Meromorphic Factorization For Matrix Functions And Applications

This paper mainly studies two classes special group of exponentials of nilpotent matrix functionand the relations between meromorphic factorization and generalized boundary value problem.Firstly, in this paper, it selects a class group of exponentials of nilpotent matrix function withindex two and its element with the sructure of quotient of two quadratic polynomials. It studies thecondition of existence for the group relative to L p()(p1)space. The necessary and sufficientconditions about existence of canonical factorizantion and the corresponding nature of Toeplitzoperators are obtained. Under the suitable normalized conditions it gets explicitly meromorphicfactorization, and by means of separating role method obtain significant factor of canonicalfactorization. On this basis, It gains its explicit Wiener-Hopf factorization factor.Secondly, by seeking to a simple33nilpotent matrix similar to general nilpotent matrix andusing Riemann-Hilbert technique. It obtains the solutions to Riemann-Hilbert problem withcoefficient matrix replaced by simply group of exponentials of nilpotent matrix function, analysehow to select appropriate three solutions for the problem in order to get conveninently canonicalfactorization. Furthermore，It introduces a new Toeplitz operator and prove that commutatorToeplitz operator is compact, by means of meromorphic factorization and discussion of so lutionto some linear systems it ultimately gets explicitly meromorphic factorization and conditions ofWiener-Hopf factorization.Lastly, it discusses the relationship between generalized boundary value problems andmeromorphic factorization. It founds solving the problem is equivalent to solving correspondingfactorization of coefficient matrix function on vector Riemann-Hilbert problem. It selects a class ofcoefficient matrix function and researches necessary and sufficient condition to meet canonicalfactorization and non-canonical factorization relative to C (R)space. By combining theRiemann-Hilbert problem with solving canonical factorization, It obtains significant factor ofmeromorphic factorization. It uses the pole separation method to obtain canonical factorization byiteration.subsequently obtain solutions to the generalized boundary value problem.