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

Abstract

Let λ be a complex number in the closed unit disc, and H be a separable Hilbert space with the orthonormal basis, say, E={en:n=0,1,2,…}. A bounded operator T on H is called a λ-Toeplitz operator if 〈Tem+1,en+1〉=λ〈Tem,en〉 (where 〈⋅,⋅〉 is the inner product on H). The L2 function φ∼∑aneinθ with an=〈Te0,en〉 for n⩾0 and an=〈Ten,e0〉 for n<0 is, on the other hand, called the symbol of T. Let us denote T by Tλ,φ. It can be verified directly that Tλ,φ is an “eigenoperator” associated with the eigenvalue λ for the following map on B(H):ϕ(A)=S⁎AS,A∈B(H), where S is the unilateral shift defined by Sen=en+1, 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⁎-algebra to show that if φ is in the class C1 and if |λ|=1 has finite order, then the essential spectrum of Tλ,φ is “rotationally invariant” with respect to λ, i.e.,σe(Tλ,φ)=λσe(Tλ,φ). In this paper, we prove that the C1 restriction for the symbol φ in the above result can be dropped entirely, and the equation actually holds for any φ in L∞ and any |λ|=1. It turns out that the key for removing the assumption on the smoothness of φ depends only on the definition of Tλ,φ and some very elementary properties of S as a Fredholm operator. The applications of this phenomenon for σe(Tλ,φ) include a generalization of A. Wintnerʼs result on the spectra of Toeplitz operators with bounded analytic symbols.