Hostname: page-component-84b7d79bbc-g5fl4 Total loading time: 0 Render date: 2024-07-25T13:54:14.514Z Has data issue: false hasContentIssue false
  • English
  • Français

On operators of class (N, k)

Published online by Cambridge University Press:  24 October 2008

P. B. Ramanujan
P. B. Ramanujan
Affiliation:
Sardar Patel University, Vallabh Vidyanagar, India
Get access Rights & Permissions [Opens in a new window]

Extract

Istrăţescu (2) has introduced a class of operators on a Banach space called operators of class (N, k). An operator T (a bounded linear transformation) on a Banach space X is said to be an operator of class (N, k), k = 2, 3,…, if ‖Tx‖k ≤ ‖Tkx‖ for all x ∈ X such that ‖x‖ = 1. If k = 2, such an operator is called an operator of class (N)(3). If X is a Hilbert space, then the class of operators of class (N) on X is an extension of the class of hyponormal operators on X (3). The object of this note is to generalize to operators of class (N, k) on a Banach space X, some results which are known to be true for normal operators on a Hilbert space, particularly with regard to their ascent and descent.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 1 , July 1970 , pp. 141 - 142
Copyright
Copyright © Cambridge Philosophical Society 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Goldberg, S.Unbounded linear operators (McGraw-Hill, 1966).Google Scholar
(2)Istrăţescu, V. and Istrăţescu, I.On Normaloid Operators. Math. Z. 105 (1968), 153–156.CrossRefGoogle Scholar
(3)Istrăţescu, V.On some hyponormal operators. Pacific J. Math. 22 (1967), 413–417.CrossRefGoogle Scholar
(4)Taylor, A. E.Introduction to Functional Analysis (Wiley and Sons; New York, 1958).Google Scholar
(5)Taylor, A. E.Theorems on ascent, descent, nullity and defect of linear operators. Math. Ann. 163 (1966), 18–49.CrossRefGoogle Scholar