论文标题
一类无限维操作员的特征值和符号特征值之间的订单关系
An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite-Dimensional Operators
论文作者
论文摘要
在本文中，我们获得了``无限尺寸符号光谱理论''的方向。我们证明，特殊类无限尺寸操作员的特征值和符号型特征值之间的不平等是不平等的。让$ t $ be Oner the Onerator，使得$ t -αi$ $ t -αi$。 $ \ {{{λ_j^r}^\ downarrow（t）\} $，$ t $的特征值一组严格撒在$α$的右侧，以减少顺序排列，让$ \ \ \ \ \ {{{λ_j^l} $α$以增加的顺序排列。 $ t $的符号特征值严格地躺在$α$的左侧，分别以增加的顺序排列（这样的安排是可以证明的，因为这表明Symplectic EigenValues唯一可能的积累点是$ a $α$）。 {λ_j^r}^\ downarrow（t），\ quad j = 1,2，\ cdots，s_r $$和$$和$$ {λ_j^l}^\ uparrow（t）\ leq {d_j^l}^l}^l}^l}^\ uparrow（t）表示严格在$α$的左右$ t $的符号特征值，这概括了Bhatia和Jain获得的有限维度（J. Math。Phys。56，112201（2015））。 （$（\ Mathcal {a} \ Mathcal {n}）_+$ operators）作为我们考虑的一组运算符的特殊情况。
In this article, we obtain some results in the direction of ``infinite dimensional symplectic spectral theory". We prove an inequality between the eigenvalues and symplectic eigenvalues of a special class of infinite dimensional operators. Let $T$ be an operator such that $T - αI$ is compact for some $α> 0$. Denote by $\{{λ_j^R}^\downarrow(T)\}$, the set of eigenvalues of $T$ lying strictly to the right side of $α$ arranged in the decreasing order and let $\{{λ_j^L}^\uparrow(T)\}$ denote the set of eigenvalues of $T$ lying strictly to the left side of $α$ arranged in the increasing order. Furthermore, let $\{{d_j^R}^\downarrow(T)\}$ denote the symplectic eigenvalues of $T$ lying strictly to the right of $α$ arranged in decreasing order and $\{{d_j^L}^\uparrow(T)\}$ denote the set of symplectic eigenvalues of $T$ lying strictly to the left of $α$ arranged in increasing order, respectively (such an arrangement is possible as it will be shown that the only possible accumulation point for the symplectic eigenvalues is $α$). Then we show that $${d_j^R}^\downarrow(T) \leq {λ_j^R}^\downarrow(T), \quad j = 1,2, \cdots, s_r$$ and $${λ_j^L}^\uparrow(T) \leq {d_j^L}^\uparrow(T), \quad j = 1,2, \cdots, s_l,$$ where $s_r$ and $s_l$ denote the number of symplectic eigenvalues of $T$ strictly to the right and left of $α$, respectively. This generalizes a finite dimensional result obtained by Bhatia and Jain (J. Math. Phys. 56, 112201 (2015)). The class of Gaussian Covariance Operators (GCO) and positive Absolutely Norm attaining Operators ($(\mathcal{A}\mathcal{N})_+$ operators) appear as special cases of the set of operators we consider.
