Spectral set
A spectral set of an operator on a normed space is a subset
such that
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for any polynomial . 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
on a Hilbert space
is defined as an operator
on a Hilbert space
such that
,
); a compact subset
is spectral for
if and only if
has a normal power dilation with spectrum in
. 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 is a closed subset of the space of maximal ideals
which is the hull of exactly one closed ideal
. In the case when
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.
Spectral set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_set&oldid=43563