Some Classes of Operators Related to p-Hyponormal Operator
Affiliation(s)
Department of Mathematics, Gaya College, Gaya, Bihar, India. Al-Ain University of Science and Technology, Al Ain & Abu Dhabi, UAE.
Department of Mathematics, Gaya College, Gaya, Bihar, India. Al-Ain University of Science and Technology, Al Ain & Abu Dhabi, UAE.
ABSTRACT
We introduce a new family of classes of operators termed as *p-paranormal operator, classes *A(p,p); p > 0 and *A(p,q); p, q > 0, parallel to p-paranormal operator and classes A(p,p); p> 0 and A(p,q); p, q > 0 introduced by M. Fujii, D. Jung, S. H. Lee, M. Y. Lee and R. Nakamoto [1]. We present a necessary and sufficient condition for p-hyponormal operator T∈B（H）to be *p-paranormal and the monotonicity of *A(p,q). We also present an alternative proof of a result of M. Fujii, et al. [1, Theorem 3.4].
We introduce a new family of classes of operators termed as *p-paranormal operator, classes *A(p,p); p > 0 and *A(p,q); p, q > 0, parallel to p-paranormal operator and classes A(p,p); p> 0 and A(p,q); p, q > 0 introduced by M. Fujii, D. Jung, S. H. Lee, M. Y. Lee and R. Nakamoto [1]. We present a necessary and sufficient condition for p-hyponormal operator T∈B（H）to be *p-paranormal and the monotonicity of *A(p,q). We also present an alternative proof of a result of M. Fujii, et al. [1, Theorem 3.4].
KEYWORDS
p-Hyponormal Operator; Monotonicity; Class of Operators *A(p, q); *Paranormal Operator; *p-Paranormal Operator
p-Hyponormal Operator; Monotonicity; Class of Operators *A(p, q); *Paranormal Operator; *p-Paranormal Operator
Cite this paper
M. Ilyas and R. Ahmad, "Some Classes of Operators Related to p-Hyponormal Operator," Advances in Pure Mathematics, Vol. 2 No. 6, 2012, pp. 419-422. doi: 10.4236/apm.2012.26063.
M. Ilyas and R. Ahmad, "Some Classes of Operators Related to p-Hyponormal Operator," Advances in Pure Mathematics, Vol. 2 No. 6, 2012, pp. 419-422. doi: 10.4236/apm.2012.26063.
References
[1] M. Fujii, D. Jung, S. H. Lee, M. Y. Lee and R. Nakamoto, “Some Classes of Operators Related to Paranormal and Log-Hyponormal Operators,” Japanese Journal of Mathematics, Vol. 51, No. 3, 2000, pp. 395-402. [2] T. Furuta, “A ≥ B ≥ 0 Assures (BrApBr)1/q≥B(p+2r)/q for r≥0, p≥0, q≥1 with (1+2r)q≥p+2r,” Proceedings of the American Mathematical Society, Vol. 101, No. 1, 1987, pp. 85-88. doi:10.2307/2046555 [3] T. Ando, “Operators with a Norm Condition,” Acta Sci- entiarum Mathematicarum, Vol. 33, 1972, pp. 169-178. [4] T. Furuta, “Elementary Proof of an Order Preserving Inequality,” Proceedings of the Japan Academy, Vol. 65, No. 5, 1989, p. 126. doi:10.3792/pjaa.65.126 [5] M. Fujii, “Furuta’s Inequality and Its Mean Theoretic Approach,” Journal of Operator Theory, Vol. 23, No. 1, 1990, pp. 67-72. [6] E. Kamei, “A Satellite to Furuta’s Inequality,” Japanese Journal of Mathematics, Vol. 33, 1988, pp. 883-886.
[1] M. Fujii, D. Jung, S. H. Lee, M. Y. Lee and R. Nakamoto, “Some Classes of Operators Related to Paranormal and Log-Hyponormal Operators,” Japanese Journal of Mathematics, Vol. 51, No. 3, 2000, pp. 395-402. [2] T. Furuta, “A ≥ B ≥ 0 Assures (BrApBr)1/q≥B(p+2r)/q for r≥0, p≥0, q≥1 with (1+2r)q≥p+2r,” Proceedings of the American Mathematical Society, Vol. 101, No. 1, 1987, pp. 85-88. doi:10.2307/2046555 [3] T. Ando, “Operators with a Norm Condition,” Acta Sci- entiarum Mathematicarum, Vol. 33, 1972, pp. 169-178. [4] T. Furuta, “Elementary Proof of an Order Preserving Inequality,” Proceedings of the Japan Academy, Vol. 65, No. 5, 1989, p. 126. doi:10.3792/pjaa.65.126 [5] M. Fujii, “Furuta’s Inequality and Its Mean Theoretic Approach,” Journal of Operator Theory, Vol. 23, No. 1, 1990, pp. 67-72. [6] E. Kamei, “A Satellite to Furuta’s Inequality,” Japanese Journal of Mathematics, Vol. 33, 1988, pp. 883-886.