On the class of $k$-quasi-$(n,m)$-power normal operators

Abstract

We introduce a family of operators called the family of $k$-quasi-$(n,m)$-power normal operators. Such family includes normal, $n$-normal and $(n,m)$-power normal operators. An operator $T \in {\mathcal B}({\mathcal H})$ is said to be $k$-quasi-$(n,m)$-power normal if it satisfies $$T^{*k}\bigg(T^nT^{*m}-T^{*m}T^n\bigg)T^k=0,$$ where $k,n$ and $m$ are natural numbers. Firstly, some basic structural properties of this family of operators are established with the help of special kind of operator matrix representation associated with such family of operators. Secondly, some properties of\linebreak algebraically $k$-quasi-$(n,m)$-power normal operators are discussed. Thirdly, we consider the study of tensor products of $k$-quasi-$(n,m)$-power normal operators. A necessary and sufficient condition for $T\otimes S$ to be a $k$-quasi-$(n,m)$-power normal is given, when $T \neq0$ and $S\neq0$.

Keywords

• $n$-normal
• $(n,m)$-normal
• $k$-quasi-$(n,m)$-normal

References

1. [1] E.H. Abood and M.A. Al-loz, On some generalization of normal operators on Hilbert space, Iraqi J. Sci. 56 (2C), 1786–1794, 2015.
2. [2] E.H. Abood and M.A. Al-loz, On some generalizations of (n,m)-normal powers operators on Hilbert space, J. Progres. Res. Math. 7 (3), 1063–1070, 2016.
3. [3] S.A. Alzuraiqi and A.B. Patel, On n-normal operators, Gen. Math. Notes 1 (2), 61–73, 2010.
4. [4] M. Cho and B.N. Nactovska, Spectral properties of n-normal operators, Filomat, 32 (14), 5063–5069, 2018.
5. [5] M. Cho, J.E. Lee, K. Tanahashic, and A. Uchiyamad, Remarks on n-normal operators, Filomat 32 (15), 5441–5451, 2018.
6. [6] R.G. Douglas, On the operator equation $S^*XT = X $ and related topics, Acta Sci. Math. (Szeged) 30, 19–32, 1969.
7. [7] B.P. Duggal, R.E. Harte, and I.H. Jeon, Polaroid operators and Weyl’s theorem, Proc. Amer. Math. Soc. 132, 1345–1349, 2004.
8. [8] B.P. Duggal, Weyl’s theorem for totally hereditarily normaloid operators, Rend. Circ. Mat. Palermo (2) 53, 417–428, 2004.

9. [9] H.G. Heuser, Functional Analysis, John Wiley and Sons., Chichester, 1982.
10. [10] I.H. Jeon and B.P. Duggal, On operators with an absolute value condition, J. Korean Math. Soc. 41, 617–627, 2004.
11. [11] I.H. Jeon and I.H. Kim, On operators satisfying $T^*|T^2|T\geq T^*|T|^2T$, Linear Algebra Appl. 418, 854–862, 2006.
12. [12] A.A. Jibril, On n-power normal operators, Arab. J. Sci. Eng. 33 (2), 247–251, 2008.
13. [13] S. Jung, E. Ko, and M.J. Leo, On class ${\mathcal{A}}$ operators, Studia Math. 198 (3), 249–260, 2010.
14. [14] I.H. Kim, Weyl’s theorem and tensor product for operators satisfying $T^{*k } |T^2| T^k \geq T^{*k} |T|^2 T^k$, J. Korean Math. Soc. 47 (2), 351–361, 2010.
15. [15] E. Ko, Algebraic and triangular n-hyponormal operator, Proc. Amer. Math. Soc. 123, 873–875, 1995.
16. [16] J. Kyu Han, H. Youl Lee, and W. Young Lee, Invertible completions of $2\times2$ upper triangular operator matrices, Proc. Amer. Math. Soc. 128, 119–123, 1999.
17. [17] K.B. Laursen and M.M. Neumann, An Introduction to Local Spectral Theory, Oxford University Press, 2000.
18. [18] S.I. Mary and P. Vijaylakshmi, Fuglede-Putnam theorem and quasi-nilpotent part of n-normal operators, Tamkang J. Math. 46 (2), 151–165, 2015.
19. [19] W. Mlak, Hyponormal contractions, Colloq. Math. 18, 137–141, 1967.
20. [20] S. Panayappan, N. Jayanthi, and D. Sumathi, Weyl’s theorem and tensor product for class $A_k$ operators, Pure Math. Sci. 1 (1), 13–23. 2012.
21. [21] M. Putinar, Quasisimilarity of tuples with Bishop’s property ($\beta$), Integral Equations Operator Theory 15, 1047–1052, 1992.
22. [22] M.A. Rosenblum, On the operator equation $BX - XA = Q$, Duke Math. J. 23, 263–269, 1956.
23. [23] T. Saito, Hyponormal operators and related topics, in: Lecture notes in Mathematics, 247, Springer-Verlag, 1971.
24. [24] D. Senthilkumar, P. Maheswari Naik, and N. Sivakumar, Generalized Weyl’s theorem for algebraically k-quasi-paranormal operators, J. Chungcheong Math. Soc. 25 (4), 655–668, 2012.
25. [25] O.A.M. Sid Ahmed and O.B. Sid Ahmed, On the classes of $(n, m)$-power $D$-normal and $(n, m)$-power $D$-quasi-normal operators, Oper. Matrices 13 (3), 705–732, 2019.
26. [26] J. Stochel, Seminormality of operators from their tensor product, Proc. Amer. Math. Soc. 124, 435–440, 1996.
27. [27] K. Tanahashi, I.H. Jeon, I.H. Kim, and A. Uchiyama, Quasinilpotent part of class A or $(p, k)$-quasihyponormal operators, Oper. Theory Adv. Appl. 187, 199–210, 2008.