Document Type : Review paper

Author

Department of Mathematics, Lorestan University, Khorramabad, Iran

Abstract

‎In this paper‎, ‎we give an explicit formula for the Moore-Penrose inverse of $W$‎, ‎denoted by $W^{\dag}$‎, ‎on $L^2(\Sigma)$‎. ‎As an application‎, ‎we give a characterization for some operator classes that are weaker than $p$-hyponormal with $W^{\dag}$‎. ‎Moreover‎, ‎we give specific examples illustrating these classes‎.

Keywords

Main Subjects

[1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Integr. Equ. Oper. Theory. 13 (1990), 307-315.
[2] M. L. Arias, G. Corach, M. C. Gonzalez, Generalized inverse and Douglas equations, Proc. Amer. Math. Soc. 136 (2008), 3177-3183.
[3] A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer-Verlag, New York,
2003.
[4] C. Burnap, I. Jung, Composition operators with weak hyponormalities, J. Math. Anal. Appl. 337 (2008), 686-694.
[5] C. Burnap, I. Jung, A. Lambert, Separating partial normality classes with composition operators, J. Operator Theory. 53 (2005), 381-397.
[6] J. Campbell, M. Embry-Wardrop, R. Fleming, Normal and quasinormal weighted composition operators, Glasgow Math. J. 33 (1991), 275-279.
[7] J. Campbell, W. Hornor, Localising and seminormal composition operators on L 2 , Proc. Roy. Soc. Edinburgh Sect. A. 124 (1994), 301-316.
[8] J. Campbell, J. Jamison, On some classes of weighted composition operators, Glasgow Math. J. 32 (1990), 87-94.
[9] S. R. Caradus, Generalized Inverse and Operator Theory, Queens Papers in Pure and Applied Mathematics, 50, Queens University, Kingston, Ont, 1978.
[10] M. Cho, T. Yamazaki, Characterizations of p-hyponormal and weak hyponormal weighted composition operators, Acta Sci. Math. 76 (2010), 173-181.
[11] A. Daniluk, J. Stochel, Seminormal composition operators induced by affine transformations, Hokkaido Math. J. 26 (1997), 377-404.
[12] D. S. Djordjevic, Further results on the reverse order law for generalized inverses, Siam J. Matrix Anal. Appl. 29 (4) (2007), 1242-1246.
[13] D. S. Djordjevic, N. C. Dincic, Reverse order law for the Moore-Penrose inverse, J. Math. Anal. Appl. 361 (2010), 252-261.
[14] H. Emamalipour, M. R. Jabbarzadeh, M. Sohrabi Chegeni, Some weak p-hyponormal classes of weighted composition operators, Filomat. 31 (9) (2017), 2643-2656.
[15] D. Harrington, R. Whitley, Seminormal composition operators, J. Operator Theory. 11 (1984), 125-135.
[16] J. Herron, Weighted conditional expectation operators, Oper. Matrices. 5 (2011), 107-118.
[17] T. Hoover, A. Lambert, J. Queen, The Markov process determined by a weighted composition operator, Studia Math. LXXII (1982), 225-235.
[18] M. R. Jabbarzadeh, M. R. Azimi, Some weak hyponormal classes of weighted composition operators, Bull. Korean. Math. Soc. 47 (2010), 793-803.
[19] M. R. Jabbarzadeh, M. Sohrabi Chegeni, Moore-Penrose inverse of conditional type operators, Oper. Matrices, 11 (2017), 289-299.
[20] F. Kimura, Analysis of non-normal operators via Aluthge transformation, Integr. Equ. Oper. Theory. 50 (2004), 375-384.
[21] A. Lambert, Hyponormal composition operators, Bull. London Math. Soc. 18 (1986), 395-400.
[22] A. Lambert, Localising sets for sigma-algebras and related point transformations, Proc. Roy. Soc. Edinburgh Sect. A. 118 (1991), 111-118.
[23] M. M. Rao, Conditional Measure and Applications, Marcel Dekker, New York, 1993.
[24] R. K. Singh, J. S. Manhas, Composition Operators on Function Spaces, North Holland Math. Studies 179, Amsterdam, 1993.
[25] R. Whitley, Normal and quasinormal composition operators, Proc. Amer. Math. Soc. 70 (1978), 114-118.
[26] T. Yamazaki, M. Yanagida, A further generalization of paranormal operators, Sci. Math. 3 (1) (2000), 23-31.