Page 100 - Textos de Matemática Vol. 44
P. 100
90 M. KARAMANLIS AND P. J. PSARRAKOS
Then,foranyδ>0,thereisaλδ ∈Csuchthat
∥χ−λδψ∥ < 1−ε2 ∥ψ∥|μ0 −λδ|+δ.
√
0+,itfollowsthat inf ∥χ−λψ∥−√1−ε2∥ψ∥|μ0−λ| =0.
1−ε2 ∥ψ∥|μ0 −λδ| is nonnegative, as δ −→ For the converse, we assume that inf ∥χ − λψ∥ − √1 − ε2 ∥ψ∥ |μ0 − λ| =
Since the quantity ∥χ−λδψ∥− λ∈C
λ∈C
0 and μ0 ∈ Int Fε (χ;ψ) . Then, by (2.3), there exists a real ρ > 0 such that
∥·∥
∥χ−λψ∥ D(μ0,ρ)⊆Int D λ,√1−ε2∥ψ∥ , ∀λ∈C.
As a consequence,
∥χ−λψ∥− 1−ε2∥ψ∥|μ0 −λ| > 1−ε2∥ψ∥ρ > 0,
This means that
inf ∥χ−λψ∥− 1−ε2 ∥ψ∥ |μ0 −λ| > 0 λ∈C
∀λ∈C.
which is a contradiction.
(ii)Foreveryδn = 1 (n∈N),thereisaλn ∈Csuchthat
or or
n
∥χ−λnψ∥ < 1−ε2 ∥ψ∥ |μ0 −λn|+δn,
|∥χ∥−∥λnψ∥| < 1−ε2 ∥ψ∥ |μ0 −λn|+ 1 , n
|λn|∥ψ∥−∥χ∥ < 1−ε2 ∥ψ∥ (∥μ0∥+∥λn∥)+ 1 . n
Since ε > 0, one can verify that
√
|λn| < ∥ψ∥ 1−√1−ε2 ,
1−ε2 ∥ψ∥|μ0|+1
∥χ∥+
i.e., the sequence {λn}n∈N is bounded and has a converging subsequence
λkn −→ λ0. As a consequence, ∥χ−λknψ∥< 1−ε2∥ψ∥|μ0−λkn|+ 1, ∀n∈N,
and as n −→ +∞,
kn ∥χ − λ0ψ∥ ≤ 1 − ε2 ∥ψ∥ |μ0 − λ0| .
This inequality can hold only as an equality because μ ∈ Fε ∥·∥
proof is complete.
(χ;ψ), and the
