On Operators of Saphar Type

Christoph Schmoeger

Abstract: A bounded linear operator $T$ on a complex Banach space $X$ is called an operator of Saphar type, if $T$ is relatively regular and if its null space is contained in its generalized range $\displaystyle\bigcap^{\infty}_{n=1}T^n(X)$. This paper contains some characterizations of operators of Saphar type. Furthermore, for a function $f$ admissible in the analytic calculus, we obtain a necessary and sufficient condition in order that $f(T)$ is an operator of Saphar type.

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