Operator theory is a subject that studies the algebraic structure and properties of linear operators. This paper mainly studies the properties, classification and representation of linear operators and their applications in various mathematical problems. The development of operator theory is of great significance for understanding many mathematical phenomena, for example, Hilbert space and Banach space in functional analysis are based on operator theory.
Operator algebra is a subject that studies the algebraic structure of linear operators. This paper mainly studies the operation rules of linear operators such as addition, multiplication and power operation, and their applications in various mathematical problems. The development of operator algebra is of great significance for understanding many mathematical phenomena. For example, the concepts of C*- algebra and B*- algebra in functional analysis are all based on operator algebra.
In a word, operator theory and operator algebra occupy a very important position in the field of mathematics. They provide powerful tools and methods for understanding and solving many complex mathematical problems. With the development of science and technology, operator theory and operator algebra will still play an important role in the future.