Spectrum of an element
of a Banach algebra
The set of numbers 
for which 
is non-invertible (the algebra is assumed to be complex, 
is a given element of it and 
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 
-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 
of a Banach algebra 
is extended to a continuous homomorphism into 
from the ring of germs of functions holomorphic in a neighbourhood of the spectrum 
. 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 
is commutative, then, by definition, the spectrum of a set 
of elements in 
is the collection 
of all 
-tuples of the form 
, where 
is a character of 
. In general, one defines the left (right) spectrum of 
to include those sets 
for which the system 
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 |
Comments
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) |
Spectrum of an element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectrum_of_an_element&oldid=48770