Approximate point spectrum of a weighted shift

Abstract

Notation. If T is a Hilbert space operator, let A(T) denote its spectrum, fl(T) its approximate point spectrum, flo(T) its point spectrum, r(T) its compression spectrum, m(T) its lower bound (i.e., inf{ lTxll/llxll: x7&0}), and r(T) its spectral radius. Let i(T) denote supn m(Tn)ln, which equals limn m(Tn)1n. Let R denote a weighted right shift on 11, defined by Ren=snen +I , where (en) is an orthonormal basis of 12, n = 1, 2, .... Let L denote its adjoint, a weighted left shift. Let B denote a weighted two-sided shift on 12, defined by Ben=sSnen+1 n=0, ? 1, ? 2, .. ., (en) here being an orthonormal basis of 12. If B has purely nonzero weights (sn), then let