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

Let λ be a complex number in the closed unit disc, and H be a separable Hilbert space with the orthonormal basis, say, E = { e n : n = 0 , 1 , 2 , … } . A bounded operator T on H is called a λ-Toeplitz operator if 〈 T e m + 1 , e n + 1 〉 = λ 〈 T e m , e n 〉 (where 〈 ⋅ , ⋅ 〉 is the inner product on H ). The L 2 function φ ∼ ∑ a n e i n θ with a n = 〈 T e 0 , e n 〉 for n ⩾ 0 and a n = 〈 T e n , e 0 〉 for n < 0 is, on the other hand, called the symbol of T. Let us denote T by T λ , φ . It can be verified directly that T λ , φ is an “eigenoperator” associated with the eigenvalue λ for the following map on B ( H ) : ϕ ( A ) = S ⁎ A S , A ∈ B ( H ) , where S is the unilateral shift defined by S e n = e n + 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 C 1 and if | λ | = 1 has finite order, then the essential spectrum of T λ , φ is “rotationally invariant” with respect to λ, i.e., σ e ( T λ , φ ) = λ σ e ( T λ , φ ) . In this paper, we prove that the C 1 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 T λ , φ and some very elementary properties of S as a Fredholm operator. The applications of this phenomenon for σ e ( T λ , φ ) include a generalization of A. Wintnerʼs result on the spectra of Toeplitz operators with bounded analytic symbols.