A Constructive Version of the Spectral Mapping Theorem

Abstract

The spectral mapping theorem in a unital Banach algebra is examined for its constructive content. Mathematics Subject Classification: 03F60, 46S30.Let B be a (complex) Banach algebra that is unital -that is, has an identity e. Let a be an element of B, R a = {λ ∈ C : a − λe has a two-sided inverse}, the resolvent set of a, and σ a = {λ ∈ C : (∀λ ∈ R a ) (λ = λ )}, the spectrum of a. The Classical Spectral Mapping Theorem says that if p is a monic polynomial of degree ≥ 1, then σ p(a) = p(σ a ). The classical proof of this theorem hangs on the decomposition of p(z) − λ, where λ ∈ C, into linear factors, and the observation that p(a) − λe has no inverse if and only if at least one of its corresponding factors has no inverse. However, this proof is not valid in constructive mathematics -that is, mathematics based on intuitionistic logic (which leads to theorems valid in a wider range of models, including recursive and other computational ones, than those proved with classical logic). Our main aims in this paper are (i) to find a constructive proof of the inclusion p(σ a ) ⊂ σ p(a) , and (ii) to recover constructively as much as is possible of the reverse inclusion. In order to fulfil these aims, we need relatively little technical background in constructive mathematics. Much of what is needed will be stated in context below; for more detailed information on constructive mathematics, see [1,2,4,7]. The first half of the spectral mapping theoremOur proof of the inclusion p(σ a ) ⊂ σ p(a) will be based on some elementary constructive semigroup theory. First, we define an inequality on a set X to be a binary relation = on X with the following two properties:x = y ⇒ ¬(x = y), x = y ⇒ y = x.

Semigroups with apartness

“…Within BISH, the history of constructive semigroups with an inequality began recently. For example, in proving a constructive version of the Spectral Mapping Theorem, Bridges and Havea [6], used such semigroups.…”

“…As we have already pointed out, our primary objective is to prove isomorphism theorems for semigroups with apartness. Our work is based on applications of ideas and notions coming from [6,10]. The paper is organized in the following way.…”

Theory of constructive semigroups with apartness -- foundations, development and practice

“…We will start with constructive analysis. The proof of one of the directions of the constructive version of the Spectral Mapping Theorem is based on some elementary constructive semigroups with inequality techniques, [11]. It is also worth mentioning the applications of commutative basic algebraic structures with tight apartness within the automated reasoning area, [15].…”

“…For the classical case see [36], [41]. Examples of applications of these theoretical concepts can be found in [2], [11], [15] [22], [40].…”

Basic notions of (constructive) semigroups with apartness

“…Within BISH the history of constructive semigroups with an inequality began recently. For example, in proving a constructive version of the Spectral Mapping Theorem, Bridges and Havea [8] have used such constructive semigroups. Isomorphism Theorems for semigroup with apartness have been proved in [11].…”