The numerical range of a tridiagonal operator

Let r be a real number and A   a tridiagonal operator defined by Aej=ej−1+rjej+1Aej=ej−1+rjej+1, j=1,2,…j=1,2,…, where {e1,e2,…}{e1,e2,…} is the standard orthonormal basis for ℓ2(N)ℓ2(N). Such tridiagonal operators arise in Rogers–Ramanujan identities. In this paper, we study the numerical ranges of these tridiagonal operators and finite-dimensional tridiagonal matrices. In particular, when r=−1r=−1, the numerical range of the finite-dimensional tridiagonal matrix is the convex hull of two explicit ellipses. Applying the result, we obtain that the numerical range of the tridiagonal operator is the square{z∈C:−1⩽R(z)⩽1,−1⩽ℑ(z)⩽1}\{1+i,1−i,−1+i,−1−i}.