Commun. Korean Math. Soc. 22 (2007), No. 3, pp. 383–389
THE HYPERINVARIANT SUBSPACE PROBLEM FOR QUASI-n-HYPONORMAL OPERATORS An-Hyun Kim and Eun-Young Kwon
Reprinted from the Communications of the Korean Mathematical Society Vol. 22, No. 3, July 2007 c °2007 The Korean Mathematical Society
Commun. Korean Math. Soc. 22 (2007), No. 3, pp. 383–389
THE HYPERINVARIANT SUBSPACE PROBLEM FOR QUASI-n-HYPONORMAL OPERATORS An-Hyun Kim and Eun-Young Kwon Abstract. In this paper we examine the hyperinvariant subspace problem for quasi-n-hyponormal operators. The main result on this problem is as follows. If T = N + K is such that N is a quasi-n-hyponormal operator whose spectrum contains an exposed arc and K belongs to the Schatten p-ideal then T has a non-trivial hyperinvariant subspace.
1. Introduction Throughout this note, H denotes an infinite-dimensional separable Hilbert space. We write B(H) for the algebra of bounded linear operators on H and K(H) for the ideal of compact operators on H. An operator T ∈ B(H) is called an n-normal operator if there exists a maximal abelian self-adjoint algebra R such that T is in the commutant of R(n) , where R(n) denotes the direct sum of n copies of R. From the definition we can see that T ∈ B(H) is n-normal if and only if it is unitarily equivalent to an n × n operator matrix (Nij ) acting on K(n) , where {Nij } is a collection of commuting normal operators on a separable Hilbert space K (cf. [5], [6, Theorem 7.17]). Moreover it was well known ([2], [1], [6, Theorem 7.2]) that each n-normal operator has an upper triangular form: i.e., if T is n-normal then T is unitarily equivalent to N11 N12 . . . N1n 0 N22 . . . N2n (1.1) .. .. , .. .. . . . . 0
...
0
Nnn
where {Nij }1≤i≤j≤n consists of mutually commuting normal operators on a separable Hilbert space K. In [4] an extended notion of n-normal operators was introduced. Received February 22, 2007. 2000 Mathematics Subject Classification. Primary 47A10, 47B20, 47B35. Key words and phrases. n-normal operators, quasi-n-hyponormal operators, hyperinvariant subspace problem. This research is financially supported by Changwon National University in 2006. c °2007 The Korean Mathematical Society
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Definition 1.1. Let K be a separable complex Hilbert space. An operator T ∈ B(H) is called a quasi-n-hyponormal operator (for n ∈ N) if it is unitarily equivalent to an n × n upper triangular operator matrix (Nij ) acting on K(n) , where the diagonal entries Njj (j = 1, 2, . . . , n) are hyponormal operators in B(K). Clearly, the classes of hyponormal and quasi-1-hyponormal operators coincide. 2. The main result If T ∈ B(H), we write ρ(T ) for the resolvent of T ; σ(T ) for the spectrum of T ; π0 (T ) for the eigenvalues of T . The following lemma is used in proving the main result. Lemma 2.1 ([4, Lemma 3]). If T is quasi-n-hyponormal and λ ∈ / σ(T ) then ||(T − λ)−1 || ≤
(1 + ||T ||)n−1 ¾. £ ¤n min 1, dist (λ, σ(T )) ½
The subspace M is said to be hyperinvariant for T ∈ B(H) if M is invariant for every operator S which commutes with T . Knowledge of the hyperinvariant subspaces of T gives information on the structure of the commutant of T . The invariant subspace problem is: does every operator have a non-trivial invariant subspace ? This problem remains still open. A related question is the hyperinvariant subspace problem: does every operator which is not a multiple of the identity have a non-trivial hyperinvariant subspace ? Also nobody knows the answer until now. Some partial solutions on those problems were given in many literature. Perhaps the most elegant results on the hyperinvariant subspace problem are the affirmative answers for non-scalar normal operators and for non-zero compact operators. Note that n-normal operators have non-trivial reducing subspaces: if T ∈ comm R(n) and P ∈ R then P (n) is a projection commuting with T . In particular one knows ([3], [6]) that if T is n-normal and is not a multiple of the identity then T has a non-trivial hyperinvariant subspace. Many authors have also examined the hyperinvariant subspace problem for some perturbations of normal operators. To see the most striking result among those attempts, we need some definitions. 1 If T is a compact operator, arrange the eigenvalues of |T | = (T ∗ T ) 2 , repeated according to multiplicity, in decreasing order, and let λn denote the n-th eigenvalues of |T |. For each p ≥ 1, define "∞ # p1 X |T |p := λpn n=1
and Cp := {T : |T |p < ∞} .
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Then Cp forms an ideal in B(H), called the Schatten p-ideal. It is well-known (cf. [6, Theorem 6.12]) that if T = N + K ∈ B(H) is such that (i) N is a normal operator and K ∈ Cp for some p ≥ 1; (ii) σ(N ) contains an exposed arc (i.e., a subset whose intersection with some open disk in C is a smooth Jordan arc), then T has a non-trivial hyperinvariant subspace. We will show that this theorem remains true if “normal” is replaced by “quasi-n-hyponormal”. If T ∈ Ck , with k an integer greater than 1, and if {λ1 , λ2 , . . .} is an enumeration of the non-zero eigenvalues of T , repeated according to multiplicity, define " Ã !# ∞ Y (−1)k−1 λk−1 λ2j j δk (T ) := + ··· + . (1 + λj )exp −λj + 2 k−1 j=1 (If π0 (T ) ⊂ {0}, define δk (T ) = 1.) It was well-known that (cf. [6, Lemmas 6.6 and 6.7]) (2.1) δk (T ) is an absolutely convergent infinite product; ¡ ¢ (2.2) there exists a constant ck > 0 such that |δk (T )| ≤ exp ck |T |kk ; (2.3) there exists a constant Mk > 0 such that ¡ ¢ ||δk (T )(1 + T )−1 || ≤ exp Mk |T |kk whenever −1 ∈ / σ(T ); ¡ ¢ (2.4) if B ∈ Ck and A ∈ B(H) is arbitrary then δk B(z − A)−1 is an analytic function of z on ρ(A). The following lemma is an extension of [6, Lemma 6.11]; the proof is similar to that of [6, Lemma 6.11]. Lemma 2.2. Let T = N + K, where N is quasi-n-hyponormal and K ∈ Ck for some k ≥ 1. Suppose π0 (T ) = ∅ and σ(N ) contains an exposed arc J. Let z0 ∈ J and let L be any closed bounded line segment with z0 as an endpoint which is not tangent to J and is such that L ∩ σ(N ) = {z0 }. Then there exists a constant c > 0 such that µ ¶ c ||(z − T )−1 || ≤ exp for all z ∈ L \ {z0 }. |z − z0 |nk+1 Proof. Observe that σ(T ) ⊆ σ(N )∪π0 (T ). So if π0 (T ) = ∅, then σ(T ) ⊂ σ(N ). Let D be an open disk such that D ∩ σ(N ) = J. Given a line segment L as in the statement of the theorem, let D0 be a disk, with radius ≤ 21 , tangent to J at z0 and contained in D which meets L and whose intersection with J is {z0 }. Define the function δ on ρ(N ) by ¢ ¡ δ(z) := δk −K(z − N )−1 . Then by (2.4), δ is analytic on ρ(N ). Observe that for z ∈ ρ(N ), ¡ ¢−1 . (z − T )−1 = (z − N )−1 1 − K(z − N )−1
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Since −1 ∈ / σ(−K(z − N )−1 ), it follows that δ(z) 6= 0 because δk converges absolutely. So, for z ∈ ρ(N ), ¡ ¢−1 (z − T )−1 = δ(z)−1 (z − N )−1 δ(z) 1 − K(z − N )−1 . By (2.2) and Lemma 2.1, ¶ µ −1 k |δ(z)| ≤ exp c1 |K(z − N ) |k (some constant c1 ) ¶ µ ≤ exp c1 |K|kk ||(z − N )−1 ||k ≤ exp
c2 ½ ¾ (c := c1 |K|kk (1 + ||N ||)(n−1)k ). £ ¤nk 2 min 1, dist (z, σ(N ))
Since for z ∈ D0 , dist (z, σ(N )) ≥ dist (z, ∂D0 ), we have ! Ã c c 2 2 ½ ¾ ≤ exp £ |δ(z)| ≤ exp ¤nk . £ ¤nk dist (z, ∂D0 ) 0 min 1, dist (z, ∂D ) Since δ(z) 6= 0 throughout the disk D¡0 , it has¢an analytic logarithm α(z), i.e., exp (α(z)) = δ(z). Since |δ(z)| = exp Re α(z) , we have c2 Re α(z) ≤ £ ¤nk . dist (z, ∂D0 ) Using an argument involved with the Borel-Carath´eodory inequality (see [6, Lemma 6.10]), we can get c3 |α(z)| ≤ for some constant c3 and z ∈ D0 ∩ L. |z − z0 |nk+1 Hence, Re α(z) ≥
−c3 , |z−z0 |nk+1
|δ(z)
−1
so that
| = exp (−Re α(z)) ≤ exp
µ
c3 |z − z0 |nk+1
¶ .
Therefore, now, for z ∈ D0 ∩ L, ||(z − T )−1 || ¡ ¢−1 = ||δ(z)−1 (z − N )−1 δ(z) 1 − K(z − N )−1 || µ ¶ ¡ ¢−1 c3 || ||(z − N )−1 || ||δ(z) 1 − K(z − N )−1 ≤ exp |z − z0 |nk+1 µ ¶ µ ¶ c3 −1 k −1 k ≤ exp ||(z − N ) || exp c4 |K|k ||(z − N ) || |z − z0 |nk+1 for some constant c4 (by (2.3)).
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Thus by Lemma 2.1 we have that for some constants c5 , c6 , ||(z − T )−1 || µ ¶ c3 ≤ exp |z − z0 |nk+1 × exp µ ≤ exp
c5 ½ ¾ £ ¤n min 1, dist (z, σ(N ))
c6 ½ ¾ £ ¤nk min 1, dist (z, σ(N ))
c3 |z − z0 |nk+1
¶
c5 £ ¤n exp dist (z, σ(N ))
Ã
c6
!
£ ¤nk dist (z, σ(N ))
since dist (z, σ(N )) ≤ dist (z, {z0 }) ≤ 1 for z ∈ L ∩ D0 . Since L is not tangent 0| 0 0 to J, dist|z−z (z, σ(N )) is bounded for z ∈ L ∩ D . It thus follows that for z ∈ L ∩ D , ||(z − T )−1 || µ ¶ µ ¶ µ ¶ c3 c7 c8 ≤ exp exp exp |z − z0 |nk+1 |z − z0 |n |z − z0 |nk ¶ µ c for some constant c > 0 (since |z − z0 | ≤ 1). ≤ exp |z − z0 |nk+1 This completes the proof.
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The following is a well-known theorem. Lemma 2.3 ([6, Theorem 6.3]). Let T ∈ B(H) be such that σ(T ) contains an exposed arc J. Suppose that for each point z0 ∈ J and each closed line segment L which meets σ(T ) only in {z0 } and which is not tangent to J, there exists a constant c > 0 such that µ ¶ c (2.5) ||(z − T )−1 || ≤ exp |z − z0 |k for some k and all z ∈ L \ {z0 }. Then T has a non-trivial hyperinvariant subspace. We are ready for: Theorem 2.4. Let T = N + K ∈ B(H) be such that (i) N is a quasi-n-hyponormal operator and K ∈ Cp for some p ≥ 1; (ii) σ(N ) contains an exposed arc J. Then T has a non-trivial hyperinvariant subspace.
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Proof. We can write N1 ∗ . . . ∗ .. 0 N2 . . . . N ≡ (Ni hyponormal). . .. .. .. . . ∗ 0 . . . 0 Nn Sn Since π0 (N ) ⊂ j=1 π0 (Nj ) and each Nj is a hyponormal operator acting on a S separable space, it follows that π0 (N ) is countable. But since σ(N ) ⊂ σ(T ) π0 (N ), we have that J ⊂ σ(T ). If π0 (T ) 6= ∅ then evidently T has a nontrivial hyperinvariant subspace. If instead π0 (T ) = ∅ then since σ(T ) ⊂ σ(N ), it follows that J is an exposed arc of σ(T ). By Lemma 2.2, the resolvent of T satisfies the condition (2.5) with k the least integer greater than np + 1. So the result follows at once from Lemma 2.3. ¤ We conclude with two corollaries. Corollary 2.5. Let T = N + K ∈ B(H) be such that (i) N is a hyponormal operator and K ∈ Cp for some p ≥ 1; (ii) σ(N ) contains an exposed arc. Then T has a non-trivial hyperinvariant subspace. Proof. Immediate from Theorem 2.4.
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Corollary 2.6. Let T = N + K ∈ B(H) be such that (i) N is an n-normal operator and K ∈ Cp for some p ≥ 1; (ii) σ(N ) is contained in a smooth Jordan curve. Then T has a non-trivial invariant subspace. Proof. Immediate from the same argument as [6, Corollary 6.14] together with Theorem 2.4. ¤ References [1] D. Deckard and C. Pearcy, On matrices over the ring of continuous complex-valued functions on a Stonian space, Proc. Amer. Math. Soc. 14 (1963), 322–328. [2] S. R. Foguel, Normal operators of finite multiplicity, Comm. Pure Appl. Math. 11 (1958), 297–313. [3] T. B. Hoover, Hyperinvariant subspaces for n-normal operators, Acta Sci. Math. (Szeged) 32 (1971), 109–119. [4] I. H. Kim and W. Y. Lee, The spectrum is continuous on the set of quasi-n-hyponormal operators, J. Math. Anal. Appl., to appear. [5] H. Radjavi and P. Rosenthal, Hyperinvariant subspaces for spectral and n-normal operators, Acta Sci. Math. (Szeged) 32 (1971), 121–126. , Invariant Subspaces, Second edition, Dover Publications, Mineda, NY, 2003. [6]
QUASI-n-HYPONORMAL OPERATORS
An-Hyun Kim Department of Mathematics Changwon National University Changwon 641–773, Korea E-mail address:
[email protected] Eun-Young Kwon Institute of Engineering Education Changwon National University Changwon 641-773, Korea E-mail address:
[email protected]
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