On the rotational invariance for the essential spectrum of λ-Toeplitz operators

Let λ   be a complex number in the closed unit disc, and HH be a separable Hilbert space with the orthonormal basis, say, E={en:n=0,1,2,…}E={en:n=0,1,2,…}. A bounded operator T   on HH is called a λ-Toeplitz operator   if 〈Tem+1,en+1〉=λ〈Tem,en〉〈Tem+1,en+1〉=λ〈Tem,en〉 (where 〈⋅,⋅〉〈⋅,⋅〉 is the inner product on HH). The L2L2 function φ∼∑aneinθφ∼∑aneinθ with an=〈Te0,en〉an=〈Te0,en〉 for n⩾0n⩾0 and an=〈Ten,e0〉an=〈Ten,e0〉 for n<0n<0 is, on the other hand, called the symbol of T. Let us denote T   by Tλ,φTλ,φ. It can be verified directly that Tλ,φTλ,φ is an “eigenoperator” associated with the eigenvalue λ   for the following map on B(H)B(H):ϕ(A)=S⁎AS,A∈B(H), where S   is the unilateral shift defined by Sen=en+1Sen=en+1, n=0,1,2,…n=0,1,2,…. In an earlier joint work, the author used a result of M.T. Jury regarding the Fredholm theory of a certain Toeplitz-composition C⁎C⁎-algebra to show that if φ   is in the class C1C1 and if |λ|=1|λ|=1 has finite order, then the essential spectrum of Tλ,φTλ,φ is “rotationally invariant” with respect to λ, i.e.,σe(Tλ,φ)=λσe(Tλ,φ).σe(Tλ,φ)=λσe(Tλ,φ). In this paper, we prove that the C1C1 restriction for the symbol φ in the above result can be dropped entirely, and the equation actually holds for any φ   in L∞L∞ and any |λ|=1|λ|=1. It turns out that the key for removing the assumption on the smoothness of φ   depends only on the definition of Tλ,φTλ,φ and some very elementary properties of S   as a Fredholm operator. The applications of this phenomenon for σe(Tλ,φ)σe(Tλ,φ) include a generalization of A. Wintnerʼs result on the spectra of Toeplitz operators with bounded analytic symbols.