Hyponormality and spectra of Toeplitz operators

Abstract

Abstract. This paper concerns algebraic and spectral properties of Toeplitz operators Tϕ, on the Hardy space H 2 (T), under certain assumptions concerning the symbols ϕ ∈ L ∞ (T). Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at Tϕ, for each quasicontinuous ϕ. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.

Some Remarks on the Spectral Properties of Toeplitz Operators

“…Since T φ satisfies Weyl's theorem, by Theorem [1, Theorem 6.52] then Weyl's theorem holds for f (T φ ) if and only if the spectral theorem holds for σ w (T φ ). As observed before, the equivalence (i) ⇔ (iii) has been proved in [13].…”

“…For instance, the operator T φ in the Example 4.8, cannot be hyponormal, since hyponormality entails SVEP. Toeplitz operators with continuous symbol which are hyponormal have been also studied by Farenick and W. Y. Lee [13].…”

“…The case (i) of Theorem 4.9 applies in particular to the case where φ is a trigonometric polynomial φ(e iθ ) =: Σ −n k=n a k e ikθ , or also in the case that T φ is hyponormal, since these operators have SVEP, and hence the index ind (λI − T φ ) on a hole is less or equal to 0. Note that if φ is a trigonometric polynomial then T φ may be not hyponormal, see [13].…”

“…It should be noted that Toeplitz operator T φ may satisfy Weyl's theorem, also if the symbol φ is not continuous, for an example see [13]. Theorem 4.4 does not apply, in general, to non commuting compact perturbations T + K of a polaroid operator.…”

Generalized Weyl's theorem and spectral continuity for (<italic>n</italic>, <italic>k</italic>)-quasiparanormal operators

“…[21] 研 究了 B(H) 上谱的连续性. 最近, Farenick 和 Lee [22] , Hwang 和 Lee [23] 研究了 Toeplitz 算子所构成的 子空间上的谱的连续性. Halmos [24] 证明了 σ 在正规算子和亚正规算子类上是连续的.…”