若干幂等投影的若干应用
Some applications of the matched projections of idempotents
AI总结：
本文探讨了幂等投影的匹配投影在距离计算、C*代数生成及数值范围描述中的应用，通过构建新的块矩阵形式，提供了新的公式和完整刻画。
摘要：
对于每个在Hilbert空间上的幂等投影Q，匹配投影m(Q)是一个已知的概念。本文探讨了匹配投影的若干应用。第一应用处理了从H上的投影到给定幂等Q的距离，利用m(Q)建立了这些距离的完整刻画，涵盖最小、最大和中间值。第二应用关注由单个非投影幂等Q生成的C*代数C*{Q}。新的4×4块矩阵表示法为Q提供了新公式，从而通过显式4×4块矩阵完全刻画了C*{Q}中的所有元素。此外，对于每个r>1，引入了一族通用的r-幂等。这些幂等具有与已知投影对不同的通用性质。提供了此类通用r-幂等的必要和充分条件。第三应用给出了数值范围的新刻画。建立了算子版本的椭圆范围定理。利用一般的非投影幂等Q及其匹配投影m(Q)，构造了一个非二次算子，并详细描述了其数值范围。此外，还引入了一个数值范围闭包不是椭圆盘，而自身既不闭合也不开放的算子。
英文摘要：
For every idempotent $Q$ on a Hilbert space $H$, the matched projection $m(Q)$ is a well-established concept. This paper explores several applications of the matched projections. The first application addresses the distances from projections on $H$ to a given idempotent $Q$. Using $m(Q)$, a complete characterization of these distances is established, covering the minimum, maximum, and intermediate values. The second application focuses on the $C^*$-algebra $C^*\{Q\}$ generated by a single non-projection idempotent $Q$. A new $4\times 4$ block matrix representation of $Q$, induced by $m(Q)$, yields novel formulas for $Q$, leading to a full characterization of all elements in $C^*\{Q\}$ via explicit $4\times 4$ block matrices. Furthermore, for each $r>1$, a family of universal $r$-idempotents is introduced. These idempotents possess a universal property distinct from known properties of projection pairs. Some necessary and sufficient conditions are provided for such universal $r$-idempotents. The third application presents new characterizations of the numerical ranges. An operator version of the elliptical range theorem is established. Using a general non-projection idempotent $Q$ and its matched projection $m(Q)$, a non-quadratic operator is constructed, and its numerical range is described in detail. Additionally, another operator is introduced whose numerical range closure is not an elliptical disk, and the numerical range itself is neither closed nor open.
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