Concavity of Eigenvalue Sums and the Spectral Shift Function
Concavity of Eigenvalue Sums and the Spectral Shift Function
复制标题
特征值和的凹性与谱位移函数
DOI：
10.1006/jfan.2000.3620
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发表时间：
2000
影响因子：
1.7
通讯作者：
V. Kostrykin
中科院分区：
文献类型：
--
作者：
V. Kostrykin
摘要 众所周知，某些有限 Hermitian 矩阵 V 的负（正）特征值之和相对于 V 是凹（凸）的。利用谱位移函数的理论，我们将此性质推广到具有任意谱的可分离希尔伯特空间上的自伴算子。更准确地说，我们证明了相对于从 −∞ 到 λ（从 λ 到 +∞）的谱参数积分的谱位移函数相对于迹类扰动是凹（凸）的。还考虑了相对迹类扰动的情况。
Abstract It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix V is concave (convex) with respect to V . Using the theory of the spectral shift function we generalize this property to self-adjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift function integrated with respect to the spectral parameter from −∞ to λ (from λ to +∞) is concave (convex) with respect to trace class perturbations. The case of relative trace class perturbations is also considered.