  We present new additive results for the pseudo core inverse in a Banach
algebra with involution. The necessary and sufficient conditions under which
the sum of two pseudo core invertible elements in Banach *-algebra is pseudo
core invertible are obtained. As an application, the pseudo core invertibility
for block complex matrices is investigated. These extend the main results of
pseudo core invertibility of Gao and Chen [Comm. Algebra, 46(2018), 38--50].
