On the existence of bases of class Cp of the kernel and the image of a matrix function

AbstractThis paper deals with the existence of bases of class Cp of the kernel and the image of a rectangular matrix function of q real variables. Fundamental results on this subject, extended from matrix functions to operator functions, have already been established by I. C. Gohberg and J. Leiterer, by means of general properties of cocycles in algebras of operator functions. This paper was prepared independently, and new elementary methods were found, using three tools of general interest: The first is a condition which guarantees the existence of global solutions defined on Rq of problems possessing local solutions. The second is the property that every subset of Cm×n consisting of all the matrices of the same rank is analytically arcwise connected. The third is the smoothness of the Moore-Penrose inverse of a matrix function of class Cp. Applications to equivalences of class Cp, to linear equations on matrix functions of class Cp, to QR decomposition of class Cp, and to simultaneous unitary diagonalization of class Cp of a basis family of projector functions are expounded. Several results are established in the more general case of matrix functions defined on an open subset of a Banach space. This paper suggests problems on the generalization of its results from matrix functions to operator functions