnormal then T<SUB>Φ</SUB> is normal. Next, we show that every pure quasinormal Toeplitz operator with a matrix-valued rational symbol is unitarily equivalent to an analytic Toeplitz operator.
SOLVABILITY OF SYLVESTER OPERATOR EQUATION WITH BOUNDED SUBNORMAL OPERATORS IN HILBERT SPACES
Bekkar, Lourabi Hariz,Mansour, Abdelouahab The Kangwon-Kyungki Mathematical Society 2019 한국수학논문집 Vol.27 No.2
The aim of this paper is to present some necessary and sufficient conditions for existence of solution of Sylvester operator equation involving bounded subnormal operators in a Hilbert space. Our results improve and generalize some results in the literature involving normal 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>
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.
A gap between hyponormality and subnormality for block Toeplitz operators
Hwang, I.S.,Kang, D.-O,Lee, W.Y. Academic Press 2011 Journal of mathematical analysis and applications Vol.382 No.2
This paper concerns a gap between hyponormality and subnormality for block Toeplitz operators. We show that there is no gap between 2-hyponormality and subnormality for a certain class of trigonometric block Toeplitz operators (e.g., its co-analytic outer coefficient is invertible). In addition we consider the extremal cases for the hyponormality of trigonometric block Toeplitz operators: in this case, hyponormality and normality coincide.
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.
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.
Hyponormal operators with rank-two self-commutators
Lee, Sang Hoon,Lee, Woo Young Elsevier 2009 Journal of mathematical analysis and applications Vol.351 No.2
<P><B>Abstract</B></P><P>In this paper it is shown that if T∈L(H) satisfies<ce:list> <ce:list-item><ce:label>(i)</ce:label><ce:para><I>T</I> is a pure hyponormal operator;</ce:para></ce:list-item><ce:list-item><ce:label>(ii)</ce:label><ce:para>[<SUP>T∗</SUP>,T] is of rank two; and</ce:para></ce:list-item><ce:list-item><ce:label>(iii)</ce:label><ce:para>ker[<SUP>T∗</SUP>,T] is invariant for <I>T</I>,</ce:para></ce:list-item></ce:list> then <I>T</I> is either a subnormal operator or the Putinar's matricial model of rank two. More precisely, if T<SUB>|ker[<SUP>T∗</SUP>,T]</SUB> has a rank-one self-commutator then <I>T</I> is subnormal and if instead T<SUB>|ker[<SUP>T∗</SUP>,T]</SUB> has a rank-two self-commutator then <I>T</I> is either a subnormal operator or the <I>k</I>th minimal partially normal extension, <SUP><SUB>Tk</SUB>ˆ(k)</SUP>, of a (k+1)-hyponormal operator <SUB>Tk</SUB> which has a rank-two self-commutator for any k∈<SUB>Z+</SUB>. Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with a rank-two self-commutator is either a subnormal operator or a finite rank perturbation of a <I>k</I>-hyponormal operator for any k∈<SUB>Z+</SUB>.</P>
Subnormality and Weighted Composition Operators on L<sup>2</sup> Spaces
AZIMI, MOHAMMAD REZA Department of Mathematics 2015 Kyungpook mathematical journal Vol.55 No.2
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 $T^{-n}{\Sigma}$.