Raul Curto, Jasang Yoon
We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.
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Let $α:1,(1,\sqrt{x},\sqrt{y})^{\wedge }$ be a weight sequence with Stampfli's subnormal completion and let $W_{α}$ be its associated weighted shift. In this paper we discuss some properties of the region $\mathcal{U}:\mathcal{=}\{(x,y):W_{α}$ is semi-cubically hyponormal$\}$ and describe the shape of the boundary of $\mathcal{U}$. In particular, we improve the results of \cite[Theorem 4.2]{LLB} with properties of $\mathcal{U}$.
In this paper we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space $H^2_{\mathbb{C}^n}$ of the unit circle. Firstly, 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. Secondly, 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 sho…
Let H_0 (resp. H_\infty denote the class of commuting pairs of subnormal operators on Hilbert space (resp. subnormal pairs), and for an integer k>=1 let H_k denote the class of k-hyponormal pairs in H_0. We study the hyponormality and subnormality of powers of pairs in H_k. We first show that if (T_1,T_2) is in H_1, then the pair (T_1^2,T_2) may fail to be in H_1. Conversely, we find a pair (T_1,T_2) in H_0 such that (T_1^2,T_2) is in H_1 but (T_1,T_2) is not. Next, we show that there exists a pair (T_1,T_2) in H_1 such that T_1^mT_2^n is subnormal (all m,n >= 1), but (T_1,T_2) is not in H_\in…
Some relationships with subnormal operators and existence of hyperinvariant subspaces
研究多项式有界算子与次正规算子关系，证明在特定不变子空间和双侧交织条件下存在非平凡超不变子空间。
A closed densely defined operator $ T $ on a Hilbert space $ \mathcal{H} $ is callled $M$-hyponormal if $\mathcal{D}(T) \subset \mathcal{D}(T^{*}) $ and there exists $ M > 0 $ for which $ \parallel(T-zI)^{*}x \parallel \leq M \parallel(T-zI)x \parallel $ for all $ z \in \mathbb{C}$ and for all $ x\in \mathcal{D}(T)$. In this paper, we prove that if bounded linear operator $ A : \mathcal{H} \rightarrow \mathcal{K}$ is such that $ AB^*\subseteq TA $, where $ B $ is a closed subnormal (resp. a closed $ M $-hyponormal) on $\mathcal{H}$, $ T $ is a closed $ M $-hyponormal (resp. a closed subnormal)…
On Invariant subspace for hyponormal operators
针对一篇声称非恒等倍数的亚正规算子都有非平凡超不变子空间的论文，指出其证明有误，说明亚正规算子不变子空间、次正规算子超不变子空间这两个算子理论重要难题仍未解决，厘清该问题具有重要意义。
We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1)-hyponormality for each k>=1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals.
Which hyponormal block Toeplitz operators are either normal or analytic?
本文通过分析矩阵值内函数的最大公因子与弱交换性，将 Curto-Hwang-Lee 关于有理符号块 Toeplitz 算子的结果推广至有界型符号，证明满足标量内函数条件时，平方也次正规的次正规 T_Ψ 必为正规或解析算子。
The subnormality for the sum of commuting subnormal operators does not guarantee the existence of commuting normal extensions.
Weyl's Theorem for pairs of commuting hyponormal operators
研究了满足准三角形性质的交换次正规算子对，证明其Weyl谱等于Taylor谱减去孤立有限重特征值集。
Semi-hyponormality of commuting pairs of Hilbert space operators
本文通过构造正2×2算子矩阵的平方根公式，研究了交换算子对的半亚正则性，分析了3参数家族W_(α,β)(a,x,y)在单位立方体内的亚正则性、亚正则性、半亚正则性和弱亚正则性区域，并指出弱亚正则性与半亚正则性不重合的子区域。
Propagation Phenomena for Operator-Valued Weighted Shifts
本文研究了2-次超norm、二次超norm和三次超norm算子值加权移位的传播现象，证明了二次超norm和三次超norm加权移位的平坦性，并提出局部平坦性概念，证明2-次超norm加权移位满足更强的传播现象，并给出结构分解定理。
Sum of self-commutators of commuting operators
本文研究复Hilbert空间上和次正规$d$元组的结构与谱性质，证明紧情形可分解为正规与拟幂零直和，并建立多变量Putnam不等式，推广了单算子情形。
The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. \ We study LPCS within the class of commuting 2-variable weighted shifts $\mathbf{T} \equiv (T_1,T_2)$ with subnormal components $T_1$ and $T_2$, acting on the Hilbert space $\ell ^2(\mathbb{Z}^2_+)$ with canonical orthonormal basis $\{e_{(k_1,k_2)}\}_{k_1,k_2 \geq 0}$ . \ The \textit{core} of a commuting 2-variable weighted shift $\mathbf{T}$, $c(\mathbf{T})$, is the restriction of $\mathbf{T}$ to the invarian…
We study the spectral pictures of (jointly) hyponormal 2-variable weighted shifts with commuting subnormal components. By contrast with all known results in the theory of subnormal single and 2-variable weighted shifts, we show that the Taylor essential spectrum can be disconnected. We do this by obtaining a simple sufficient condition that guarantees disconnectedness, based on the norms of the horizontal slices of the shift. We also show that for every k >= 1 there exists a k-hyponormal 2-variable weighted shift whose horizontal and vertical slices have 1- or 2-atomic Berger measures, and who…
A linear operator $S$ in a complex Hilbert space $\hh$ for which the set $\dzn{S}$ of its $C^\infty$-vectors is dense in $\hh$ and $\{\|S^n f\|^2\}_{n=0}^\infty$ is a Stieltjes moment sequence for every $f \in \dzn{S}$ is said to generate Stieltjes moment sequences. It is shown that there exists a closed non-hyponormal operator $S$ which generates Stieltjes moment sequences. What is more, $\dzn{S}$ is a core of any power $S^n$ of $S$. This is established with the help of a weighted shift on a directed tree with one branching vertex. The main tool in the construction comes from the theory of in…
Quasinormality of powers of commuting pairs of bounded operators
研究Hilbert空间上交换算子对的准正常性和球面准正常性及其幂，证明了准正常算子的性质，并探讨了子正常算子的平方为准正常时的条件。
In this paper, we study the operator equation $AB=λBA$ for a bounded operator $A,B$ on a complex Hilbert space. We focus on algebraic relations between different operators that include normal, $M$-hyponormal, quasi $*$-paranormal and other classes.
The Cauchy dual subnormality problem in two variables
研究解析左逆交换对环面 2 - 等距算子，满足联合核条件时与可逆权重算子值加权移位酉等价，初始权重交换时柯西对偶联合次正规，还构造反例说明初始权重不交换时柯西对偶非联合次正规。
Weighted shifts on directed trees are a decade old generalisation of classical shift operators in the sequence space $\ell^2$. In this paper we introduce the joint backward extension property (JBEP) for classes of weighted shifts on directed trees. If a class satisfies JBEP, the existence of a common backward extension within the class for a family of weighted shifts on rooted directed trees does not depend on the additional structure of the big tree (of fixed depth). We decide whether several classes of operators have JBEP. For subnormal or power hyponormal weighted shifts the property is sat…
Let $\mathfrak{H}_{0}$ denote the class of commuting pairs of subnormal operators on Hilbert space, and let $\mathcal{TC}:=\{\mathbf{T}\in \mathfrak{% H}_{0}:c(\mathbf{T)}$ is of tensor form$\}$, where $c(\mathbf{T})$ is the core of $\mathbf{T}$. We obtain a concrete necessary and sufficient condition for the subnormality of $\mathbf{T}\equiv (T_{1},T_{2})\in \mathcal{TC}$ in terms of $c(\mathbf{T})$, the marginal measures of $T_{1}$ and $T_{2}$, and the weight $α_{01}$.
A commuting family of subnormal operators need not have a commuting normal extension. We study when a representation of an abelian semigroup can be extended to a normal representation, and show that it suffices to extend the set of generators to commuting normals. We also extended a result due to Athavale to representations on abelian lattice ordered semigroups.
We show that a densely defined closable operator $A$ such that the resolvent set of $A^2$ is not empty is necessarily closed. This result is then extended to the case of a polynomial $p(A)$. We also generalize a recent result by Sebestyén-Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given, one of them being a proof that if $T$ is a quasinormal (unbounded) operator such that $T^n$ is normal for some $n\geq2$, then $T$ is normal. By a recent result by Pietrzycki-Stochel, we infer that a closed subnormal operator such that $T^n$ is normal,…
Dilation of normal operators associated with an annulus
本文研究圆环谱集约束下正规压缩算子的同时酉膨胀问题，通过求解多圆环上的Dirichlet问题构造膨胀，并推广到双重交换的次正规及2×2标量压缩情形。
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