In
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operat ...
, a set
is said to be a spectral set for a (possibly unbounded) linear operator
on a Banach space if the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of
is in
and
von-Neumann's inequality holds for
on
- i.e. for all rational functions
with no
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
on
:
This concept is related to the topic of analytic functional calculus
of operators. In general, one wants to get more details about the operators constructed from functions with the original operator as the variable.
For a detailed discussion between Spectral Sets and von Neumann's inequality, see.
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