# Spectrum of an element

of a Banach algebra

The set of numbers $\lambda \in \mathbf C$ for which $a - \lambda e$ is non-invertible (the algebra is assumed to be complex, $a$ is a given element of it and $e$ is the identity of the algebra). The spectrum is a non-empty compact set (the Gel'fand–Mazur theorem). In the case of a commutative algebra, the spectrum coincides with the set of values on this element of all the characters of the algebra (cf. Character of a $C ^ {*}$- algebra).

This concept can be used as a basis for developing a functional calculus for the elements of a Banach algebra. The natural calculus of polynomials in an element $a$ of a Banach algebra $A$ is extended to a continuous homomorphism into $A$ from the ring of germs of functions holomorphic in a neighbourhood of the spectrum $\sigma ( a)$. The necessity of considering functions in several variables leads to the concept of the joint spectrum of a system of elements of a Banach algebra. If $A$ is commutative, then, by definition, the spectrum of a set $\{ a _ {i} \} _ {i=} 1 ^ {n}$ of elements in $A$ is the collection $\sigma ( \{ a _ {i} \} ) \subset \mathbf C ^ {n}$ of all $n$- tuples of the form $\{ \phi ( a _ {i} ) \} _ {i=} 1 ^ {n}$, where $\phi$ is a character of $A$. In general, one defines the left (right) spectrum of $\{ a _ {i} \} _ {i=} 1 ^ {n}$ to include those sets $\{ \lambda _ {i} \} _ {i=} 1 ^ {n} \in \mathbf C ^ {n}$ for which the system $\{ a _ {i} - \lambda _ {i} e \}$ is contained in a non-trivial left (respectively, right) ideal of the algebra. The spectrum is then defined as the union of the left and right spectra. For the basic results of multi-parametric spectral theory, and also for other approaches to the concept of the spectrum of a set of elements, see [1][4].

#### References

 [1] N. Bourbaki, "Theories spectrales" , Eléments de mathématiques , 32 , Hermann (1967) [2] R. Harte, "The spectral mapping theorem in several variables" Bull. Amer. Math. Soc. , 78 (1972) pp. 871–875 [3] J. Taylor, "A joint spectrum for several commuting operators" J. Funct. Anal. , 6 (1970) pp. 172–191 [4] W. Zhelazko, "An axiomatic approach to joint spectra I" Studia Math. , 64 (1979) pp. 249–261