SKEW COMPLEX SYMMETRIC OPERATORS AND WEYL TYPE THEOREMS

Title & Authors
SKEW COMPLEX SYMMETRIC OPERATORS AND WEYL TYPE THEOREMS
KO, EUNGIL; KO, EUNJEONG; LEE, JI EUN;

Abstract
An operator $\small{T{{\in}}{\mathcal{L}}({\mathcal{H}})}$ is said to be skew complex symmetric if there exists a conjugation C on $\small{{\mathcal{H}}}$ such that $\small{T=-CT^*C}$. In this paper, we study properties of skew complex symmetric operators including spectral connections, Fredholmness, and subspace-hypercyclicity between skew complex symmetric operators and their adjoints. Moreover, we consider Weyl type theorems and Browder type theorems for skew complex symmetric operators.
Keywords
skew complex symmetric operator;subspace-hypercyclicity;Weyl type theorems;
Language
English
Cited by
1.
On $${m}$$ m -Complex Symmetric Operators II, Mediterranean Journal of Mathematics, 2016, 13, 5, 3255
References
1.
P. Aiena, Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer Academic Pub. 2004.

2.
P. Aiena, M. T. Biondi, and C. Carpintero, On Drazin invertibility, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2839-2848.

3.
P. Aiena and P. Pena, Variations on Weyl's theorem, J. Math. Anal. Appl. 324 (2006), no. 1, 566-579.

4.
M. Amouch and H. Zguitti, On the equivalence of Browder's and generalized Browder's theorem, Glasg. Math. J. 48 (2006), no. 1, 179-185.

5.
I. J. An, Weyl type theorems for $2{\times}2$ operator matrices, Kyung Hee Univ., Ph.D. Thesis. 2013.

6.
M. Berkani, On a class of quasi-Fredholm operators, Integral Equations Operator Theory 34 (1999), no. 2, 244-249.

7.
M. Berkani and J. J. Koliha, Weyl type theorems for bounded linear operators, Acta Sci. Math. 69 (2003), no. 1-2, 359-376.

8.
M. Berkani and H. Zariouh, Extended Weyl type theorems, Math. Bohem. 134 (2009), no. 4, 369-378.

9.
J. B. Conway, A Course in Functional Analysis, Springer Verlag, Berlin, Heidelberg, New York, 1990.

10.
S. V. Djordjevic and Y. M. Han, Browder's theorems and spectral continuity, Glasg. Math. J. 42 (2000), no. 3, 479-486.

11.
S. V. Djordjevic and Y. M. Han, a-Weyl's theorem for operator matrices, Proc. Amer. Math. Soc 130 (2002), no. 3, 715-722.

12.
S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006), no. 3, 1285-1315.

13.
S. R. Garcia and M. Putinar, Complex symmetric operators and applications. II, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3913-3931.

14.
S. R. Garcia and W. R. Wogen, Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362 (2010), no. 11, 6065-6077.

15.
P. R. Halmos, A Hilbert Space Problem Book, Second edition, Springer-Verlag, New York, 1982.

16.
R. Harte, Invertibility and Singularity for Bounded Linear Operators, Marcel Dekker Inc., New York and Basel, 1988.

17.
R. E. Harte and W. Y. Lee, Another note on Weyl's theorem, Trans. Amer. Math. Soc. 349 (1997), no. 5, 2115-2124.

18.
S. Jung, E. Ko, and J. Lee, On scalar extensions and spectral decompositions of complex symmetric operators, J. Math. Anal. Appl. 382 (2011), no. 2, 252-260.

19.
S. Jung, E. Ko, and J. Lee, On complex symmetric operator matrices, J. Math. Anal. Appl. 406 (2013), no. 2, 373-385.

20.
S. Jung, E. Ko, M. Lee, and J. Lee, On local spectral properties of complex symmetric operators, J. Math. Anal. Appl. 379 (2011), no. 1, 325-333.

21.
J. J. Koliha, A generalized Drazin inverse, Glasg. Math J. 38 (1996), no. 3, 367-381.

22.
P. Lancaster and L. Rodman, The Algebraic Riccati Equation, Oxford University Press, Oxford, 1995.

23.
K. Laursen, Operators with finite ascent, Pacific J. Math. 152 (1992), no. 2, 323-336.

24.
K. Laursen and M. Neumann, An Introduction to Local Spectral Theory, Clarendon Press, Oxford, 2000.

25.
C. G. Li and S. Zhu, Skew symmetric normal operators, Proc. Amer. Math. Soc. 141 (2013), no. 8, 2755-2762.

26.
B. F. Madore and R. A. Martinez-Avendano, Subspace hypercyclicity, preprint.

27.
V. Mehrmann and H. Xu, Numerical methods in control, J. Comput. Appl. Math. 123 (2000), no. 1-2, 371-394.

28.
C. Sun, X. Cao, and L. Dai, Property ($w_1$) and Weyl type theorem, J. Math. Anal. Appl. 363 (2010), no. 1, 1-6.