M. C. Camara, W. T. Ross
For an inner function u we discuss the dual operator for the well-known compressed shift. We establish conditions for two dual compressed shifts to be unitarily equivalent/similar and we describe the invariant subspace structure for the dual.
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Near Invariance of The Dual Compressed Shift
该研究提出模型空间正交补空间的近双压缩移位不变子空间概念，借助Hitt算法获得其结构。
Spectra and invariant subspaces of compressed shifts on nearly invariant subspaces
本文研究压缩位移在近不变子空间上的谱性质和不变子空间结构，通过单位等价性与变换理论揭示S*不变性放松对谱结构的影响，拓展了经典模型空间理论的应用范围。
A geometric approach to the compressed shift operator on the Hardy space over the bidisk
本文通过几何方法研究双圆盘上Hardy空间压缩位移算子的谱性质及广义Cowen-Douglas算子的特征化。
Given a purely contractive matrix-valued analytic function $Θ$ on the unit disc $\bm{D}$, we study the $\mc{U} (n)$-parameter family of unitary perturbations of the operator $Z_Θ$ of multiplication by $z$ in the Hilbert space $L^2_Θ$ of $n-$component vector-valued functions on the unit circle $\bm{T}$ which are square integrable with respect to the matrix-valued measure $\Om_Θ$ determined uniquely by $Θ$ and the matrix-valued Herglotz representation theorem. In the case where $Θ$ is an extreme point of the unit ball of bounded $\bm{M}_n$-valued functions we verify that the $\mc{U} (n)$-paramet…
Characterizations of Matrix Valued Asymmetric Truncated Toeplitz Operators
本文利用压缩移位、修正压缩移位及移位不变性，刻画了矩阵值非对称截断Toeplitz算子，该算子是作用于两个可能不同模型空间的乘法算子的压缩。
Two-Variable Compressions of Shifts, Toeplitz Operators, and Numerical Ranges
本文研究双变量压缩的移位、托普利兹算子和数值范围，证明有理内函数由托普利兹算子符号决定，但数值范围不一定决定其唯一性。
In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Cesàro operator.
In this paper we develop a general theory of compressed sensing for analog signals, in close similarity to prior results for vectors in finite dimensional spaces that are sparse in a given orthonormal basis. The signals are modeled by functions in a reproducing kernel Hilbert space. Sparsity is defined as the minimal number of terms in expansions based on the kernel functions. Minimizing this number is under certain conditions equivalent to minimizing an atomic norm, the pre-dual of the supremum norm for functions in the Hilbert space. The norm minimizer is shown to exist based on a compactnes…
In this paper we consider compressions of $k^{th}$--order slant Toeplitz operators to the backward shift invariant subspaces of the classical Hardy space $H^2$. In particular, we characterize these operators using compressed shifts and finite rank operators of special kind.
A complete solution to the Cauchy dual subnormality problem for torally expansive toral $3$-isometric weighted $2$-shifts
本文通过求解低双次数二变量多项式对应的一对Hausdorff矩问题，完整解决了环面扩张的环面3-等距加权2-移位的柯西对偶次正规性问题。
We introduce an extension of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in $L^2(\R^k)$. We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restriction…
Dual Truncated Hankel Operators: Characterizations and Properties
本研究引入对偶截断汉克尔算子（DTHO）概念，利用对偶压缩移位算子给出其算子方程特征刻画，提供相关算子方程的全面解法，并建立其适用于一般及特定条件符号的基本算子理论性质。
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。
The property of being shift invariant and being reflexive or transitive in the case of the space of (asymmetric) truncated Toeplitz operators, and the space of (asymmetric) dual truncated operators is investigated. Most of the results obtained are new even for the symmetric case. A characterization of asymmetric dual truncated Toeplitz operators is also given.
We consider two-variable model spaces associated to rational inner functions $Θ$ on the bidisk, which always possess canonical $z_2$-invariant subspaces $\mathcal{S}_2.$ A particularly interesting compression of the shift is the compression of multiplication by $z_1$ to $\mathcal{S}_2$, namely $ S^1_Θ:= P_{\mathcal{S}_2} M_{z_1} |_{\mathcal{S}_2}$. We show that these compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators with well-behaved symbols and characterize their numerical ranges and radii. We later specialize to particularly simple rational inner functions and st…
The numerical range of a bounded, linear operator on a Hilbert space is a set in $\mathbb{C}$ that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space $H^2(\mathbb{D})$, and compressions of the shift operator to model spaces, i.e.~spaces of the form $H^2 \ominus θH^2$ where $θ$ is inner. For these compressions of the shift operator, we provide a survey of results on the…
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