This thesis is based on some problems posed in \cite{BaHa} and
\cite{NazarovShapiro}, and in some cases, we find the
solutions and investigate their properties.
Chapter 1 presents the prerequisites for the rest of the chapters.
These materials can be found with more details in the books cited in
the references. In the last section, we find a concrete formula for
the adjoint of the \mathbb{U}-automorphic composition operators, acting on the monomial basis for $H^2(\mathbb{U})$; which is used as one of the major tools later in Chapter 3.
In Chapter 2, we introduce the class of Parametric Toepliz operators
(PTOs) as the solutions to the operator-equation
$T_{e^{-\imath\theta}}XT_{e^{\imath\theta}}=\lambda X$ (for a given
complex number \lambda) on
\mathscr{B}\big(H^{2}(\partial\mathbb{U})\big), where
$T_{e^{\imath\theta}}$ and $T_{e^{-\imath\theta}}$ are unilateral
forward and backward shifts respectively. This operator-equation was
first introduced in \cite{BaHa}, but not studied. We investigate the
algebraic and operator-theoretic properties of PTOs. In most cases,
it is shown that PTOs behave in the same way as the classical
Toeplitz operators on $H^2(\partial\mathbb{U})$.
In the first section of Chapter 3, after introducing the notions of
asymptotic Toeplitzness and asymptotic Hankelness, which were first
introduced in \cite{BaHa} and \cite{Feintuch} respectively, we study
some of their algebraic properties along with a distance formula. In
the next section, building on techniques developed by
Nazarov-Shapiro \cite{NazarovShapiro} and the adjoint formula given
in Chapter 1, we show that the adjoint of a composition operator,
induced by a unit disk-automorphism, is not strongly asymptotically
Toeplitz. This result answers Nazarov-Shapiro's question in
\cite{NazarovShapiro}. In the other direction, we also study the
asymptotic Toeplitzness of the product of a composition operator
with its adjoint, and Toeplitz-Composition operators.