We give a sufficient condition for a family of pseudoresolvents in a Banach algebra to be trivially zero. As an important consequence, we provide an alternate proof of the classical result that the spectrum of any linear bounded operator on a Banach space is nonempty. The proofs are elementary, requiring only a basic knowledge of real and complex analysis.
"Pseudoresolvents in Banach Algebras." Missouri J. Math. Sci. 16 (3) 168 - 172, Fall 2004. https://doi.org/10.35834/2004/1603168