GBD and $ \mathcal{L} $-positive semidefinite elements in $ C^* $-algebras

This paper focused on the generalized Bott-Duffin (GBD) inverse and the $ {\rm GBD} $ elements in Banach algebra with involution and $ C^* $-algebra, as well as on the property of the $ p $-positive semidefinite elements that are a generalization of the $ \mathcal{L} $-positive semidefinite matrices closely related to the GBD inverse. Also, using matrix equalities, inclusion relations of subspaces, and projectors, we established various characterizations of the GBD property in the matrix sets, especially on the set of $ \mathcal{L} $-positive semidefinite matrices. Additionally, we compared the methods and tools that we have at our disposal in the matrix set on one side and in Banach and $ C^* $-algebras on the other. Using the GBD inverse as an example, we would like to compare the results and their proofs in both sets and explain steps to quite easily skip from one set to the other, as well as situations in which we must pay additional attention in order to avoid mistakes.