The spectrum of certain lower triangular matrices as operators on the l_p spaces

James D. Stafney

Vol. 42 (1972), No. 2, 515–525

DOI: 10.2140/pjm.1972.42.515

Abstract

In this paper we compute the spectrum of the lower triangular matrices A_a = (e_m,n), where c_mn = (n + 1)^a∕(m + 1)^a+1, m ≧ n ≧ 0,a is real and the corresponding operator on l_p is bounded (see 4.1). This result and other lemmas are used to determine the spectrum of lower triangular matrices p(n)lq(m), m ≧ n ≧ 0 as operators on l_p where p is a monic polynomial of degree a,q is a monic polynomial of degree a + 1 and q(m)≠0 for m = 0,1,⋯ . The spectrum is the diagonal together with the set C_a−p − 1₊₁ when a − p⁻¹ + 1 > 0, where C_b = {λ;|λ − (2b)⁻¹|≦ (2b)⁻¹} (see 4.3).

Mathematical Subject Classification 2000

Primary: 47A10

Secondary: 47B35

Milestones

Received: 1 April 1971

Revised: 20 December 1971

Published: 1 August 1972