Spectral set




In operator theory, a set X⊆C{displaystyle Xsubseteq mathbb {C} }Xsubseteq {mathbb {C}} is said to be a spectral set for a (possibly unbounded) linear operator T{displaystyle T}T on a Banach space if the spectrum of T{displaystyle T}T is in X{displaystyle X}X and von-Neumann's inequality holds for T{displaystyle T}T on X{displaystyle X}X - i.e. for all rational functions r(x){displaystyle r(x)}r(x) with no poles on X{displaystyle X}X


‖r(T)‖≤‖r‖X=sup{|r(x)|:x∈X}{displaystyle leftVert r(T)rightVert leq leftVert rrightVert _{X}=sup left{leftvert r(x)rightvert :xin Xright}}leftVert r(T)rightVert leq leftVert rrightVert _{{X}}=sup left{leftvert r(x)rightvert :xin Xright}

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.








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