Subnormality and 2-hyponormality for Toeplitz Operators

Let H and K be complex Hilbert spaces, let L(H,K) be the set of bounded linear operators from H to K and write L(H) := L(H,H). An operator T ∈ L(H) is said to be normal if T ∗T = TT ∗, hyponormal if T ∗T ≥ TT ∗, and subnormal if T = N |H, where N is normal on some Hilbert space K ⊇ H. If T is subnormal then T is also hyponormal. Recall that the Hilbert space L2(T) has a canonical orthonormal basis given by the trigonometric functions en(z) = z, for all n ∈ Z, and that the Hardy space H2(T) is the closed linear span of {en : n = 0, 1, · · · }. An element f ∈ L2(T) is said to be analytic if f ∈ H2(T), and co-analytic if f ∈ L2(T) H2(T). If P denotes the orthogonal projection from L2(T) to H2(T), then for every φ ∈ L∞(T) the operators Tφ and Hφ on H2(T) defined by Tφg := P (φg) and Hφ(g) := (I − P )(φg) (g ∈ H2(T))