Some new inequalities for the number of operators

This master's thesis consists of four main headings. In the introduction, the reproducing kernel, the Berezin symbol and the history of inequalities are mentioned. In the resources section, previous studies on the Berezin number are included. In the basic concepts section, some terms and notations are included so that the concepts of Berezin symbol and Berezin number can be easily perceived. Then, in order to define the concept of Berezin number, some concepts such as Hilbert space with reproducing kernel, normalized reproducing kernel, Berezin symbol are discussed. Some features of the Berezin number are mentioned and examples are given. At the same time, some auxiliary theorems are given to prove the theorems. The mathematical theory of inequalities in general, and Hilbert-type inequalities in particular, and their applications have developed significantly over the last century. With the emergece of new ideas and evidence, new results and applications, studies are constantly being added to the subject of mathematical inequalities, which in turn improve the subject of inequalities. The Berezin symbol and the Berezin number have had wide application in the study of various problems of operator theory in Hilbert space with reproducing kernels, in quantum physics and in non-commutative geometry. In addition, the concepts of numerical radius of an operator and Berezin number have a comprehensive place due to their applications in the fields of differential equations, engineering, numerical analysis and quantum mechanics. In the last few years, significant work has been done to find the inverse force inequality for the Berezin number of operators. Many researchers have used Hardy-Hilbert type inequalities to do this. In the main conclusion ∑_(n=0)^∞▒∑_(m=0)^∞▒〖(a_m b_n)/(m+n+1)<π/√2〗 {∑_(n=0)^∞▒a_n^2 ∑_(n=0)^∞▒b_n^2 +(∑_(n=0)^∞▒a_n b_n )^2 }^(1/2) by making use of discrete type Hardy-Hilbert type inequalities, the theorem gives inequality for Berezin symbols of self-adjoint operators on Hilbert spaces with reproducing kernels, using positive continuous fuctions. Theorem 4.1.2. for the number of Berezins as an application of this theorem has been obtained. In theorem 4.1.3, Berezin number inequality is given for self-adjoint operators by using convex and continuous functions on Hilbert spaces with H=H(Ω) generating kernels, again making use of discrete type Hardy-Hilbert type inequality. In addition , the functions f and g are non-negative continuous functions on [0,├ ∞)┤ such that f(t)g(t)=t ,t≥0 . Using the auxiliary theorems given in the basic concepts section, some improved uper bounds have been obtained (Theorem 4.1.5, Theorem 4.1.8, Theorem 4.1.10)