Some results of n×n block matrices of linear relations in Banach spaces

Abstract

In this paper, we consider in the product of Banach spaces X1⊗X2⊗...⊗Xn, the n×n-block matrices of linear relations in the form,
M :=
      
A1,1 ... A1,n
. . .
... . . .
An,1 ... An,n
      
,
where the entries of the matrix are in general unbounded linear relations and satisfy the following conditions: Ai,j : Xj → Xi, ∀i,j ∈{1,...,n}. Studying the spectral properties of M, it is natural to take stability of closedness for this matrix. So, we have to study this problem in the present paper. In addition, we show under some suitable conditions that M is a Fredholm linear relation. Keywords: Closed linear relation, n×n matrix linear relation, Fredholm linear relation.