A SHORT PROOF OF A THEOREM OF G. M. PETERSEN
PETERSEN ([5], theorem 3) gave sufficient and necessary conditions for a finite number of regular matrices to have no nontrivial singularity Sj(s) . In this paper the mentioned theorem is proved by functional analytical methods and without using the supposition that the matrices have no singularity S^ . AMS-Classification: 40D20, 40H05, 46A45 In the following we consider sequences χ = (]ί)]ί=ι matrices A = (a , ) . . with complex values. Let m , c ηκ η fκ—ι and Cq be the set of bounded sequences, convergent sequences respectively sequences converging to 0 . Given some matrix A and some sequence space Ε , we define E a := {x = (xk) I Ax := ( nkk)ñ=1 e x i s t s a n d A x € E } and call c A and m η c^ the domain respectively the bounded domain of A ; especially we use the notation c Q A := (cQ)A . I f A is regular (that means C £ C A and lim^x : = lim Ax = lim χ for any χ e c ) the term αο IIAxn. := IÏS I I a χ ι η—~ k=1 is defined for any χ 6 m .