Hyponormal block Toeplitz operators with finite rank self-commutators

In this paper, we identify a large class of hyponormal block Toeplitz operators whose self-commutators are of finite rank. Recall that an operator T_φ is hyponormal and [T_φ^, T_φ] is a finite rank operator if and only if there exists a finite Blaschke product b in E(φ), where E(φ) := ∈ H^∞(T): \|k\|_∞ ≤ 1 and φ-k· φ̄ ∈ H^∞(T) \. An analogous set E(Φ) can be defined for a matrix-valued symbol Φ. In the block Toeplitz operator case, we first establish that if a symbol Φ is in L^∞(T,Mₙ) and if E(Φ) contains a constant unitary matrix U, then T_Φ is normal. We then obtain a suitable converse, under a mild assumption on the symbol. Next, we provide a partial answer to a conjecture recently posed by R.E. Curto, I.S. Hwang and W.Y. Lee [10, Conjecture 6.1]. Concretely, assume that Φ ∈ H∞(T, Mₙ) is such that Φ^ is of bounded type and T_Φ is hyponormal. Then [T_Φ^, T_Φ] is a finite rank operator if and only if there exists a finite Blaschke–Potapov product in E(Φ), where Φ:=Φ^* ; and ; Φ(eiθ):=Φ(e-iθ).