Abstract/Overview
A closed densely defined operator H, on a Banach space X, whose spectrum is contained in R and satisfies
(z −H)−1
≤ c hziα |=z|β ∀ z 6∈ R (0.1) for some α , β ≥ 0; c > 0, is said to be of (α, β)−type R . If instead of (0.1) we have
(z −H)−1
≤ c |z|α |=z|β ∀ z 6∈ R, (0.2) then H is of (α, β)0−type R . Examples of such operators include self-adjoint operators, Laplacian on L1(R), Schro¨dinger operators on Lp(Rn) and operators H whose spectra lie in R and permit some control on
eiHt
. In this paper we will characterise the (α, β)−type R operators. In particular we show that property (0.1) is stable under dialation by real numbers in the interval (0,1) and perturbation by positive reals. We will also show that is H is of (α, β)−type R then so is H2.
O.</div>, < (2024). Operators with slowly growing resolvents towards the spectrum. Afribary. Retrieved from https://afribary.com/works/operators-with-slowly-growing-resolvents-towards-the-spectrum
O.</div>, <div>Ongati "Operators with slowly growing resolvents towards the spectrum" Afribary. Afribary, 04 Jun. 2024, https://afribary.com/works/operators-with-slowly-growing-resolvents-towards-the-spectrum. Accessed 16 Sep. 2024.
O.</div>, <div>Ongati . "Operators with slowly growing resolvents towards the spectrum". Afribary, Afribary, 04 Jun. 2024. Web. 16 Sep. 2024. < https://afribary.com/works/operators-with-slowly-growing-resolvents-towards-the-spectrum >.
O.</div>, <div>Ongati . "Operators with slowly growing resolvents towards the spectrum" Afribary (2024). Accessed September 16, 2024. https://afribary.com/works/operators-with-slowly-growing-resolvents-towards-the-spectrum