ON THE SPECTRUM OF THE UPPER TRIANGULAR DOUBLE BAND MATRIX U(a0, a1, a2; b0, b1, b2) OVER THE SEQUENCE SPACE c

The upper triangular double band matrix U(a(0), a(1), a(2); b(0), b(1), b(2)) is defined on a Banach sequence space by U(a(0), a(1), a(2); b(0), b(1), b(2)) (x(n)) = (a(n)x(n) + b(n)x(n)+1)(n-0)(infinity) where a(x) = a(y), b(x) = b(y) for x equivalent to y(mod3). The class of the operator U(a(0), a(1), a(2); b(0), b(1), b(2)) includes, in particular, the operator U (r, s) when a(k) - r and b(k) - s for all k epsilon N, with r, s epsilon R and s not equal 0. Also, it includes the upper difference operator; a(k) = 1 and b(k) = -1 for all k epsilon N. In this paper, we completely determine the spectrum, the fine spectrum, the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator U(a(0), a(1), a(2); b(0), b(1), b(2)) over the sequence space c.

Automated summary

This paper investigates various spectral properties of the upper triangular double band matrix in the sequence space, including spectrum, fine spectrum, approximate point spectrum, defect spectrum, and compression spectrum.