J. Aust. Math. Soc. 88 (2010), 193–203 doi:10.1017/S1446788710000091 L.n/-HYPONORMALITY: A MISSING BRIDGE BETWEEN SUBNORMALITY AND PARANORMALITY IL BONG JUNG ˛, SUN HYUN PARK and JAN STOCHEL (Received 13 March 2009; accepted 17 January 2010) Communicated by J. J. Koliha Abstract A new notion of L.n/-hyponormality is introduced in order to provide a bridge between subnormality and paranormality, two concepts which have received considerable attention from operator theorists since the 1950s. Criteria for L.n/-hyponormality are given. Relationships to other notions of hyponormality are discussed in the context of weighted shift and composition operators. 2000 Mathematics subject classification: primary 47B20; secondary 47B33, 47B37. Keywords and phrases: subnormal operator, paranormal operator, L.n/-hyponormal operator, weighted shift operator, composition operator. 1. Towards L.n/-hyponormality Let H be a complex Hilbert space and let B.H/ be the set of all bounded linear operators on H. Denote by I the identity operator on H. We write N .T / and R.T / for the kernel and the range of T 2 B.H/. Given two operators A; B 2 B.H/, we denote by TA; BU their commutator, that is, TA; BU VD AB − BA. Recall that an operator T 2 B.H/ is said to be subnormal if there exists a complex Hilbert space K and a normal operator N 2 B.K/ such that H ⊆ K (isometric embedding) and Th D Nh for all h 2 H. The celebrated Halmos–Bram characterization of subnormality (see T5, 19U) states that an operator T 2 B.H/ is subnormal if and only if n X i j hT f j ; T fi i ≥ 0 (1.1) i; jD0 The work of the first author was supported by a Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (No. R01-2008-000-20088-0). The work of the third author was supported by MNiSzW grant N201 026 32/1350. c 2010 Australian Mathematical Publishing Association Inc. 1446-7887/2010 $16.00 193 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.8, on 29 Sep 2021 at 01:43:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788710000091 194 I. B. Jung, S. H. Park and J. Stochel [2] for all finite sequences f0;:::; fn 2 H: To build a bridge between subnormality and hyponormality, McCullough and Paulsen introduced the notion of strong n-hyponormality (T23U; see also T4, 8, 9, 11U): T is said to be (strongly) n-hyponormal (n ≥ 1) if inequality (1.1) holds for all f0;:::; fn 2 H; or, equivalently, the operator ∗ j i n matrix .T T /i; jD0 is positive; this turns out to be equivalent to the positivity of the T ∗ j i U n T U operator matrix . T ; T /i; jD1 (see 23, 24 ). Hence, 1-hyponormality coincides with hyponormality. Two decades later a more subtle characterization of subnormality was described by Embry (see T12U). It states that an operator T 2 B.H/ is subnormal if and only if n X iC j iC j hT f j ; T fi i ≥ 0 (1.2) i; jD0 for all finite sequences f0;:::; fn 2 H. Based on Embry’s characterization, Mc- Cullough and Paulsen introduced in T24U a new class of operators which, follow- ing T17U, will be called E.n/-hyponormal: T is said to be E.n/-hyponormal (n ≥ 1) if inequality (1.2) holds for all f0;:::; fn 2 H, or, equivalently, the operator matrix ∗i ∗ j i j n T U .T .T T /T /i; jD0 is positive. As shown in 24 , E.1/-hyponormality is essentially weaker than 1-hyponormality. Moreover, in view of T17U, T is E.1/-hyponormal if and only if jT j4 ≤ jT 2j2, and so, by the Heinz inequality, such T must be an A-class operator, that is, jT j2 ≤ jT 2j (see T14, p. 166U). Hence, E.n/-hyponormality can be thought of as a bridge between subnormal operators and A-class operators. The class of E.n/-hyponormal composition operators on L2-spaces was completely characterized in terms of Radon–Nikodym derivatives in T17U. Let us recall the well-known characterization of positivity of a two by two operator AB V ! V ! V ! matrix B∗ C , where A H H, B K H and C K K are bounded linear operators, K is a complex Hilbert space and A ≥ 0 (see T26U): AB if A is invertible; then is positive if and only if B∗ A−1 B ≤ C: (1.3) B∗ C D D ∗ ∗n n D ∗iC j iC j n Applying this to A I , B .T T;:::; T T / and C .T T /i; jD1, we get the following proposition. PROPOSITION 1.1. An operator T 2 B.H/ is E.n/-hyponormal if and only if the ∗i T ∗ j i U j n operator matrix .T T ; T T /i; jD1 is positive. The following fact (which was mentioned in T17, p. 3956U) can be deduced either from the original definition of E.n/-hyponormality or from Proposition 1.1 by using the fact that powers of an operator with dense range have dense range. COROLLARY 1.2. An operator T 2 B.H/ with dense range is n-hyponormal if and only if it is E.n/-hyponormal. Embry’s characterization of subnormality was essentially simplified by Lambert in T21U. The original characterization by Lambert was proved only for injective operators. Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.8, on 29 Sep 2021 at 01:43:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788710000091 [3] L.n/-hyponormality 195 The version formulated below gets rid of this unnecessary restriction (see T27, Theorem 7U): an operator T 2 B.H/ is subnormal if and only if n X iC j 2 N kT f k λi λ j ≥ 0 (1.4) i; jD0 for every vector f 2 H and for all finite sequences λ0; : : : ; λn 2 C. By analogy with previous definitions, we give the following one. DEFINITION 1.3. An operator T 2 B.H/ is L.n/-hyponormal (n ≥ 1) if inequal- ity (1.4) holds for all λ0; : : : ; λn 2 C and for every f 2 H. Clearly, the class of L.n/-hyponormal operators is closed under the operations of taking (finite or infinite) orthogonal sums and multiplication by scalars. However, it is not closed under addition and multiplication in B.H/ (this disadvantage is shared by other types of hyponormality including subnormality; see T11, 22U). Replacing f by T −2n f in (1.4), we see that the following proposition holds. PROPOSITION 1.4. If an operator T 2 B.H/ is L.n/-hyponormal and T is invertible in B.H/, then T −1 is L.n/-hyponormal. We now show that an inductive-type limit procedure preserves L.n/-hyponormality. Since the same property is valid for other kinds of hyponormality, we formulate the result for all of them. PROPOSITION 1.5. Let fHσ gσ 2Σ be a monotonically increasing net of .closed linear/ W subspaces of H such that H D σ2Σ Hσ , let fTσ gσ 2Σ be a net of operators Tσ 2 B.Hσ / and let T 2 B.H/ be an operator such that [ sup kTτ k < 1 and T f D lim Tτ f 8 f 2 Hσ : (1.5) τ2Σ τ2Σ σ 2Σ If for every σ 2 Σ the operator Tσ is L.n/-hyponormal (E.n/-hyponormal, n-hyponormal, subnormal), then so is T . PROOF. A standard approximation argument reduces the proof to showing that m m [ T f D lim Tτ f 8m ≥ 0; f 2 Hσ : (1.6) τ2Σ σ 2Σ We do so by induction on m. The cases when m D 0; 1 are obvious due to the equality in (1.5). Suppose that (1.6) holds for a fixed m ≥ 1. It follows from the inequality in (1.5) that for all τ; τ0; σ 2 Σ such that τ ≥ τ0 ≥ σ and for every f 2 Hσ , kT mC1 f − T mC1 f k ≤ kT .T m/ f − T .T m f /k τ τ τ τ τ0 C kT .T m f / − T .T m f /k C kT .T m f / − T .T m f /k τ τ0 τ0 τ0 ≤ sup kT k kT m f − T m f k ρ τ τ0 ρ2Σ C kT .T m f / − T .T m f /k C kT kkT m f − T m f k; τ τ0 τ0 τ0 which by the equality in (1.5) and the induction hypothesis completes the proof. 2 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.8, on 29 Sep 2021 at 01:43:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788710000091 196 I. B. Jung, S. H. Park and J. Stochel [4] 1 COROLLARY 1.6. Let fHkg D be a monotonically increasing sequence of invariant k 1 W1 subspaces for an operator T 2 B.H/ such that kD1 Hk D H. Then T is L.n/- hyponormal (E.n/-hyponormal, n-hyponormal, subnormal) if and only if T jHk is L.n/-hyponormal (E.n/-hyponormal, n-hyponormal, subnormal) for every k ≥ 1.