On quasinormal, subnormal, and hyponormal Toeplitz operators

Abstract

Let ϕ be a non-constant function inL ^∞(D) such thatφ=φ ₁+φ ₂, whereφ ₁ is an element in the Bergman spaceL ² _a (D), and$\phi _2 \in \overline {L_a^2 (D)}$, the space of all complex conjugates of functions inL ² _a (D). In this paper, it is shown that if 1 is an element in the closure of the range of the self-commutator ofT _ϕ,$T_{\bar \phi } T_\phi - T_\phi T\phi$, then the Toeplitz operatorT _ϕ defined ofL ² _a (D) is not quasinormal. Moreover, if$\phi = \psi + \lambda \bar \psi$, whereψ∈ H ^∞(D), and λεC, it is proved that ifT _ϕ is quasinormal, thenT _ϕ is normal. Also, the spectrum of a class of pure hyponormal Toeplitz operators is determined.