Generalized inverses of tridiagonal operators

Abstract

Let H be a Hilbert space with { e n : n ∈ N } as an orthonormal basis. Let T : H → H be a bounded linear operator defined by Te n = e n - 1 + λ sin ( 2 nr ) e n + e n + 1 , where λ is real and r is a rational multiple of π. In this short note it is established that the Moore–Penrose inverse of T is not bounded. We also show that the same conclusion is valid for a few related classes of operators.