Title:

Structure of scalartype operators on Lp spaces and wellbounded operators on Hilbert spaces

It is known that every scalartype spectral operator on a Hilbert space H is similar to a multiplication operator on some L^{2} space. The purpose of the main theorem in Chapter 2 of this thesis is to show that every scalartype spectral operator on an L^{1} space whose spectral measure has finite multiplicity is similar to a multiplication operator on the same L^{1} space. Then we prove a similar result for scalartype spectral operators on L^{p} (Ω, S_{Ω}, m), p ≠ 2, 1 < p < ∞, with spectral measure E(^{.}) of finite uniform multiplicity provided an extra condition is satisfied. Also, we give conditions that make a scalartype spectral operator on L^{2}(Ω, S_{Ω}, m) similar to a multiplication operator on the same L^{2}(Ω, S_{Ω}, m). In 1954, Dunford proved that a bounded operator T on a Banach space X is spectral if and only if it has the canonical decomposition T = S +Q, where S is a scalartype operator and Q is a quasinilpotent operator which commutes with S. In Chapter 3, we prove that any wellbounded operator T on a Hilbert space H has the form T = A + Q, where A is a selfadjoint operator and Q is a quasinilpotent operator such that AQ  QA is quasinilpotent. Then we prove that a trigonometrically wellbounded operator T on H can be decomposed as T = U(Q + I) where U is a unitary operator and Q is quasinilpotent such that UQ = QU is also quasinilpotent. In Chapter 4 we prove that an ACoperator with discrete spectrum on H can be decomposed as a sum of a normal operator N and a quasinilpotent Q such that NQ  QN is quasinilpotent. However, the converse of each of the last three theorems is not true in general. In the final chapter we introduce a new class of operators on L^{2}([a,b]) which is larger than the class of wellbounded operators on L^{2}([a,b]) and we call them operators with an AC_{2}functional calculus. Then we give an example of an operator with an AC_{2}functional calculus on L^{2}([0,1]) which can be decomposed as a sum of a selfadjoint operator and a quasinilpotent. We also discuss the possibility of decomposing every operator T with an AC_{2}functional calculus on L^{2}([a,b]) into the sum of a selfadjoint operator A and a quasinilpotent operator Q such that AQ  QA is quasinilpotent.
