Bounds for the spectral abscissa of an element in a Banach algebra

Abstract

For an arbitrary element x with spectrum sp(x) in a Banach algebra with identity e ≠ 0 we define the upper (lower) spectral abscissa$\mathop {\sigma + (x)}\limits_{( - )} = \mathop {\max }\limits_{(\min )} \operatorname{Re} \lambda ,\lambda \in sp(x)$. With the aid of the spectral radius$\rho (x) = \mathop {\max }\limits_{\lambda \in sp(x)} \left| \lambda \right| = \mathop {\lim }\limits_{n \to + \infty } \parallel x^n {{1 - } \mathord{\left/ {\vphantom {{1 - } n}} \right. \kern-\nulldelimiterspace} n}$ we prove the following bounds: γ₋(x)⩽σ₋(x)⩽Γ₋(x)⩽₊(x)⩽σ₊(x)⩽γ₊(x), Γ_(±)(x)=(2δ_(±))⁻¹ (ρ ²_δ )_(±)−δ ²_(±) −ρ ²₀ )(δ_(±)≠0), γ_(±)(x)= (±)ρ_δ(±)−δ_(±), δ₊⩾0, δ₋⩽0 ρ ^δ_(±) = ρ(x+eδ_(±)). We mention a case where equality is achieved, some corollaries,and discuss the sharpness of the bounds: for every ɛ > 0 there is a δ: ¦δ¦ ≥ρ ²₀ /2ɛ, such that Δ: = ¦γ_(±) x−Γ_(±) x¦⩽ε and conversely, if the bounds are computed for some δ ≠ 0, then △ ≤ρ ²₀ /2 ¦δ¦. An example is considered.