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 *

KEYWORDS

*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

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 (B^{r}A^{p}B^{r})^{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 (B

[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.