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''of a Banach algebra''
 
''of a Banach algebra''
  
The set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s0866001.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s0866002.png" /> is non-invertible (the algebra is assumed to be complex, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s0866003.png" /> is a given element of it and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s0866004.png" /> 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|Character of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s0866005.png" />-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|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|Banach algebra]]. The natural calculus of polynomials in an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s0866006.png" /> of a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s0866007.png" /> is extended to a continuous homomorphism into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s0866008.png" /> from the ring of germs of functions holomorphic in a neighbourhood of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s0866009.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660010.png" /> is commutative, then, by definition, the spectrum of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660011.png" /> of elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660012.png" /> is the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660013.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660014.png" />-tuples of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660016.png" /> is a character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660017.png" />. In general, one defines the left (right) spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660018.png" /> to include those sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660019.png" /> for which the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086600/s08660020.png" /> 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 [[#References|[1]]]–[[#References|[4]]].
+
This concept can be used as a basis for developing a functional calculus for the elements of a [[Banach algebra|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 [[#References|[1]]]–[[#References|[4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Theories spectrales" , ''Eléments de mathématiques'' , '''32''' , Hermann  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Harte,  "The spectral mapping theorem in several variables"  ''Bull. Amer. Math. Soc.'' , '''78'''  (1972)  pp. 871–875</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Taylor,  "A joint spectrum for several commuting operators"  ''J. Funct. Anal.'' , '''6'''  (1970)  pp. 172–191</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Zhelazko,  "An axiomatic approach to joint spectra I"  ''Studia Math.'' , '''64'''  (1979)  pp. 249–261</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Theories spectrales" , ''Eléments de mathématiques'' , '''32''' , Hermann  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Harte,  "The spectral mapping theorem in several variables"  ''Bull. Amer. Math. Soc.'' , '''78'''  (1972)  pp. 871–875</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Taylor,  "A joint spectrum for several commuting operators"  ''J. Funct. Anal.'' , '''6'''  (1970)  pp. 172–191</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Zhelazko,  "An axiomatic approach to joint spectra I"  ''Studia Math.'' , '''64'''  (1979)  pp. 249–261</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.E. Rickart,  "General theory of Banach algebras" , v. Nostrand  (1960)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.E. Rickart,  "General theory of Banach algebras" , v. Nostrand  (1960)</TD></TR></table>

Latest revision as of 08:22, 6 June 2020


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

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)
How to Cite This Entry:
Spectrum of an element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectrum_of_an_element&oldid=48770
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article