×

On bv-reversible matrices. (English) Zbl 0765.15016

Summary: S. Banach [Théorie des opérations linéaires (1932; Zbl 0005.20901)] stated that if \(A\) is a reversible matrix, then the system of equations \(Y_ n=A_ n(x)=\sum^ \infty_{k=0}a_{nk}x_ k\) has a unique solution given by \(x=vl+By\), where \(B=(b_{nk})\) is the unique right inverse of \(A\), \(By=\left(\sum^ \infty_{k=0}b_{nk}y_ k\right)^ \infty_{n=0}\), \(y\in c\), \(x\in c_ A\) and \(v=(v_ n)^ \infty_ 0\in\ell_ \infty\). M. S. Macphail [Proc. Am. Math. Soc. 5, 120-121 (1954; Zbl 0055.289)] showed that \(v\) need not belong to \(\ell_ \infty\) by giving a simple reversible matrix with \(v\) unbounded.
It is the purpose of this paper to extend Banach’s work on \(c\)-reversible matrices, to \(bv\)-reversible matrices and construct matrices which are \(bv\)-reversible matrices but not \(c\)-reversible; the first one with \(v\) bounded and the second one with \(v\) unbounded.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A06 Linear equations (linear algebraic aspects)
15A09 Theory of matrix inversion and generalized inverses
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
PDFBibTeX XMLCite