Let \((S, {\mathcal {B}}, m)\) be a finite measure space. In this paper we show that every bounded linear operator T from \(L^{p_{1}}(S)\) into \(L^{p_{2}}(S)\) is an S-operator (or a generalized pseudo-differential operator) with the symbol \(\sigma \) for some \( 1\le \alpha<p_{1}, p_{2}<\beta \le \infty \). We make use of this symbolic representation to study various functional analytical properties of T. First, we present necessary and sufficient conditions for a function to be the symbol of the adjoint of T in terms of the symbol of T. Then, we give necessary and sufficient conditions to guarantee that the bounded linear operators on \(L^p(S)\) posses a particular complex number (function) as its eigenvalue (eigenfunction). As an application, we obtain necessary and sufficient conditions on the symbols to ensure that corresponding bounded linear operators on \(L^{2}(S)\) are compact, self-adjoint, or compact self-adjoint. Lastly, we give a result concerning to factorization of compact operators.

令\（（S，{\ mathcal {B}}，m）\）为有限度量空间。在本文中，我们证明了从\（L ^ {p_ {1}}（S）\）到\（L ^ {p_ {2}}（S）\）的每个有界线性算子T都是S-算子（或某个\（1 \ le \ alpha <p_ {1}，p_ {2} <\ beta \ le \ infty \）的符号\（\ sigma \）的广义伪微分算子）。我们利用这种符号表示来研究T的各种功能分析性质。首先，我们介绍必要和充分条件的函数是的伴随的符号Ť中的符号方面Ť。然后，我们给出必要和充分的条件，以保证\（L ^ p（S）\）上的有界线性算子具有特定的复数（函数）作为其特征值（特征函数）。作为一种应用，我们在符号上获得了必要的充分条件，以确保\（L ^ {2}（S）\）上的相应有界线性算子是紧凑的，自伴随的或紧凑的自伴随的。最后，给出关于紧算子分解的结果。