View source for Spectral set

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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 $S\in\mathbf C$ 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. von Neumann,   "Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes"  ''Math. Nachr.'' , '''4'''  (1951)  pp. 258–281</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.E. Katznelson,   V.I. Matsaev,   ''Teor. Funkts. Funktsional. Anal. i Prilozhen.'' , '''3'''  (1966)  pp. 3–10</TD></TR></table>



====Comments====
Cf. also [[Spectral synthesis|Spectral synthesis]].