The numerical range of a tridiagonal operator

Let r be a real number and A a tridiagonal operator defined by A e j = e j − 1 + r j e j + 1 , j = 1 , 2 , … , where { e 1 , e 2 , … } is the standard orthonormal basis for ℓ 2 ( 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 = − 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 } .