交换算子自交换子之和
Sum of self-commutators of commuting operators
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中文总结 AI 辅助
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。
中文摘要 AI 辅助
众所周知，复Hilbert空间上谱具有平面Lebesgue测度零的每个次正规算子都是正规的。特别地，复Hilbert空间上的每个紧次正规算子都是正规的。本文研究复Hilbert空间$\mathcal H$上所谓的和次正规$d$元组（即满足$\sum_{j=1}^d [T^*_j, T_j] \geqslant 0$的交换$d$元组${\bf T}=(T_1, \ldots, T_d)$）的类似结构和谱现象。我们证明每个紧算子的和次正规$d$元组可分解为一个正规$d$元组与一个拟幂零和次正规$d$元组的直和。作为推论，有限维Hilbert空间上的每个和次正规$d$元组都是正规的。与单算子情形相反，和次正规$d$元组不必是正规oid的。然而，我们为具有交换虚部的和次正规$d$元组建立了Putnam不等式的多变量类比。作为推论，我们证明当$s(\sigma({\bf T}))$的平面Lebesgue测度为零时${\bf T}$是正规的，其中$\sigma({\bf T})$表示${\bf T}$的Taylor谱，$s$是系数交替为$1$和$i$的复线性多项式。
英文摘要
It is well known that every hyponormal operator on a complex Hilbert space whose spectrum has planar Lebesgue measure zero is normal. In particular, every compact hyponormal operator on a complex Hilbert space is normal. In this paper, we investigate analogous structural and spectral phenomena for so-called sum-hyponormal $d$-tuples on a complex Hilbert space $\mathcal H$, namely, commuting $d$-tuples ${\bf T}=(T_1, \ldots, T_d)$ satisfying $\sum_{j=1}^d [T^*_j, T_j] \geqslant 0$. We show that every sum-hyponormal $d$-tuple of compact operators decomposes as the direct sum of a normal $d$-tuple and a quasinilpotent sum-hyponormal $d$-tuple. As a consequence, every sum-hyponormal $d$-tuple on a finite-dimensional Hilbert space is normal. In contrast to the single-operator case, a sum-hyponormal $d$-tuple need not be normaloid. Nevertheless, we establish a multivariable analogue of Putnam's inequality for sum-hyponormal $d$-tuples with commuting imaginary parts. As a consequence, we prove that ${\bf T}$ is normal whenever the planar Lebesgue measure of $s(σ({\bf T}))$ is zero, where $σ({\bf T})$ denotes the Taylor spectrum of ${\bf T}$ and ${s}$ is the complex-linear polynomial with alternating coefficients $1$ and $i$.
发表机构
- IIT Kanpur（印度理工学院坎普尔分校）
- Aliah University（阿利亚大学）
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