### Abstract:

Since its inception, Fredholm theory has become an important aspect of
spectral theory. Among the spectra arising within Fredholm theory is the
Weyl spectrum which has been intensively studied by several authors, both
in the operator case and in the general situation of Banach algebras.
The Weyl spectrum of a bounded linear operator T on a Banach space
X is the set T
K∈K(X) σ(T + K), where σ(T) denotes the spectrum of T and
K(X) the closed ideal of all compact operators on X. A recent result by E.
A. Alekhno shows that, if “Banach space" is replaced by an arbitrary complex
Banach lattice E, then the Weyl spectrum of T on E can be made more
precise, and takes on the form T
0≤K∈K(E) σ(T + K).
By an ordered Banach algebra (OBA) we mean a complex unital Banach algebra
A containing an algebra cone; that is, a subset C which contains the unit
of A and is closed under addition, multiplication and positive scalar multiplication.
As is well-known, the algebra of all bounded linear operators on
a complex Banach lattice is an important example of an OBA.
If A denotes an arbitrary OBA with algebra cone C, B a Banach algebra and
T : A → B a homomorphism with N(T) = {a ∈ A : Ta = 0} indicating the
null space of T, then the Weyl spectrum T
c∈N(T) σ(a + c) of a ∈ A is in general
strictly contained in the set T
c∈C∩N(T) σ(a + c) — see Example 4.1.13.
As a result of this, we investigate the latter set, which we shall refer to as
the upper Weyl spectrum of a ∈ A. In this work the concept of the upper Browder
spectrum of a will also be introduced and results related to these spectra
and the underlying sets of elements on which these spectra are defined will
be given.
This thesis aims to present initial steps taken in the effort of unifying the
theory of positivity in OBAs with the general Fredholm theory in Banach
algebras.