Spectral set

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

[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.