Geodesic
Linear Extensions Associated with Abstract Functional Operators
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 1, pp. 78-82

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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.
Keywords: $C^*$-algebra, functional operator, weighted shift operator, spectrum of an operator, linear extension, hyperbolicity. 
@article{FAA_2008_42_1_a6,
     author = {A. B. Antonevich and I. Yu. Trubnikov},
     title = {Linear {Extensions} {Associated} with {Abstract} {Functional} {Operators}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {78--82},
     publisher = {mathdoc},
     volume = {42},
     number = {1},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2008_42_1_a6/}
}
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A. B. Antonevich; I. Yu. Trubnikov. Linear Extensions Associated with Abstract Functional Operators. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 1, pp. 78-82. http://geodesic.mathdoc.fr/item/FAA_2008_42_1_a6/