上三角闭算子矩阵的本质谱和Weyl谱性质

青梅, 黄俊杰, 阿拉坦仓

数学学报 ›› 2021, Vol. 64 ›› Issue (5) : 761-772.

PDF(421 KB)
PDF(421 KB)
数学学报 ›› 2021, Vol. 64 ›› Issue (5) : 761-772. DOI: 10.12386/A2021sxxb0064
论文

上三角闭算子矩阵的本质谱和Weyl谱性质

    青梅, 黄俊杰, 阿拉坦仓
作者信息 +

The Essential and Weyl Spectral Properties of Closed Upper Triangular Operator Matrices

    Mei QING, Jun Jie HUANG, Alatancang
Author information +
文章历史 +

摘要

本文主要研究了上三角闭算子矩阵TB=(A0BD):DA)⊕DD)⊂HKHK的本质谱和Weyl谱的性质,其中HK都是无穷维复可分的Hilbert空间.首先,对给定的稠定闭算子AD,得到了存在可闭算子B使得TB是半Weyl和半Fredholm算子的充分必要条件,其中B满足DB)⊃DD).进一步,刻画了TB的固有本质谱和Weyl谱集合.最后,给出了等式σ*TB)=σ*A)∪σ*D)成立的充分必要条件,其中σ*TB)包含TB的本质谱和Weyl谱.

Abstract

In this paper, the properties of essential and Weyl spectrum of the upper triangular closed operator matrix TB=(A0 BD):D(A)⊕D(D) ⊂ HKHK are studied, where H and K are complex infinite-dimensional separable Hilbert spaces. Fisrtly, for given closed operator A and D with dense domains, we obtain the necessary and sufficient conditions for TB to be semi-Weyl and semi-Fredholm for some closable operator B with D(B) ⊃ D(D). Moreover, the intersections of the essential and Weyl spectrum of TB are determined. Finally, the necessary and sufficient conditions for σ*(TB)=σ*(A) ∪ σ*(D) are described, where σ*(TB) includes the essential and Weyl spectrum of TB.

关键词

闭算子 / 上三角算子矩阵 / 本质谱 / Weyl 谱 / 补问题

Key words

closed operator / upper triangular operator matrix / essential spectrum / Weyl spectrum / completion

引用本文

导出引用
青梅, 黄俊杰, 阿拉坦仓. 上三角闭算子矩阵的本质谱和Weyl谱性质. 数学学报, 2021, 64(5): 761-772 https://doi.org/10.12386/A2021sxxb0064
Mei QING, Jun Jie HUANG, Alatancang. The Essential and Weyl Spectral Properties of Closed Upper Triangular Operator Matrices. Acta Mathematica Sinica, Chinese Series, 2021, 64(5): 761-772 https://doi.org/10.12386/A2021sxxb0064

参考文献

[1] Bai Q., Huang J., Chen A., Essential, Weyl and Browder spectra of unbounded upper triangular operator matrices, Linear and Multilinear Algebra, 2016, 64:1583-1594.
[2] Bai Q., Huang J., Chen A., The property of essential, Weyl and Browder spectrum of upper triangular operator matrices, Filomat, Accepted.
[3] Bai Q., Huang J., Chen A., Weyl type theorems of 2×2 upper triangular operator matrices, J. Math. Anal. Appl., 2016, 434:1065-1076.
[4] Ben-Israel A., Greville T. N. E., Generalized Inverse:Theory and Applications, Second Edition, Springer, New York, 2003.
[5] Cao X. H., Guo M. Z., Meng B., Semi-Fredholm spectrum and Weyl's theory for operator matrices, Acta Math. Sinica, Engl. Ser., 2006, 22:167-178.
[6] Cao X. H., Meng B., Essential approximate point spectra and Weyl's theorem for upper triangular operator matrices, J. Math. Anal. Appl., 2005, 304:759-771.
[7] Cvetković-Ilić D. S., Completions of upper-triangular matrices to left-Fredholm operators with non-positive index, J. Oper. Theory, 2016, 76:101-114.
[8] Djordjević D. S., Perturbations of spectra of operator matrices, J. Oper. Theory, 2002, 48:467-486.
[9] Du H., Pan J., Perturbation of spectrum of 2×2 operator matrices, Proc. Amer. Math. Soc., 1994, 121:761-766.
[10] Gohberg I., Goldberg S., Kaashoek M. A., Classes of Linear Operators, Vol. 1, Birkhäuser-Verlag, Basel, 1990.
[11] Hai G. J., Chen A., On the right (left) invertible completions for operator matrices, Integr. Equ. Oper. Theory, 2010, 67:79-93.
[12] Han J. K., Lee H. Y., Lee W. Y., Invertible completions of 2×2 upper triangular operator matrices, Proc. Amer. Math. Soc., 1999, 128:119-123.
[13] Li Y., Du H., The intersection of essential approximate point spectra of operator matrices, J. Math. Anal. Appl., 2006, 323:1171-1183.
[14] Li Y., Sun X., Du H., The intersection of left and right essential spectra of 2×2 upper triangular operator matrices, Bull. Lond. Math. Soc., 2004, 36:811-819.
[15] Schechter M., Principles of Functional Analysis (Second Edition), American Mathematical Society, Providence, Rhode Island, 2001.
[16] Zhang S., Wu Z., Characterizations of perturbations of spectra of 2×2 upper triangular operator matrices, J. Math. Anal. Appl., 2012, 392:103-110.

基金

国家自然科学基金(11961022,11761029,11961052);内蒙古自然科学基金(2018BS01005);呼和浩特民族学院博士基金(HMBS1901);呼和浩特民族学院科研创新团队建设计划内蒙古自治区本级引进高层次人才项目
PDF(421 KB)

Accesses

Citation

Detail

段落导航
相关文章

/