Norm Estimation Of Solutions To A Class Of Sylvester Equations

Sylvester equation comes from many practical problems and is an important mathematical research object.This thesis studies the upper norms and lower norms estimation of the solutions of matrix equations and operator equations of the form AX +XB = AC +DB.They have important applications in perturbation analysis of matrix and operator polar decomposition.First,we consider the case of the matrix.Under the premise that both A and B are positive definite matrices,we give several upper and lower bounds on the Frobenius norm of the solution X to this kind of matrix equation,and prove theoretically that the upper bounds obtained are sharper than those obtained in some literatures.Then we compare the size relations of the two new obtained lower bounds and get the necessary and su cient conditions for both of them to be positive.We also demonstrate the sharpness of the upper bound and the positivity of the lower bound by numerical tests and numerical examples,respectivelyThen,we consider the case of operators on separable Hilbert Spaces.Under the premise that all operators in the equation are Hilbert-Schmidt operators and A and B are positive definite operators.We study the upper bounds and lower bounds of the Schatten-2norm of the solution X to this kind of operator equation,and extend the results obtained in the matrix case in parallel.