Linear Extensions Associated with Abstract Functional Operators

Abstract functional operators are defined as elements of a $C^*$-algebra $B$ with a structure consisting of a closed $C^*$-subalgebra $A\subset B$ and a unitary element $T\in B$ such that the mapping $\widehat{T}\colon a \to TaT^{-1}$ is an automorphism of $A$ and the set of finite sums $\sum a_kT^k$, $a_k\in A$, is norm dense in $B$.
We give a new construction of a linear extension associated with the abstract weighted shift operator $aT$ and obtain generalizations of known theorems about the relationship between the invertibility of operators and the hyperbolicity of the associated linear extensions to the case of abstract functional operators.