Namespaces
Variants
Actions

Difference between revisions of "Spectral set"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
 
(2 intermediate revisions by the same user not shown)
Line 1: Line 1:
A spectral set of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s0865001.png" /> on a normed space is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s0865002.png" /> such that
+
{{TEX|done}}
 +
A spectral set of an operator $A$ on a normed space is a subset $S\subset\mathbf C$ such that
  
<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/s086500/s0865003.png" /></td> </tr></table>
+
$$\|p(A)\|\leq\sup\{|p(z)|:z\in S\}$$
  
for any polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s0865004.png" />. Thus, the unit circle is a spectral set for any contraction (an operator whose norm does not exceed one) on a Hilbert space (von Neumann's theorem). This result is closely connected with the existence of a unitary power dilation for any contraction (a power dilation of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s0865005.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s0865006.png" /> is defined as an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s0865007.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s0865008.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s0865009.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s08650010.png" />); a compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s08650011.png" /> is spectral for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s08650012.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s08650013.png" /> has a normal power dilation with spectrum in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s08650014.png" />. The minimal radius of the circle which is a spectral set for every contraction in a Banach space is equal to one.
+
for any polynomial $p(z)$. Thus, the unit circle is a spectral set for any contraction (an operator whose norm does not exceed one) on a Hilbert space (von Neumann's theorem). This result is closely connected with the existence of a unitary power dilation for any contraction (a power dilation of an operator $A$ on a Hilbert space $H$ is defined as an operator $A_1$ on a Hilbert space $H_1\supset H$ such that $P_HA_1^n|_H=A^n$, $n\in\mathbf Z^+$); a compact subset $S\subset\mathbf C$ is spectral for $A$ if and only if $S$ has a normal power dilation with spectrum in $\partial S$. The minimal radius of the circle which is a spectral set for every contraction in a Banach space is equal to one.
  
A spectral set, or set of spectral synthesis, for a commutative Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s08650015.png" /> is a closed subset of the space of maximal ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s08650016.png" /> which is the hull of exactly one closed ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s08650017.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086500/s08650018.png" /> is the group algebra of a locally compact Abelian group, spectral sets are also called sets of harmonic synthesis.
+
A spectral set, or set of spectral synthesis, for a commutative Banach algebra $\mathfrak A$ is a closed subset of the space of maximal ideals $\mathfrak M_{\mathfrak A}$ which is the hull of exactly one closed ideal $I\subset\mathfrak A$. In the case when $\mathfrak A$ is the group algebra of a locally compact Abelian group, spectral sets are also called sets of harmonic synthesis.
  
 
====References====
 
====References====

Latest revision as of 15:46, 29 December 2018

A spectral set of an operator $A$ on a normed space is a subset $S\subset\mathbf C$ such that

$$\|p(A)\|\leq\sup\{|p(z)|:z\in S\}$$

for any polynomial $p(z)$. Thus, the unit circle is a spectral set for any contraction (an operator whose norm does not exceed one) on a Hilbert space (von Neumann's theorem). This result is closely connected with the existence of a unitary power dilation for any contraction (a power dilation of an operator $A$ on a Hilbert space $H$ is defined as an operator $A_1$ on a Hilbert space $H_1\supset H$ such that $P_HA_1^n|_H=A^n$, $n\in\mathbf Z^+$); a compact subset $S\subset\mathbf C$ is spectral for $A$ if and only if $S$ has a normal power dilation with spectrum in $\partial S$. The minimal radius of the circle which is a spectral set for every contraction in a Banach space is equal to one.

A spectral set, or set of spectral synthesis, for a commutative Banach algebra $\mathfrak A$ is a closed subset of the space of maximal ideals $\mathfrak M_{\mathfrak A}$ which is the hull of exactly one closed ideal $I\subset\mathfrak A$. In the case when $\mathfrak A$ is the group algebra of a locally compact Abelian group, spectral sets are also called sets of harmonic synthesis.

References

[1] J. von Neumann, "Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes" Math. Nachr. , 4 (1951) pp. 258–281
[2] V.E. Katznelson, V.I. Matsaev, Teor. Funkts. Funktsional. Anal. i Prilozhen. , 3 (1966) pp. 3–10


Comments

Cf. also Spectral synthesis.

How to Cite This Entry:
Spectral set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_set&oldid=12656
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