由正常算子所决定的双线性泛函的极值定理

An intimum theorem Derived from the Bounded Normal Operator

本文用复Hilbert空间正常算子理论,导出了由正常算子所决定的双线性泛函的一条极值定理,并给出了它的应用。

In this paper, we obtain the following infimum theorem, Let H be a real Hilbert space and H~#=H+iH denote its complexification, Suppose that N_2H~#|→H~# isa bounded normal linear operator and denoted by the same letter N, the standard extension H~#|→H~# Spec (N)denotes the spectrum of the extension. Thcn we have inf{Re<Nu,u>:u∈H~#, ‖u‖=1} =-inf{<Nu, u>:u∈H,‖u‖=1} =inf {Re(λ):λ∈spec(N)} in particular, if A=circ (α_1,α_2……a_n) is a n×n real circlant matrix, then inf{<Ay, y>: y∈R^n ‖Y‖R^n=1} in{Re(sum from k=1 to n akZ^(k-1)):Z^n=1} Its application can be seen from the problem of ordingry differential equations.