Theory of Linear Operator is intended to provide the basics regarding the mathematical key features of unbounded operators to readers that are not familiar with such technical aspects. In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. The aim of first chapter is to give a general method to provide accurate estimates of the constants (A(??), ??) satisfying the inequalities. The purpose of second chapter is to collect some of these tools in a common toolbox for the study of general classes of functional equations by introducing notion of a good linear operator, which satisfies certain regularity conditions in terms of value distribution theory. In third chapter, we introduce the notion of weighted A-statistical convergence of a sequence, where A represents the non regular matrix. We also prove the Korovkin approximation theorem by using the notion of weighted A-statistical convergence. Fourth chapter highlights the fact that the criteria acquire purely geometrical form in the more general case of a spectral operator of scalar type (scalar operators) in a complex Banach space. Fifth chapter focuses on a class of meromorphic P-valent starlike functions involving certain linear operators. In sixth chapter, we investigate a perturbation of the Drazin inverse AD of a closed linear operator A; the main tool for obtaining the estimates is the gap between subspaces and operators. The seventh chapter focuses on a pair of sequences of linear positive operators. In eighth chapter, we study some ideal convergence results of ??-positive linear operators defined on an appropriate subspace of the space of all analytic functions on a bounded simply connected domain in the complex plane. Ninth chapter reveals on completely positive linear operators for Banach spaces. In tenth chapter, we show pointwise approximation of functions from Lp(W)Β by linear operators of their Fourier series. Eleventh chapter introduces some new subclasses of multivalent functions and investigate various inclusion properties of these subclasses. Some radius problems are also discussed. Twelfth chapter proposes on an alternative notion of orbit for ??-linear operators that is inspired by difference equations. In thirteenth chapter, we introduce the definition of linear relative ??-width and find estimates of linear relative ??-widths for linear operators preserving the intersection of cones of ??-monotonicity functions. Two important techniques to achieve the Jackson type estimation by Kantorovich type positive linear operators in spaces are introduced in fourteenth chapter, and three typical applications are given. In fifteenth chapter, we consider some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence. Properties of (S) and (Gs) for bounded linear operators are explained in sixteenth chapter. Seventeenth chapter describes on asymptotic behavior of the iterates of positive linear operators. The aim of eighteenth chapter is to construct a class of linear operators in more general conditions and last chapter discloses on approximation of signals (functions) belonging to the weighted W(Lp,Ξ(T))-class by linear operators.
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