Spectra of Upper-Triangular Operator Matrices

doi:10.12386/A2011sxxb0006

Abstract

Abstract:

Let X and Y be Banach spaces, A∈B(X),B∈B(Y), C∈B(Y,X), M_C=(^A₀ ^C _B) be the operator matrix acting on the Banach space X⊕Y. In this paper, we give out 20 kind spectra structure of M_C, decide 18 kind spectra filling-in-hole properties of M_C and present some interesting examples of these problems.

Key words: Banach spaces, upper-triangular operator matrices, spectra

鎽樿锛�

璁�X,Y鏄疊anach绌洪棿, 瀵�A∈B(X),B∈B(Y), C∈B(Y,X),浠�M_C璁�X⊕Y涓婄殑绠楀瓙 (^A₀ ^C _B). 鏈枃缁欏嚭浜嗙畻瀛�M_C鐨�20绉嶈氨鐨勭粨鏋勮〃绀�, 18绉嶈氨鐨勫～娲炴�ц川浠ュ強鍏充簬杩欎簺闂鐨勬湁瓒ｄ緥瀛�.

鍏抽敭璇�: Banach绌洪棿, 涓婁笁瑙掔畻瀛愮煩闃�, 璋�

CLC Number:

O177.2

Shi Fang ZHANG, Huai Jie ZHONG, Jun De WU. Spectra of Upper-Triangular Operator Matrices[J]. Acta Mathematica Sinica, Chinese Series, 2011, 54(1): 41-60.

寮犱笘鑺�, 閽熸��鏉�, 姝︿繆寰�. 涓婁笁瑙掔畻瀛愮煩闃电殑璋盵J]. 鏁板瀛︽姤, 2011, 54(1): 41-60.

References

[1] Djordjevi'c D. S., Perturbations of spectra of operator matrices, J. Operator Theory, 2002, 48: 467--486.

[2] Zhang S. F., Zhong H. J., Jiang Q. F., Drazin spectrum of operator matrices on the Banach space, Linear Algebra Appl., 2008, 429: 2067--2075.

[3] Barraa M., Boumazgour M., On the perturbations of spectra of upper triangular operator matrices, J. Math. Anal. Appl., 2008, 347: 315--322.

[4] Zerouali E. H., Zguitti H., Perturbation of spectra of operator matrices and local spectral theory, J. Math. Anal. Appl., 2006, 324: 992--1005.

[5] Djordjevi'c S. V., Zguitti H., Essential point spectra of operator matrices through local spectral theory, J. Math. Anal. Appl., 2008, 338: 285--291.

[6] 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.

[7] Lee W. Y., Weyl spectra of operator matrices, Proc. Amer. Math. Soc., 2000, 129: 131--138.

[8] Zhang Y. N., Zhong H. J., Lin L. Q., Browder spectra and essential spectra of operator matrices, Acta Math. Sinica, 2008, 24: 947--954.

[9] Hwang I. S., Lee W. Y., The boundedness below of 2×2 upper triangular operator matrices, Integr. Equ. Oper. Theory, 2001, 39: 267--276.

[10] Cao X. H., Guo M. Z., Meng B., Semi-Fredholm spectrum and Weyl's theory for operator matrices, Acta Mathematica Sinica, English Series, 2006, 22(2): 169--178.

[11] Cao X. H., Guo M. Z., Meng B., Weyl's theorem for upper triangular operator matrices, Linear Algebra Appl., 2005, 402: 61--73.

[12] Chen X. L., Zhang S. F., Zhong H. J., On the filling in holes problem of operator matrices, Linear Algebra Appl., 2009, 430: 558--563.

[13] Duggal B. P., Upper triangular operator matrices, SVEP and Browder, Weyl Theorems, Integr. Equ. Oper. Theory, 2009, 63: 17--28.

[14] Barraa M., Boumazgour M., A note on the spectrum of an upper triangular operator matrix, Proc. Amer. Math. Soc., 2003, 131: 3083--3088.

[15] Benhida C., Zerouali E. H., Zguitti H., Spectral properties of upper-triangular block operators, Acta Sci. Math.(Szeged), 2005, 71: 607--616.

[16] Cao X. H., Browder spectra for upper triangular operator matrices,J. Math. Anal. Appl., 2008, 342: 477--484.

[17] Cao X. H., Guo M. Z., Meng B., Drazin spectrum and Wely's theorem for operator matrices, J. Math. Res. Exposition, 2006, 26: 413--422.

[18] Djordjevi'c S. V., Han Y. M., A note on Weyl's theorem for operator matrices, Proc. Amer. Math. Soc., 2003, 131: 2543--2547.

[19] Djordjevi'c S. V., Han Y. M., Spectral continuity for operator matrices, Glasg. Math. J., 2001, 43: 487--490.

[20] Du H. K., Pan J., Perturbation of spectrums of 2×2 operator matrices, Proc. Amer. Math. Soc., 1994, 121: 761--766.

[21] Elbjaoui H., Zerouali E. H., Local spectral theory for 2×2 operator matrices, Int. J. Math. Math. Sci., 2003, 42: 2667--2672.

[22] Lee W. Y., Weyl's theorem for operator matrices, Integr. Equ. Oper. Theory, 1998, 32: 319--331.

[23] Li Y., Sun X. H., Du H. K., A note on the left essential spectra of operator matrices, Acta Mathematica Sinica, English Series, 2007, 23(12): 2235--2240.

[24] Li Y., Sun X. H., Du H. K., The intersection of left (right) spectra of 2 × 2 upper triangular operator matrices, Linear Algebra Appl., 2006, 418: 112--121.

[25] Li Y., Du H. K, The intersection of essential approximate point spectra of operator matrices, J. Math. Anal. Appl., 2006, 323: 1171--1183.

[26] Zhang H. Y., Du H. K., Browder spectra of upper-triangular operator matrices, J. Math. Anal. Appl., 2006, 323: 700--707.

[27] Zhang S. F., Zhong H. J., A note of Browder spectrum of operator matrices, J. Math. Anal. Appl., 2008, \bf 344: 927--931.

[28] Li Y., Sun X. H., Du H. K., The intersection of left (right) weyl spectrum of 2×2 upper triangular operator matrices, Acta Mathematica Sinica, Chinese Series, 2005, 48(4): 653--660.

[29] Zhang S. F., Wu Z. Y., Zhong H. J., Continuous spectrum, point spectrum and residual spectrum of operator matrices, Linear Algebra Appl., 2010, 433: 653--661.

[30] Aiena P., Fredholm and Local Spectral Theory, with Applications to Multipliers, London: Kluwer Academic Publishers, 2004.

[31] Berkani M., Index of B-Fredholm operators and generalization of a Weyl theorem, Proc. Amer. Math. Soc., 2002, 130: 1717--1723.

[32] Berkani M., On a class of quasi-Fredholm operators, Integr. Equ. Oper. Theory, 1999, 34: 244--249.

[33] Mbekhta M., Müller V., On the axiomatic theory of spectrum II., Studia Math., 1996, 119: 129--147.

[34] Schmoeger C., Perturbation properties of some classes of operators, Rendiconti di Matematica, 1994, 14: 533--541.

[35] Burgos M., Kaidi A., Mbekhta M., Oudghiri M., The descent spectrum and perturbations, J. Operator Theory, 2006, 56: 259--271.