International Journal of Nonlinear Analysis and Applications
On the Cauchy dual and complex symmetric of composition operators
Document Type : Research Paper
Author
Department of Mathematics, Lorestan University, Khorramabad, Iran
Abstract
In this paper, firstly we show that some classical properties for Cauchy dual and Moore-Penrose inverse of composition operators, such as complex symmetric and Aluthge transform on $L^{2}(\Sigma)$. Secondly we give a characterization for some operator classes of weak $p$-hyponormal via Moore-Penrose inverse of composition operators. Finally, some examples are then presented to illustrate that, the Moore-Penrose inverse of composition operators lie between these classes.
Keywords
[1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integ. Eq. Oper. Theo. 13 (1990) 307–315.
[2] M. L. Arias, G. Corach and M. C. Gonzalez, Generalized inverse and Douglas equations, Proc. Amer. Math. Soc. 136 (2008) 3177–3183.[3] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer-Verlag, New York,2003.
[4] N. L. Braha and I. Hoxha, (p, r, q)-∗-Paranormal and absolute-(p, r)-∗-paranormal operators, J. Math. Anal. 4 (2013) 14–22.
[5] C. Burnap and I. Jung, Composition operators with weak hyponormalities, J. Math. Anal. Appl. 337 (2008) 686–694.
[6] C. Burnap, I. Jung and A. Lambert, Separating partial normality classes with composition operators, J. Oper. Theo. 53 (2005) 381–397.
[7] J. Campbell and J. Jamison, On some classes of weighted composition operators, Glasgow Math. J. 32 (1990) 87–94.
[8] J. Campbell, M. Embry-Wardrop and R. Fleming, Normal and quasinormal weighted composition operators, Glasgow Math. J. 33 (1991) 275–279.
[9] J. Campbell and W. Hornor, Localising and seminormal composition operators on L2, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 301–316.
[10] S. R. Caradus,Generalized Inverse and Operator Theory, Queens Papers in Pure and Applied Mathematics, 50,
Queens University, Kingston, Ont, 1978.
[11] S. Chavan, On operators Cauchy dual to 2-hyperexpansive operators, Proc. Edinburgh Math. Soc. 50 (2007) 637–552.
[12] M. Cho and T. Yamazaki, Characterizations of p-hyponormal and weak hyponormal weighted composition operators, Acta Sci. Math. (Szeged) 76 (2010) 173–181.
[13] A. Daniluk, and J. Stochel, Seminormal composition operators induced by affine transformations, Hokkaido Math. J. 26 (1997) 377–404.
[14] D. S. Djordjevic, Further results on the reverse order law for generalized inverses, SIAM J. Matrix Anal. Appl. 29(4) (2007) 1242–1246.
[15] D. S. Djordjevic and N. C. Dincic, Reverse order law for the Moore–Penrose inverse, J. Math. Anal. Appl. 361 (2010) 252–261.
[16] S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006) 1285–1315.
[17] D. Harrington and R. Whitley, Seminormal composition operators, J. Oper. Theo. 11 (1984) 125–135.
[18] J. Herron, Weighted conditional expectation operators, Oper. Mat. 5 (2011) 107–118.
[19] T. Hoover, A. Lambert and J. Queen, The Markov process determined by a weighted composition operator, Studia Math. LXXII (1982), 225-235.
[20] M. R. Jabbarzadeh and M. R. Azimi, Some weak hyponormal classes of weighted composition operators, Bull. Korean Math. Soc. 47 (2010) 793–803.
[21] S. Jung, Y. Kim, E. Ko and J. E. Lee, Complex symmetric weighted composition operators on H2 (D), J. Funct. Anal. 267 (2014) 323–351.
[22] F. Kimura, Analysis of non-normal operators via Aluthge transformation, Integ. Eq. Oper. Theo. 50 (2004) 375–384.
[23] A. Lambert, Localising sets for sigma-algebras and related point transformations, Proc. Roy. Soc. Edinburgh Sect. A 118 (1991) 111–118.
[24] A. Lambert, Hyponormal composition operators, Bull. London Math. Soc. 18 (1986) 395–400.
[25] A. Olofsson, Wandering subspace theorems, Integral Equ. Oper. Theo. 51 (2005) 395–409.
[26] M. M. Rao, Conditional Measure and Applications, Marcel Dekker, New York, 1993.
[27] D. Senthilkumar and P. Maheswari Naik, Weyl’s theorem for algebraically absolute-(p, r)-paranormal operators, Banach J. Math. Anal. 5 (2011) 29–37.
[28] R. K. Singh and J. S. Manhas, Composition Operators on Function Spaces, North Holland Math. Studies 179, Amsterdam 1993.
[29] S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001) 147–189.
[30] R. Whitley, Normal and quasinormal composition operators, Proc. Amer. Math. Soc. 70 (1978) 114–118.
[31] T. Yamazaki and M. Yanagida, A further generalization of paranormal operators, Sci. Math. 3.1 (2000) 23–31.
[32] S. Panayappan and A. Radharamani, A note on p-paranormal operators and absolute k-paranormal operators, Int. J. Math. Anal. 2 (2008) 25–28.
[33] M. R. Jabbarzadeh, H. Emamalipour and M. Sohrabi Chegeni, Parallelism between Moore-Penrose inverse and Aluthge transformation of operators, Appl. Anal. Discrete Math. 12(2) (2018) 318–335.
[2] M. L. Arias, G. Corach and M. C. Gonzalez, Generalized inverse and Douglas equations, Proc. Amer. Math. Soc. 136 (2008) 3177–3183.[3] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer-Verlag, New York,2003.
[4] N. L. Braha and I. Hoxha, (p, r, q)-∗-Paranormal and absolute-(p, r)-∗-paranormal operators, J. Math. Anal. 4 (2013) 14–22.
[5] C. Burnap and I. Jung, Composition operators with weak hyponormalities, J. Math. Anal. Appl. 337 (2008) 686–694.
[6] C. Burnap, I. Jung and A. Lambert, Separating partial normality classes with composition operators, J. Oper. Theo. 53 (2005) 381–397.
[7] J. Campbell and J. Jamison, On some classes of weighted composition operators, Glasgow Math. J. 32 (1990) 87–94.
[8] J. Campbell, M. Embry-Wardrop and R. Fleming, Normal and quasinormal weighted composition operators, Glasgow Math. J. 33 (1991) 275–279.
[9] J. Campbell and W. Hornor, Localising and seminormal composition operators on L2, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 301–316.
[10] S. R. Caradus,Generalized Inverse and Operator Theory, Queens Papers in Pure and Applied Mathematics, 50,
Queens University, Kingston, Ont, 1978.
[11] S. Chavan, On operators Cauchy dual to 2-hyperexpansive operators, Proc. Edinburgh Math. Soc. 50 (2007) 637–552.
[12] M. Cho and T. Yamazaki, Characterizations of p-hyponormal and weak hyponormal weighted composition operators, Acta Sci. Math. (Szeged) 76 (2010) 173–181.
[13] A. Daniluk, and J. Stochel, Seminormal composition operators induced by affine transformations, Hokkaido Math. J. 26 (1997) 377–404.
[14] D. S. Djordjevic, Further results on the reverse order law for generalized inverses, SIAM J. Matrix Anal. Appl. 29(4) (2007) 1242–1246.
[15] D. S. Djordjevic and N. C. Dincic, Reverse order law for the Moore–Penrose inverse, J. Math. Anal. Appl. 361 (2010) 252–261.
[16] S. R. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358 (2006) 1285–1315.
[17] D. Harrington and R. Whitley, Seminormal composition operators, J. Oper. Theo. 11 (1984) 125–135.
[18] J. Herron, Weighted conditional expectation operators, Oper. Mat. 5 (2011) 107–118.
[19] T. Hoover, A. Lambert and J. Queen, The Markov process determined by a weighted composition operator, Studia Math. LXXII (1982), 225-235.
[20] M. R. Jabbarzadeh and M. R. Azimi, Some weak hyponormal classes of weighted composition operators, Bull. Korean Math. Soc. 47 (2010) 793–803.
[21] S. Jung, Y. Kim, E. Ko and J. E. Lee, Complex symmetric weighted composition operators on H2 (D), J. Funct. Anal. 267 (2014) 323–351.
[22] F. Kimura, Analysis of non-normal operators via Aluthge transformation, Integ. Eq. Oper. Theo. 50 (2004) 375–384.
[23] A. Lambert, Localising sets for sigma-algebras and related point transformations, Proc. Roy. Soc. Edinburgh Sect. A 118 (1991) 111–118.
[24] A. Lambert, Hyponormal composition operators, Bull. London Math. Soc. 18 (1986) 395–400.
[25] A. Olofsson, Wandering subspace theorems, Integral Equ. Oper. Theo. 51 (2005) 395–409.
[26] M. M. Rao, Conditional Measure and Applications, Marcel Dekker, New York, 1993.
[27] D. Senthilkumar and P. Maheswari Naik, Weyl’s theorem for algebraically absolute-(p, r)-paranormal operators, Banach J. Math. Anal. 5 (2011) 29–37.
[28] R. K. Singh and J. S. Manhas, Composition Operators on Function Spaces, North Holland Math. Studies 179, Amsterdam 1993.
[29] S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001) 147–189.
[30] R. Whitley, Normal and quasinormal composition operators, Proc. Amer. Math. Soc. 70 (1978) 114–118.
[31] T. Yamazaki and M. Yanagida, A further generalization of paranormal operators, Sci. Math. 3.1 (2000) 23–31.
[32] S. Panayappan and A. Radharamani, A note on p-paranormal operators and absolute k-paranormal operators, Int. J. Math. Anal. 2 (2008) 25–28.
[33] M. R. Jabbarzadeh, H. Emamalipour and M. Sohrabi Chegeni, Parallelism between Moore-Penrose inverse and Aluthge transformation of operators, Appl. Anal. Discrete Math. 12(2) (2018) 318–335.
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Volume 12, Issue 2
November 2021Pages 85-97
- Receive Date: 06 July 2020
- Accept Date: 08 August 2020