AI 中文总结
该数学研究针对近不变子空间上的加权对偶截断Toeplitz算子，证明其范数公式、紧性等性质，推导代数恒等式及相关校正、位移公式。
AI 中文摘要
设$\boldsymbol{\textit{M}} = h\boldsymbol{\textit{K}}_u$为$H^2$的近$S^*$-不变子空间，其中$\boldsymbol{\textit{K}}_u = H^2 \u2296 uH^2$，$h$为极值乘子。对$\boldsymbol{\textit{\u03c6}} \u2208 L^\u221e(\boldsymbol{\textit{T}})$，研究压缩算子$D_\boldsymbol{\textit{\u03c6}}^\boldsymbol{\textit{M}} = P_{\boldsymbol{\textit{M}}^\u22a5}M_\boldsymbol{\textit{\u03c6}}|_{\boldsymbol{\textit{M}}^\u22a5}$，称之为加权对偶截断Toeplitz算子。当$h \u2261 1$时，该算子退化为经典对偶截断Toeplitz算子。利用Hartmann-Ross投影公式，证明$\boldsymbol{\u2016}D_\boldsymbol{\textit{\u03c6}}^\boldsymbol{\textit{M}}\boldsymbol{\u2016} = \boldsymbol{\u2016}\boldsymbol{\textit{\u03c6}}\boldsymbol{\u2016}_\u221e，刻画其紧性，并证明仅当$h$为内函数时，由$h$诱导的自然乘法映射可精确对应加权理论与经典理论。还建立了其复对称性，通过加权截断Hankel算子得到块矩阵、亏格及半交换子恒等式；作为主要代数结论，证明$D_\boldsymbol{\textit{\u03c6}}^\boldsymbol{\textit{M}} D_\boldsymbol{\textit{\u03c8}}^\boldsymbol{\textit{M}} = 0$等价于$\boldsymbol{\textit{\u03c6}} = 0$或$\boldsymbol{\textit{\u03c8}} = 0$在$\boldsymbol{\textit{T}}$上几乎处处成立。最后，推导了广义对偶位移$D_z^\boldsymbol{\textit{M}}$的秩至多为2的校正公式及有限秩位移恒等式。
英文摘要
Let $\mathcal{M}=h\mathcal{K}_u$ be a nearly $S^*$-invariant subspace of $H^2$, where $\mathcal{K}_u=H^2\ominus uH^2$ and $h$ is the extremal multiplier. For $\vp\in L^\infty(\T)$, we study the compression \[ D_\vp^\mathcal{M} = \restr{P_{\mathcal{M}^\perp}M_\vp}{\mathcal{M}^\perp}, \] called a weighted dual truncated Toeplitz operator. When $h\equiv1$, this reduces to the classical dual truncated Toeplitz operator. Using the Hartmann--Ross projection formula, we prove \[ \|D_\vp^\mathcal{M}\|=\|\vp\|_\infty, \] characterize compactness, and show that the natural multiplication map by $h$ identifies the weighted and classical theories precisely when $h$ is inner. We also establish complex symmetry and obtain block matrix, defect, and semi-commutator identities via weighted truncated Hankel operators. As a main algebraic consequence, we prove \[ D_\vp^\mathcal{M} D_ψ^\mathcal{M}=0 \quad\Longleftrightarrow\quad \vp=0\ \text{or}\ ψ=0 \quad\text{a.e. on }\T. \] Finally, we derive a a rank-at-most-two correction formula and a finite-rank displacement identity for the generalized dual shift $D_z^\mathcal{M}$.
Comments28 pages
