Finite sums of commutators

Spectral characterization of sums of commutators II

For countably generated ideals, $\Jc$, of $B(\Hil)$, geometric stability is necessary for the canonical spectral characterization of sums of $(\Jc,B(\Hil))$--commutators to hold. This answers a question raised by Dykema, Figiel, Weiss and Wodzicki. There are some ideals, $\Jc$, having quasi--nilpotent elements that are not sums of $(\Jc,B(\Hil))$--commutators. Also, every trace on every geometrically stable ideal is a spectral trace.

Spectral characterization of sums of commutators I

Suppose $\Cal J$ is a two-sided quasi-Banach ideal of compact operators on a separable infinite-dimensional Hilbert space $\Cal H$. We show that an operator $T\in\Cal J$ can be expressed as finite linear combination of commutators $[A,B]$ where $A\in\Cal J$ and $B\in\Cal B(\Cal H)$ if and only its eigenvalues $(λ_n)$ (arranged in decreasing order of absolute value, repeated according to algebraic multiplicity and augmented by zeros if necessary) satisfy the condition that the diagonal operator $\diag\{\frac1n(λ_1+\cdots +λ_n)\}$ is a member of $\Cal J.$ This answers (for quasi-Banach ideals) a…

有限冯·诺依曼代数中的一致交换子界

A uniform commutator bound in finite von Neumann algebras

本文证明有限冯·诺依曼代数中零迹元素可表示为交换子，且因子范数乘积受绝对常数控制。

布尔二次型与正切律

The Boolean quadratic forms and tangent law

本文研究布尔交换子与反交换子加权和的极限，证明平移广义正切函数出现在极限定理中，并给出二次型的任意累积量公式，应用于布尔概率论。

Ranks of Commutators and Generalized Semicommutators of Quasihomogeneous Toeplitz Operators

We study the ranks of commutators and generalized semicommutators of Toeplitz operators with quasihomogeneous symbols on both the harmonic Bergman space and the Bergman Space. In particular, when one of quasihomogeneous symbols is the the form of $e^{ikθ}r^{m}$, we first obtain specific sufficient and necessary conditions for commutators and generalized semicommutators to be finite rank. Then we make further efforts to determine the range of each finite rank commutator and generalized semicommutators, and consequently the explicit canonical form and the rank are obtained. Thus, the finite rank…

本质上交换的移位不变子空间上的投影

Finite-rank commutators of projections onto shift-invariant subspaces

利用Halmos双投影定理，通过内函数局部条件完全刻画了Hardy空间上移位不变子空间正交投影的本质可交换性，并应用于截断Toeplitz算子的紧性刻画。