Date of Award

2018

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

Anton Schep

Abstract

In this dissertation we investigate the algebra numerical range defined by the Banach algebra of regular operators on a Dedekind complete complex Banach lattice, i.e., V (Lr(E), T) = {Φ(T) : Φ ∈ Lr(E)∗, ||Φ|| = 1 = Φ(I)}. For T in the center Z(E) of E we prove that V (Lr(E), T) = co(σ(T)). For T ⊥ I we prove that V (Lr(E), T) is a disk centered at the origin. We then consider the part of V (Lr(E), T) obtained by restricting ourselves to positive states Φ ∈ Lr(E)∗. In this case we show that we get a closed interval on the real line. Next we consider the problem of characterizing the linear maps on Lr(E) which preserve V (Lr(E), T). For this we first describe the regular states on Lr(E), in particular for the case E = `p(n) for 1 ≤ p ≤ ∞. This description allows us to show that any map Ψ on Lr(`p(n)) preserving V (Lr(`p(n)), T) for all T ∈ Lr(`p(n)) is of the form Ψ(T) = U ∗ (PtQTP) where U consists of elements of modulus 1, (∗) represents Hadamard multiplication, P is a permutation, and Q is a map that permutes off-diagonal entries of T. Furthermore, special conditions are given for Q for the cases p = 1, p = ∞ and p = 2. Finally, some extensions of these results to more general finite dimensional Banach lattices and infinite dimensional ` p’s are considered.

Included in

Mathematics Commons

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