DOI: https://doi.org/10.36719/2663-4619/125/206-214
Rugayya Nuriyeva
Azərbaijan State Oil and Industry University
Master student
https://orcid.org/0009-0001-4183-5464
rugeyyenuriyeva3002@gmail.com
Power Bounded Operators on Reflexive Banach Spaces
Abstract
Let 𝑋 be a complex Banach space and let 𝑋∗ be its dual. A linear operator 𝑇: 𝑋 → 𝑋 is said to be power bounded if
𝑠𝑢𝑝 𝑛≥0 ‖𝑇𝑛‖ < ∞.
Let 𝔻 and 𝕋 respectively, be the open unit disc and the unit circle in the complex plane. As usual, by 𝜎(𝑇) we denote the spectrum of 𝑇. If 𝑇 is a power bounded operator, then clearly, 𝜎𝑢(𝑇) ⊆ 𝔻
The set 𝜎𝑢(𝑇) ≔ 𝜎(𝑇)⋂𝕋 will be called unitary spectrum of 𝑇. We prove that if 𝑇 is a power bounded operator on reflexive Banach space with 𝜎𝑢(𝑇) = {1}, then, there is a projection 𝑃 such that
lim 𝑛→∞ ⟨𝑇𝑛𝑥, 𝑥∗⟩ = ⟨𝑃𝑥, 𝑥∗⟩ for all 𝑥 ∈ 𝑋, 𝑥∗ ∈ 𝑋∗.
Keywords: mean ergodic theorem, locally compact group, power-bounded mean