Integr. equ. oper. theory 59 (2007), 173–187 c 2007 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/020173-15, published online June 27, 2007 DOI 10.1007/s00020-007-1519-8
Integral Equations and Operator Theory
Kth Roots of p-Hyponormal Operators are Subscalar Operators of Order 4k Eungil Ko Abstract. In this paper, we consider the special case of the question raised by Halmos (see below). In particular, we show that if T k is p-hyponormal, then T is a subscalar operator of order 4k. As a corollary, we obtain that if T k is p-hyponormal and σ(T ) has nonempty interior in the plane, then T has a nontrivial invariant subspace. Mathematics Subject Classification (2000). 47B20, 47B38. Keywords. p-hyponormality, subscalarity, the property (β), invariant subspace.
1. Introduction Let H and K be separable, complex Hilbert spaces and L(H, K) denote the space of all bounded linear operators from H to K. If H = K, we write L(H) in place of L(H, K). If T ∈ L(H), we write σ(T ), σap (T ), and σe (T ) for the spectrum, the approximate point spectrum, and the essential spectrum of T , respectively. ¯ denote the closure of Let D be an open disc in C the complex plane. Let D ¯ denote the space of continuous complex valued functions D in C. And let C m (D) ¯ The notation f ∞ will be used to denote the sup norm of a function f on on D. ¯ Then define for f ∈ C m (D), ¯ D f ≡ f ∞ + fx ∞ + fy ∞ + · · · + fyy···y ∞ where for example fx denotes the partial with respect to the coordinate variable ¯ into a Banach space. Note that the pointwise product of x. This makes C m (D) ¯ is again in C m (D) ¯ (in fact C m (D) ¯ with this norm is a two functions in C m (D) topological algebra). The work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00461).
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A bounded linear operator S on H is called scalar of order m if for some open disc D in C there exists a map ¯ −→ L(H) Φ : C m (D) such that 1) Φ is an algebra homomorphism, and 2) Φ is continuous when the ¯ and the operator norm is placed on L(H), and 3) above norm is used on C m (D) Φ(z) = S, where z stands for identity function on C, and 4) Φ(1) = I. The map Φ is called a spectral resolution for S. An operator is called subscalar if it is similar to the restriction of a scalar operator to a closed invariant subspace. Recall that an operator T is called p-hyponormal, 0 < p ≤ 1, if (T ∗ T )p ≥ ∗ p (T T ) where T ∗ is the adjoint of T . If p = 1, T is called hyponormal and if p = 12 , T is called semihyponormal. p-Hyponormal operators were introduced by Aluthge (see [1]). There is a vast literature concerning p-hyponormal operators. In particular, Aluthge proved in [1] that if T = U |T | (polar decomposition) is 1 p-hyponormal with 0 < p < 12 where |T | = (T ∗ T ) 2 and U is the appropriate partial isometry satisfying kerU = ker|T | = kerT and ker U ∗ = kerT ∗ , then T˜ is ˜ is hyponormal where T˜ = |T | 21 U |T | 21 . Note that T˜ is (p + 12 )-hyponormal and T˜ called the Aluthge transform of T and we will use this notation throughout this paper. We say that an operator T ∈ L(H) is a kth root of a p-hyponormal operator √ if T k is p-hyponormal for some positive integer k. We denote this class by ( k P H) where k ≥ 2. If T is p-hyponormal, it is known that T k is kp -hyponormal (see [2]). Hence T k is not p-hyponormal. On the other hand, if T is a nilpotent operator of order k, then T k is p-hyponormal, but T is not p-hyponormal. Also, let T be a p-hyponormal operator. If 0 1H , A= T 0 then A is not p-hyponormal, but A2 = T ⊕ T is p-hyponormal. P.R. Halmos raised in [9] the following question. (P) If T ∈ L(H) and T 2 has a nontrivial invariant subspace, must T have a nontrivial invariant subspace too? In this paper, we will consider the special case of the question raised by Halmos. In particular, we show that if T k is p-hyponormal, then T is a subscalar operator of order 4k. As a corollary, we obtain that if T k is p-hyponormal and σ(T ) has interior in the plane, then T has a nontrivial invariant subspace.
2. Preliminaries An operator T ∈ L(H) is said to satisfy the single valued extension property if for any open set U in C, the function T − z : O(U, H) −→ O(U, H)
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defined by the obvious pointwise multiplication is one-to-one where O(U, H) denote the Fr´echet space of H-valued analytic functions on U with respect to uniform topology. If T has the single valued extension property, then for any x ∈ H there exists a unique maximal open set ρT (x)(⊃ ρ(T ), the resolvent set) and a unique H-valued analytic function f defined in ρT (x) such that (T − z)f (z) = x,
z ∈ ρT (x).
Moreover, if F ⊂ C is a closed set and σT (x) = C\ρT (x), then HT (F ) = {x ∈ H : σT (x) ⊂ F } is a linear subspace (not necessarily closed) of H and obviously HT (F ) = HT (F ∩ σ(T )). An operator T ∈ L(H) is said to satisfy the property (β) if for every open subset G of C and every sequence fn : G → H on H-valued analytic function such that (T − z)fn (z) converges uniformly to 0 in norm on compact subsets of G, fn (z) converges uniformly to 0 in norm on compact subsets of G. Let z be the coordinate in C and let dµ(z) denote the planar Lebesgue measure. Fix a separable, complex Hilbert space H and a bounded (connected) open subset U of C. We shall denote by L2 (U, H) the Hilbert space of measurable functions f : U → H, such that 1 f 2,U = { f (z)2 dµ(z)} 2 < ∞. U
The space of functions f ∈ L2 (U, H) which are analytic functions in U (i.e., ¯ = 0) is denoted by ∂f A2 (U, H) = L2 (U, H) ∩ O(U, H). A2 (U, H) is called the Bergman space for U . We will use the following version of Green’s formula for the plane, also known ¯ , H) in the exactly the same way as as the Cauchy-Pompeiu formula. Define C p (U p ¯ C (U ) except that the functions in the space are now H-valued. Cauchy-Pompeiu formula 2.1. Let D be an open disc in the plane, let z ∈ D and ¯ H). Then f ∈ C 2 (D, 1 f (ζ) ¯ ∗ (− 1 ) f (z) = dζ + ∂f 2πi ∂D ζ − z πz where ∗ denotes the convolution product.
Remark 2.2. The function
f (ζ) dζ ζ −z ∂D appearing in Cauchy-Pompeiu formula is analytic in D 2and extends continuously ¯ H). ¯ as can be seen by examining the term. So, g ∈ A (D, H) for f ∈ C 2 (D, to D D ¯ H) and F ∈ Remark 2.3. Let D be an open disc in C. Then for φ ∈ C ∞ (D, ∞ ¯ C (D, L(H)) we have the following relation. ¯ ∗ (− 1 ). ¯ ) · φ) ∗ (− 1 ) = F φ − F (∂φ) ((∂F πz πz g(z) =
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Let us define now a special Sobolev type space, called W m (D, H) where, as ¯ H), let before, D is a bounded open disc in C. For f ∈ C m (D, f 2W m =
m
∂¯i f 22,D
i=0
¯ H) under this norm. Note that Then let W m (D, H) be the completion of C m (D, m W (D, H) is a Hilbert space contained continuously in L2 (D, H). We next discuss the fact concerning the multiplication operator by z on W m (D, H). The linear operator M of multiplication by z on W m (D, H) is continuous and it has a spectral resolution, defined by the relation ¯ −→ L(W m (D, H)), ΦM : C m (D)
ΦM (f ) = Mf .
Therefore, M is a scalar operator of order m. Moreover, let V : W m (D, H) → 2 ⊕m 0 L (D, H) be the operator defined by ¯ . . . , ∂¯m f ). V (f ) = (f, ∂f, Since V f 2 = f 2W m =
m
∂¯i f 22,D ,
i=0
an operator V is an isometry such that V M = (⊕m 0 Nz )V , where Nz is the multiplication operator on L2 (D, H). Since ⊕m N is normal, M is a subnormal operator. z 0
3. Subscalarity In this section we show that every kth root of a p-hyponormal operator has a scalar extension. We begin with the following theorem. Theorem 3.1. For every bounded disk D in C there is a constant CD , such that for an arbitrary operator A ∈ L(H) and f ∈ W 2k (D, H) we have (I − P )f 2,D ≤ CD
2k
(A − z k )∗ ∂¯i f 2,D
i=k
where P denotes the orthogonal projection of L2 (D, H) onto the Bergman space A2 (D, H). ¯ H) such that si ≡ 1 on D − D for i = 1, 2, . . . , k. Let Proof. Let si be in C ∞ (D, ∞ ¯ fn ∈ C (D, H) be a sequence which approximates f in the norm W 2k . Then for
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a fixed n we have 1 ∂¯k [fn + (A − z k )∗ ∂¯k fn ] k! k 1 k k ¯ = ∂ fn + ∂¯j (A − z k )∗ ∂¯2k−j fn k! j=0 j =
k−1 1 k ∂¯j (A − z k )∗ ∂¯2k−j fn . j k! j=0
(3.1)
By the Cauchy-Pompeiu formula and (3.1), we get
=
1 ∂¯k−1 [fn + (A − z k )∗ ∂¯k fn ] k! 1 (A − ζ k )∗ ∂¯k fn (ζ)] ∂¯k−1 [fn (ζ) + k! 1 dζ 2πi ∂D ζ −z k−1 1 k s1 +[ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ). k! j=0 j πz
Set 1 g1,n (z) = 2πi
∂¯k−1 [fn (ζ) +
∂D
(3.2)
1 k! (A
− ζ k )∗ ∂¯k fn (ζ)] dζ. ζ −z
Then g1,n ∈ A2 (D, H) by Remark 2.2. Thus
=
1 ∂¯k−1 [fn + (A − z k )∗ ∂¯k fn ] k! k−1 1 k s1 g1,n + [ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ). j k! πz j=0
In order to complete our proof, we need the following claims. Claim I. For t = 1, . . . , k we have the following relation.
=
1 ∂¯k−t [fn + (A − z k )∗ ∂¯k fn ] k! st st s2 gt,n + gt−1,n ∗ (− ) + · · · · · · + g1,n ∗ (− ) ∗ · · · ∗ (− ) πz πz πz k−1 1 st s1 k +[ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− ) j k! j=0 πz πz
where gr,n (z) =
1 2πi
∂¯k−r [fn (ζ) + ∂D
if r > 0 and gr,n (z) = 0 if r ≤ 0.
1 k! (A
− ζ k )∗ ∂¯k fn (ζ)] dζ ζ−z
(3.3)
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Proof of Claim I. We prove this claim by induction. If t = 1 it is true from (3.3). Assume it holds when t = r − 1. If we apply the Cauchy-Pompeiu formula when t = r, then 1 ∂¯k−r [fn + (A − z k )∗ ∂¯k fn ] k! 1 (A − ζ k )∗ ∂¯k fn (ζ)] ∂¯k−r [fn (ζ) + k! 1 dζ = 2πi ∂D ζ −z 1 sr +∂¯k−(r−1) [fn + (A − z k )∗ ∂¯k fn ] ∗ (− ). k! πz Set 1 (A − ζ k )∗ ∂¯k fn (ζ)] ∂¯k−r [fn (ζ) + k! 1 gr,n (z) = dζ. 2πi ∂D ζ −z Then by induction assumption we have 1 ∂¯k−r [fn + (A − z k )∗ ∂¯k fn ] k! sr−1 sr−1 s2 ) + · · · + g1,n ∗ (− ) ∗ · · · ∗ (− ) +gr,n + {gr−1,n + gr−2,n ∗ (− πz πz πz k−1 1 k sr−1 sr s1 +[ )} ∗ (− ). ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− k! j=0 j πz πz πz So we complete the proof of Claim I.
By Claim I, we get 1 fn + (A − z k )∗ ∂¯k fn k! sk sk s2 = gk,n + gk−1,n ∗ (− ) + · · · + g1,n ∗ (− ) ∗ · · · ∗ (− ) πz πz πz k−1 1 k sk s1 +[ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− ). j k! j=0 πz πz sk s2 sk Set gn = gk,n + gk−1,n ∗ (− πz ) + · · · · · · + g1,n ∗ (− πz ) ∗ · · · ∗ (− πz ). Then gn ∈ 2 A (D, H). Hence we have 1 fn + (A − z k )∗ ∂¯k fn k! k−1 sk 1 k s1 = gn + [ ∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− ). j k! πz πz j=0
Claim II. For j = 0, 1, . . . , k − 1, the following relation holds. sk s1 [∂¯j (A − z k )∗ ∂¯2k−j fn ] ∗ (− ) ∗ · · · ∗ (− ) πz πz j sk sj−t+1 j t ) ∗ · · · ∗ (− )]. = [(−1) (A − z k )∗ ∂¯2k−(j−t) fn ∗ (− t πz πz t=0
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Proof of Claim II. We prove this claim by induction. If j = 0, it is trivial. Assume that Claim II holds when j = r. Then by applications of Remark 2.3, we obtain the following; sk s1 ) ∗ · · · ∗ (− ) πz πz s1 r k ∗ ¯2k−r−1 r k ∗ ¯2k−r ¯ ¯ [∂ (A − z ) ∂ fn − ∂ (A − z ) ∂ fn ∗ (− )] πz sk s2 ∗(− ) ∗ · · · ∗ (− ) πz πz r sk sr−t+2 r ) ∗ · · · ∗ (− )] [(−1)t (A − z k )∗ ∂¯2k−(r+1−t) fn ∗ (− t πz πz t=0 r sk sr−t+1 r ) ∗ · · · ∗ (− )] − [(−1)t (A − z k )∗ ∂¯2k−(r−t) fn ∗ (− t πz πz t=0 r+1 sk sr−t+2 r+1 ) ∗ · · · ∗ (− )]. [(−1)t (A − z k )∗ ∂¯2k−(r+1−t) fn ∗ (− t πz πz t=0 [∂¯r+1 (A − z k )∗ ∂¯2k−r−1 fn ] ∗ (−
=
=
=
So we complete the proof of Claim II. By Claim II, we get fn − gn
1 (A − z k )∗ ∂¯k fn k! k−1 j 1 k j + (−1)t (A − z k )∗ ∂¯2k−(j−t) fn j t k! j=0 t=0
= −
∗(−
sk sj−t+1 ) ∗ · · · ∗ (− ). πz πz
Taking the norm, we get fn − gn 2,D
≤
1 (A − z k )∗ ∂¯k fn 2,D k! k−1 j 1 k j + (A − z k )∗ ∂¯2k−(j−t) fn 2,D j t k! j=0 t=0
sj−t+1 sk (− ) ∗ · · · ∗ (− )2,D . πz πz Let CD be the maximum among coefficients of {(A − z k )∗ ∂¯i fn 2,D }2k i=k . Then f − g2,D
≤ f − fn 2,D + fn − gn 2,D ≤ f − fn 2,D + CD
2k i=k
(A − z k )∗ ∂¯i fn 2,D .
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By passing to the limit we conclude f − P f 2,D ≤ CD
2k
(A − z k )∗ ∂¯i f 2,D .
i=k
So we complete our proof.
Corollary 3.2. Let T ∈ L(H) be a semihyponormal operator. If {fn } is a sequence in W 2k (D, H) such that limn→∞ (T − z k )fn W 2k = 0 for all z ∈ D, then there exists a sequence of analytic functions {gn } such that lim fn − gn 2,D = 0.
n→∞
Proof. From [10, Lemma 4.3] and the hypothesis we get that lim (T − z k )∗ ∂¯j fn 2,D = 0
n→∞
for j = 0, 1, 2, . . . , 2k. Hence Theorem 3.1 implies that lim (I − P )fn 2,D = 0.
n→∞
Set gn = P fn . Then {gn } is the desired sequence.
Recall that for bounded open disc D containing 0 in C the Bergman operator for D is the operator S defined on A2 (D, H) by (Sf )(z) = zf (z). Then the Bergman operator for D has the following property. Lemma 3.3. ([7, Corollary 10.7]) If S is the Bergman operator for the bounded open disc D containing 0, then S is bounded below. Next we generalize [11, Lemma 3.4]. Lemma 3.4. Let T be any kth roots of a semihyponormal operator. Then for a bounded disk D which contains σ(T ), the operator V : H → H(D) defined by V h = 1 ⊗ h (= 1 ⊗ h + (T − z)W 2k (D, H)) is one-to-one and has closed range, where H(D) = W 2k (D, H)/(T − z)W 2k (D, H) and 1 ⊗ h denotes the constant function sending any z ∈ D to h. Proof. If hn ∈ H and fn ∈ W 2k (D, H) are sequences such that lim (T − z)fn + 1 ⊗ hn W 2k = 0,
n→∞
(3.4)
it suffices to show that limn→∞ hn = 0. Now by the definition of the norm of Sobolev space, (3.4) implies lim (T − z)∂¯j fn 2,D = 0
n→∞
for j = 1, . . . , 2k. From (3.5), we get lim (T k − z k )∂¯j fn 2,D = 0
→∞
(3.5)
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for j = 1, . . . , 2k. Since T k is semihyponormal, by [11, Lemma 4.3] lim (T k − z k )∗ ∂¯j fn 2,D = 0.
n→∞
(3.6)
for j = 1, . . . , 2k. Then by Theorem 3.1, we have lim (I − P )fn 2,D = 0
n→∞
(3.7)
where P denotes the orthogonal projection of L2 (D, H) onto A2 (D, H). By (3.4) and (3.7), we have lim (T − z)P fn + 1 ⊗ hn 2,D = 0.
n→∞
Let Γ be a curve in D surrounding σ(T ). Then for z ∈ Γ lim P fn (z) + (T − z)−1 (1 ⊗ hn ) = 0
n→∞
uniformly. Hence, by Riesz-Dunford functional calculus, 1 P fn (z)dz + hn = 0. lim n→∞ 2πi Γ But since Γ P fn (z)dz = 0 by Cauchy’s theorem, limn→∞ hn = 0.
The next proposition is essential for the proof of our main theorem. √ Proposition 3.5. Let T be in ( k P H). Then for a bounded disk D which contains σ(T ) ∪ {0}, the operator V : H → H(D) defined by V h = 1 ⊗ h (= 1 ⊗ h + (T − z)W 4k (D, H)) is one-to-one and has closed range, where H(D) = W 4k (D, H)/(T − z)W 4k (D, H) and 1 ⊗ h denotes the constant function sending any z ∈ D to h. Proof. Let hn ∈ H and fn ∈ W 4k (D, H) be sequences such that lim (T − z)fn + 1 ⊗ hn W 4k = 0.
n→∞
(3.8)
Then by the definition of the norm of Sobolev space, the equation (3.8) implies (3.9) lim (T − z)∂¯i fn 2,D = 0 n→∞
for j = 1, 2, . . . , 4k. Hence from (3.9) we get lim (T k − z k )∂¯i fn 2,D = 0
n→∞
(3.10)
for j = 1, 2, . . . , 4k. a) If 12 ≤ p < 1, then T k is semihyponormal. Therefore, it is true from Lemma 3.4. b) If 0 < p < 12 , then T˜k is semihyponormal from [1]. Since T˜k |T k |1/2 = |T k |1/2 T k , from (3.10) we get 1 lim (T˜k − z k )∂¯i |T k | 2 fn 2,D = 0
n→∞
(3.11)
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for j = 1, 2, . . . , 4k. By applications of [10, Lemma 4.3] and (3.11), we obtain 1 lim (T˜k − z k )∗ ∂¯i |T k | 2 fn 2,D = 0 (3.12) n→∞
for j = 1, 2, . . . , 4k. Then Theorem 3.1 and (3.12) imply 1
lim (I − P )∂¯i |T k | 2 fn 2,D = 0
n→∞
(3.13)
for j = 1, 2, . . . , 2k where P denotes the orthogonal projection of L2 (D, H) onto A2 (D, H). From (3.11) and (3.13) we get 1 (3.14) lim (T˜k − z k )P ∂¯i |T k | 2 fn 2,D = 0 n→∞
for j = 1, 2, . . . , 2k. Let T k = Uk |T k | be the polar decomposition of T k . Since Uk |T k |1/2 T˜k = T k Uk |T k |1/2 , from (3.14) we have 1 1 lim (T k − z k )Uk |T k | 2 P ∂¯i |T k | 2 fn 2,D = 0
n→∞
(3.15)
for j = 1, 2, . . . , 2k. Since T k is p-hyponormal, it is known from [12] that T k has the property (β). Hence it is easy to show that 1 1 lim Uk |T k | 2 P ∂¯i |T k | 2 fn 2,D = 0
n→∞
for j = 1, 2, . . . , 2k. Since T k = Uk |T k |, from (3.16) 1 lim T˜k P ∂¯i |T k | 2 fn 2,D = 0 n→∞
(3.16)
(3.17)
for j = 1, 2, . . . , 2k. Then (3.14) and (3.17) imply 1 lim z k P ∂¯i |T k | 2 fn 2,D = 0
n→∞
(3.18)
for j = 1, 2, . . . , 2k. By applications of Lemma 3.3, there exists a constant c > 0 such that 1 1 (3.19) z k P ∂¯i |T k | 2 fn 2,D ≥ cP ∂¯i |T k | 2 fn 2,D for j = 1, 2, . . . , 2k. Then from (3.18) and (3.19) we obtain 1
lim P ∂¯i |T k | 2 fn 2,D = 0
n→∞
(3.20)
for j = 1, 2, . . . , 2k. Hence it follows from (3.13) and (3.20) that 1
lim ∂¯i |T k | 2 fn 2,D = 0
n→∞
(3.21)
for j = 1, 2, . . . , 2k. T k = Uk |T k |, from (3.21) we get lim T k ∂¯i fn 2,D = 0
(3.22)
for j = 1, 2, . . . , 2k. By (3.10) and (3.22), we have lim z k ∂¯i fn 2,D = 0
(3.23)
n→∞
n→∞
for j = 1, 2, . . . , 2k. Now applying Theorem 3.1 with A = (0), we obtain lim (I − P )fn 2,D = 0.
n→∞
(3.24)
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By (3.8) and (3.24), we get lim (T − z)P fn + 1 ⊗ hn 2,D = 0.
n→∞
(3.25)
Let Γ be a curve in D surrounding σ(T ). Then for z ∈ Γ lim P fn (z) + (T − z)−1 (1 ⊗ hn ) = 0
n→∞
uniformly. Hence, by Riesz-Dunford functional calculus, 1 lim P fn (z)dz + hn = 0. n→∞ 2πi Γ But since Γ P fn (z)dz = 0 by Cauchy’s theorem, limn→∞ hn = 0.
Now we are ready to prove our main theorem. √ Theorem 3.6. An arbitrary operator T in ( k P H) is a subscalar operator of order 4k. Proof. Consider an arbitrary bounded open disk D in C which contains σ(T ) ∪ {0} and the quotient space H(D) = W 4k (D, H)/(T − z)W 4k (D, H) endowed with the Hilbert space norm. The class of a vector f or an operator A on ˜ Let M (= Mz ) be the multiplication H(D) will be denoted by f˜, respectively A. 4k operator by z on W (D, H). Then M is a scalar operator of order 4k and its spectral resolution is ¯ −→ L(W 4k (D, H)), Φ : C 4k (D)
Φ(f ) = Mf ,
where Mf is the multiplication operator with f . Since M commutes with T − z, ˜ on H(D) is still a scalar operator of order 4k, with Φ ˜ as a spectral resolution. M Let V be the operator V h = 1 ⊗ h (= 1 ⊗ h + (T − z)W 4k (D, H)), ˜V. from H into H(D), denoting by 1 ⊗ h the constant function h. Then V T = M Since V is one-to-one and has closed range by Proposition 3.5, T is subscalar of order 4k. √ Corollary 3.7. Let T ∈ ( k P H). If N is a nilpotent operator of order k, then αT , V T V ∗ , and T ⊕ N are subscalar operators of order 4k, where α ∈ C and V is an isometry. √ Proof. Since αT and V T V ∗ are in ( k P H), they are trivial from Theorem 3.6. Since (T ⊕ N )k = T k ⊕ 0 ∈ (P H), T ⊕ N is a subscalar operator from Theorem 3.6.
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√ Next we can easily observe from Theorem 3.6 that the restriction of T ∈ ( k P H) is also subscalar. √ Corollary 3.8. If T ∈ ( k P H), then T |M is a subscalar operator of order 4k, where M is a nontrivial invariant subspace for T . Corollary 3.9. Let T ∈ L(H) be a unilateral weighted shift with positive weight sequence {αn }∞ n=0 . If αn−k . . . αn−1 ≤ αn · · · αn+k−1 for n = k, k + 1, . . ., then T is a subscalar operator of order 4k. k Proof. Let {en }∞ n=0 be an orthonormal basis of a Hilbert space H. Since T en = ∗k αn · · · αn+k−1 en+k and T en = αn−1 · · · αn−k en−k , it is easy to calculate that T k is p-hyponormal for n = k, k + 1, . . .. Hence the proof follows from Theorem 3.6.
Recall that if U is a non-empty open set in C and if Ω ⊂ U has the property that supλ∈Ω |f (λ)| = supβ∈U |f (β)| for every function f in H ∞ (U ) (i.e. for all f bounded and analytic on U ), then Ω is said to be dominating for U . The following corollary is an extension of S. Brown’s beautiful theorem (see [5]). √ Corollary 3.10. Let T be in ( k P H). If σ(T ) has the property that there exists some non-empty open set U such that σ(T ) ∩ U is dominating for U , then T has a nontrivial invariant subspace. Proof. This follows from Theorem 3.6 and [8].
Recall that an operator T ∈ L(H) is said to be power regular if 1 limn→∞ T nx n exists for every x ∈ H. √ Corollary 3.11. If T is in ( k P H), it is power regular. Proof. It is known from Theorem 3.6 that every kth root of a p-hyponormal operator is similar to the restriction of a scalar operator to one of its invariant subspace. Since a scalar operator is power regular and the restriction of power regular operators to their invariant subspaces clearly remains power regular, every kth root of a p-hyponormal operator is power regular. √ Corollary 3.12. If T is in ( k P H), it satisfies the property (β). Hence it satisfies the single valued extension property. Proof. Since every scalar operator satisfies the property (β) and the property (β) is transmitted from an operator to its restriction to closed invariant subspaces, it follows from Theorem 3.6 that every kth root of a p-hyponormal operator satisfies the property (β). Hence it satisfies the single valued extension property.
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Recall that an X ∈ L(H, K) is called a quasiaffinity if it has trivial kernel and dense range. An operator A ∈ L(H) is said to be a quasiaffine transform of an operator T ∈ L(K) there exists a quasiaffinity X ∈ L(H, K) such that XA = T X. Furthermore, operators A and T are said to be quasisimilar if there are quasiaffinities X and Y such that XA = T X and AY = Y T . √ Corollary 3.13. Let A and T be in ( k P H). If they are quasisimilar, then σ(A) = σ(T ) and σe (A) = σe (T ). Proof. Since A and T satisfy the property (β) by Corollary 3.12, the proof follows from [14]. √ Proposition 3.14. If T is in ( k P H), then for any bounded open disk D containing σ(T ) ∪ {0} and any sequence fn ∈ W 4k (D, H), we have limn→∞ fn 2,D = 0 whenever limn→∞ (T − z)fn W 4k = 0. Proof. If limn→∞ (T − z)fn W 4k = 0 for any sequence fn ∈ W 4k (D, H), by applications of the proof in Proposition 3.5 we get (cf, (3.24)) lim (I − P )fn 2,D = 0.
n→∞
Hence we have lim (T − z)P fn 2,D = 0.
n→∞
Since T satisfies the property (β) by Corollary 3.12, it is easy to show that lim P fn 2,D = 0.
n→∞
Hence limn→∞ fn 2,D = 0.
The following corollary is the special case of Proposition 3.14. √ Corollary 3.15. If T is in ( k P H), then for any bounded open disk D the operator T − z : W 4k (D, H) −→ W 4k (D, H) is one-to-one.
√ Corollary 3.16. Let T1 and T3 be in ( k P H). Then T1 − z T2 : ⊕W 4k (D, H) −→ ⊕W 4k (D, H) A−z = 0 T3 − z is one-to-one. Proof. Let f = f1 ⊕ f2 ∈ ⊕W 4k (D, H) be such that (A − z)f = 0. Then f1 (T1 − z)f1 + T2 f2 0 T1 − z T2 = = . 0 T3 − z f2 (T3 − z)f2 0 So we have (T1 − z)f1 + T2 f2 = 0 and
(3.26)
(T3 − z)f2 = 0.
(3.27)
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By Corollary 3.15 and (3.27), f2 = 0. Hence from (3.26) we have (T1 − z)f1 = 0. Again by Corollary 3.15, f1 = 0. Thus f = 0. √ Theorem 3.17. Let T be in ( k P H). If 0 1H A= , T 0 then A is not p-hyponormal, but is subscalar of order 8k. Proof. Since (A∗ A)p −(AA∗ )p = {(T ∗ T )p −1H }⊕{1H −(T T ∗)p } is not positive, A is not p-hyponormal. But since T k is p-hyponormal, A2k = T k ⊕ T k is p-hyponormal. By Theorem 3.6, A is a subscalar operator of order 8k. Since A2k is p-hyponormal, we remark that if σ(A) is rich, then A2k has a nontrivial invariant subspace from [12]. Next, we will consider the spcial case of the question raised by Halmos. Corollary 3.18. With the notation of Theorem 3.17, if σ(A) has the property that there exists some non-empty open set U such that σ(A) ∩ U is dominating for U , then A has a nontrivial invariant subspace. Proof. Since A is subscalar from Theorem 3.17, it follows from [8] that A has a nontrivial invariant subspace. Corollary 3.19. With the notation of Theorem 3.17, if σ(A) has the property that there exists some non-empty open set U such that σ(A) ∩ U is dominating for U , 0 Tm has a nontrivial invariant subspace for any positive integer then T m+1 0 m. Proof. Since
Tm = A T m+1 0 for any positive integer m and A has a nontrivial invariant subspace by Corollary 3.18, we get that A2m+1 has a nontrivial invariant subspace. So we complete the proof. 2m+1
0
References [1] A. Aluthge, On p-hyponormal operators for 0 < p < 1, Int. Eq. Op. Th. 13(1990), 307–315. [2] A. Aluthge and D. Wang, Powers of p-hyponormal operators, J. Inequality Appl. 3(1999), 279–284. [3] S. Brown and E. Ko, Operators of Putinar type, Op. Th. Adv. Appl. 104(1998), Birkh¨ auser Verlag, Boston, 49–57. [4] P. Bourdon, Orbits of hyponormal operators, Michigan Math. J. 44(1997), 345–353. [5] S. Brown, Hyponormal operators with thick spectrum have invariant subspaces, Ann. of Math. 125(1987), 93–103.
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[6] I. Colojoar˘ a and C. Foia¸s, Theory of generalized spectral operators, Gordon and Breach, New York, 1968. [7] J. Conway, Subnormal operators, Pitman, London, 1981. [8] J. Eschmeier, Invariant subspaces for subscalar operators, Arch. math. 52(1989), 562–570. [9] P. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76(1970), 887–933. [10] E. Ko, On w-hyponormal operators, Studia Math. 156(2003), 165–175. [11] E. Ko, Square roots of semihyponormal operators have scalar extensions, Bull. Sci. Math. 127(2003), 557–567. [12] E. Ko, w-Hyponormal operators have scalar extensions, Int. Eq. Op. Th. 53(2005), 363–372. [13] M. Putinar, Hyponormal operators are subscalar, J. Operator Th. 12(1984), 385–395. [14] M. Putinar, Quasisimilarity of tuples with Bishop’s property (β), Int. Eq. Op. Th. 15(1992), 1047–1052. [15] H. Radjavi and P. Rosenthal, On roots of normal operators, J. Math. Anal. Appl. 34(1971), 653–664. [16] D. Xia, Spectral theory of hyponormal operators, Op. Th. Adv. Appl. 10, Birkh¨ auser Verlag, Boston, 1983. Eungil Ko Department of Mathematics Ewha Women’s University Seoul 120-750 Korea e-mail:
[email protected] Submitted: March 8, 2007 Revised: May 8, 2007
