Generalized inverses of tridiagonal operators

Let H   be a Hilbert space with {en:n∈N}{en:n∈N} as an orthonormal basis. Let T:H→HT:H→H be a bounded linear operator defined by Ten=en-1+λsin(2nr)en+en+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.