DOI: https://doi.org/10.36719/2663-4619/125/206-214
Rugayya Nuriyeva
AzΙrbaijan 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 π be a complex Banach space and let π∗ be its dual. A linear operator π: π → π is said to be power bounded if
π π’π π≥0 βππβ < ∞.
Let π» and π respectively, be the open unit disc and the unit circle in the complex plane. As usual, by π(π) we denote the spectrum of π. If π is a power bounded operator, then clearly, ππ’(π) ⊆ π»
The set ππ’(π) β π(π)βπ will be called unitary spectrum of π. We prove that if π is a power bounded operator on reflexive Banach space with ππ’(π) = {1}, then, there is a projection π such that
lim π→∞ β¨πππ₯, π₯∗β© = β¨ππ₯, π₯∗β© for all π₯ ∈ π, π₯∗ ∈ π∗.
Keywords: mean ergodic theorem, locally compact group, power-bounded mean