英語訳：Rを複素平面の連結領域とし、HをHilbert space...

2018/01/25 07:35

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R is a connected domain in the complex plane, H is a Hilbert space, and T(λ) (where λ belongs to R) is a closed operator on H with a non-empty resolvent set. The set of operators {T(λ)}_(λ∈R) is called an analytic family in the sense of Kato if it has the following properties.
For each element λ0 in R, there exists an element z0 in ρ(T(λ0)) such that for all λ in the vicinity of λ0, z0 is an element of ρ(T(λ)) and (T(λ)-z0)^(-1) is an operator-valued analytic function of λ near z0.
In other words, the set of operators {T(λ)}_(λ∈R) is analytically dependent on the parameter λ, and for each λ0, there exists a corresponding element z0 that satisfies certain conditions.

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2018/01/25 08:40 回答No.1

Let R be the connected region of complex planes, let H be Hilbert space, and let T(λ) (where λ is an element of R) be the closed operator on H with a resolvent set which is not an empty set. When it contains the following property, {T(λ)}_(λ∈R), an accumulation of operators, is called "analytic family in the sense of Kato." The element z0, corresponding ρ(T(λ0)), exists for each element λ0 of :R. For all λ sufficiently close to λ0, z0 is a element of ρ(T(λ)). (T(λ)-z0)^(-1), when close to z0, is an operator valued analytic function of λ.

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