Logarithms and Antilogarithms of Operators Having Either Finite Nullity or Finite Deficiency

Abstract

Let A ∈ L(X). Recall that the nullity and the deficiency of A are α _A = dim ker A, β _A = codim A(dom A) =

$$ \alpha _A = \dim \,\ker \,A,\,\beta _A = \,co\dim \,A(dom\,A) = \dim X/A(dom\,A), $$

respectively.respectively. The index κA of an operator A ∈ L(X) having either finite nullity or finite deficiency is defined as follows:

$$ \kappa A = \left\{ {\begin{array}{*{20}c} {\beta _A - \alpha _{A\,} if\,\alpha _A < + \infty ,\,\beta _A < + \infty ,} \\ { + \infty \,if\,\alpha _A \, < + \infty ,} \\ { - \infty \,if\,\beta _A < + \infty .} \\ \end{array} \,} \right. $$