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Difference between revisions of "Spectral radius"

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''of an element of a Banach algebra''
 
''of an element of a Banach algebra''
  
The radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864601.png" /> of the smallest closed disc in the plane that contains the spectrum of this element (cf. [[Spectrum of an element|Spectrum of an element]]). The spectral radius of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864602.png" /> is connected with the norms of its powers by the formula
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The radius $\rho$ of the smallest closed disc in the plane that contains the spectrum of this element (cf. [[Spectrum of an element|Spectrum of an element]]). The spectral radius of an element $a$ is connected with the norms of its powers by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864603.png" /></td> </tr></table>
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$$\rho(a)=\lim_{n\to\infty}\|a^n\|^{1/n}=\inf\|a^n\|^{1/n},$$
  
which, in particular, implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864604.png" />. The spectral radius of a bounded linear operator on a Banach space is the spectral radius of it regarded as an element of the Banach algebra of all operators. In a Hilbert space, the spectral radius of an operator is equal to the greatest lower bound of the norms of the operators similar to it (see [[#References|[2]]]):
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which, in particular, implies that $\rho(a)\leq\|a\|$. The spectral radius of a bounded linear operator on a Banach space is the spectral radius of it regarded as an element of the Banach algebra of all operators. In a Hilbert space, the spectral radius of an operator is equal to the greatest lower bound of the norms of the operators similar to it (see [[#References|[2]]]):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864605.png" /></td> </tr></table>
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$$\rho(A)=\inf_X\|XAX^{-1}\|.$$
  
If the operator is normal, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864606.png" /> (cf. [[Normal operator|Normal operator]]).
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If the operator is normal, then $\rho(A)=\|A\|$ (cf. [[Normal operator|Normal operator]]).
  
As a function of the elements of a Banach algebra, the spectral radius is upper semi-continuous (but not, in general, continuous). The subharmonicity of the spectral radius has been proved [[#References|[3]]]. (This means that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864607.png" /> is a holomorphic mapping of some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864608.png" /> into a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s0864609.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086460/s08646010.png" /> is a [[Subharmonic function|subharmonic function]].)
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As a function of the elements of a Banach algebra, the spectral radius is upper semi-continuous (but not, in general, continuous). The subharmonicity of the spectral radius has been proved [[#References|[3]]]. (This means that if $z\mapsto h(z)$ is a holomorphic mapping of some domain $D\subset\mathbf C$ into a Banach algebra $\mathfrak A$, then $z\mapsto\rho(h(z))$ is a [[Subharmonic function|subharmonic function]].)
  
 
====References====
 
====References====

Latest revision as of 14:46, 21 August 2014

of an element of a Banach algebra

The radius $\rho$ of the smallest closed disc in the plane that contains the spectrum of this element (cf. Spectrum of an element). The spectral radius of an element $a$ is connected with the norms of its powers by the formula

$$\rho(a)=\lim_{n\to\infty}\|a^n\|^{1/n}=\inf\|a^n\|^{1/n},$$

which, in particular, implies that $\rho(a)\leq\|a\|$. The spectral radius of a bounded linear operator on a Banach space is the spectral radius of it regarded as an element of the Banach algebra of all operators. In a Hilbert space, the spectral radius of an operator is equal to the greatest lower bound of the norms of the operators similar to it (see [2]):

$$\rho(A)=\inf_X\|XAX^{-1}\|.$$

If the operator is normal, then $\rho(A)=\|A\|$ (cf. Normal operator).

As a function of the elements of a Banach algebra, the spectral radius is upper semi-continuous (but not, in general, continuous). The subharmonicity of the spectral radius has been proved [3]. (This means that if $z\mapsto h(z)$ is a holomorphic mapping of some domain $D\subset\mathbf C$ into a Banach algebra $\mathfrak A$, then $z\mapsto\rho(h(z))$ is a subharmonic function.)

References

[1] M.A. Naimark, "Normed rings" , Reidel (1959) (Translated from Russian)
[2] P.R. Halmos, "A Hilbert space problem book" , Springer (1980)
[3] E. Vesentini, "On the subharmonicity of the spectral radius" Boll. Union. Mat. Ital. , 1 (1968) pp. 427–429
[4] V. Ptak, "On the spectral radius in Banach algebras with involution" Bull. London Math. Soc. , 2 (1970) pp. 327–334


Comments

References

[a1] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
How to Cite This Entry:
Spectral radius. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_radius&oldid=15490
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