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

References

[a1]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[a2]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
[a3]	C.E. Rickart, "General theory of Banach algebras" , v. Nostrand (1960)