Normal operators in L2 spaces and almost everywhere convergence

In this thesis, we investigate whether the degrees of spectral operators acting in L^p spaces converge in different topologies. Let us denote (Ω,Σ,μ) a measurement space, and L^2(Ω,Σ,μ) the space of integrable functions in the second order Lebesgue sense. In previous studies, the convergence of the degrees of the normal operators acting in these spaces according to the strong topology and the convergence almost everywhere were investigated. In the thesis, we consider the normal N operator acting in L^2 (Ω,Σ,μ) spaces and the ϕ(N) operator function connected to this operator. Here, ϕ is a continuous function defined in the spectrum of the N operator. Next, we examine the convergence and almost-everywhere convergence of {ϕ(N)^k f}_(k∈N) sequences, 〖f∈L〗^2 (Ω,Σ,μ). In the other part of the thesis, we consider spectral operators, which are the generalizations of normal operators in Hilbert spaces in Banach spaces. We examine the convergence and almost-everywhere convergence of {T^n f}_(n∈N) sequences in these spaces, with the T operator the spectral operator acting on L^p (Ω,Σ,μ) spaces. Keywords: Birkhoff's theorem, Contraction operators Ergodic theorems, Halmosh's theorem, İnvariant vectors, Normal vectors, Nearly-convergence in L^2-space, Unitary operators, von-Neumann's theorem,