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Absolutely norm attaining paranormal operators
Accepted Manuscript Absolutely Norm attaining Paranormal operators
G. Ramesh
PII: DOI: Reference:
S0022-247X(18)30417-7 https://doi.org/10.1016/j.jmaa.2018.05.024 YJMAA 22256
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
14 February 2017
Please cite this article in press as: G. Ramesh, Absolutely Norm attaining Paranormal operators, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.05.024
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ABSOLUTELY NORM ATTAINING PARANORMAL OPERATORS G. RAMESH Abstract. A bounded linear operator T : H1 → H2 , where H1 , H2 are Hilbert spaces is said to be norm attaining if there exists a unit vector x ∈ H1 such that T x = T . If for any closed subspace M of H1 , the restriction T |M : M → H2 of T to M is norm attaining, then T is called an absolutely norm attaining operator or AN -operator. We prove the following characterization theorem: A positive operator T defined on an infinite dimensional Hilbert space H is an AN -operator if and only if the essential spectrum of T is a single point and [m(T ), me (T )) contains atmost finitely many points. Here m(T ) and me (T ) are the minimum modulus and essential minimum modulus of T . As a consequence we obtain a sufficient condition under which the AN -property of an operator implies AN -property of its adjoint. We also study the structure of paranormal AN -operators and give a necessary and sufficient condition under which a paranormal AN operator is normal.
1. Introduction In this article we continue the study of absolutely norm attaining operators of the earlier work from [7]. The class of absolutely norm attaining operators is introduced in [3] and further the detailed study of these operators is appeared in [6, 10, 7]. In the present article first we prove a characterization theorem for positive AN -operators. In general if T is an AN -operator, it may not be true that T ∗ is also an AN -operator (See [10, Example 6.3] for more details). We give a sufficient condition under which this result holds true. Next, we study the structure of absolutely norm attaining paranormal operators. Specifically, we show that if T is a paranormal AN -operator, then there exists pairs (Hα , Uα ), where Hα is a reducing subspace of T and Uα is an isometry on Hα such that Date: 08:09 Wednesday 9th May, 2018. 1991 Mathematics Subject Classification. Primary 47A15; Secondary 47B07, 47B20, 47B40. Key words and phrases. Compact operator, norm attaining operator, AN -operator, Weyl’s theorem, paranormal operator, reducing subspace. 1
2
G. RAMESH
H=
β∈σ(|T |)
Hβ
and
T =
βUβ .
β∈σ(|T |)
Here σ(|T |) is the spectrum of |T |, the modulus of T . Among all Hα ’s atmost one can have infinite dimension. That means atmost of the Uβ ’s can be an isometry and the remaining must be unitary. We describe a necessary and sufficient condition which ensure the normality of a paranormal AN -operator. We organize the article as follows: In the remaining part of this section we describe basic definitions and notations needed to prove our main results. In the second section, we prove a characterization of positive AN -operators and as a consequence we give a sufficient condition which guarantees the AN -property of the adjoint operator when the operator has AN -property. In the final section, we discuss the structure of paranormal AN -operators. As a consequence we show that every such operator has countably many finite dimensional reducing subspaces. Another important consequence is that a paranormal compact operator is normal. Throughout we consider complex Hilbert spaces which will be denoted by H, H1 , H2 etc. The inner product and the induced norm on H are designated by ·, · and · . If M is a subspace of H, then SM := {x ∈ M : x = 1} is the unit sphere of M and the orthogonal complement of M in H is denoted by M ⊥ . If M is a closed subspace of M , then the orthogonal projection onto M is denoted by PM . We denote the space of all bounded linear operators between H1 and H2 by B(H1 , H2 ). In case if H1 = H2 = H, then we denote B(H1 , H2 ) by B(H). For T ∈ B(H1 , H2 ), there exists a unique operator denoted by T ∗ : H2 → H1 satisfying T x, y = x, T ∗ y for all x ∈ H1 and for all y ∈ H2 . This operator T ∗ is called the adjoint of T . The null space and the range spaces of T are denoted by N (T ) and R(T ), respectively. Let T ∈ B(H). Then T is said to be normal if T ∗ T = T T ∗ , self-adjoint if T = T ∗ . If T is self-adjoint and T x, x ≥ 0 for all x ∈ H, then T is called positive. It is well known that for a positive operator T , there exists a unique positive operator S ∈ B(H) such that S 2 = T . The operator S is 1 called as the positive square root of T and is denoted by T 2 . If S, T ∈ B(H) are self-adjoint and S − T ≥ 0, then we write this by S ≥ T. If P ∈ B(H) is such that P 2 = P , then P is called a projection. If N (P ) and R(P ) are orthogonal to each other, then P is called an orthogonal projection. A projection P is an orthogonal projection if and only if it is self-adjoint if and only if it is normal. We call an operator V ∈ B(H1 , H2 ) to be an isometry if V x = x for each x ∈ H1 . An operator V ∈ B(H1 , H2 ) is said to be a partial isometry
ON AN -OPERATORS
3
if V |N (V )⊥ is an isometry. That is V x = x for all x ∈ N (V )⊥ . If V ∈ B(H) is isometry and onto, then V is said to be a unitary operator. 1 If T ∈ B(H1 , H2 ), then T ∗ T ∈ B(H1 ) is positive and |T | := (T ∗ T ) 2 is called the modulus of T . In fact, there exists a unique partial isometry V ∈ B(H1 , H2 ) such that T = V |T | and N (V ) = N (T ). This factorization is called the polar decomposition of T . A closed subspace M of H is said to be invariant under T ∈ B(H) if T M ⊆ M and reducing if both M and M ⊥ are invariant under T . For T ∈ B(H), the set ρ(T ) := {λ ∈ C : T − λI : H → H is invertible and (T − λI)−1 ∈ B(H)} is called the resolvent set and the complement σ(T ) = C \ ρ(T ) is called the spectrum of T . It is well known that σ(T ) is a non empty compact subset of C. The point spectrum of T is defined by σp (T ) = {λ ∈ C : T − λI is not one-to-one}. Note that σp (T ) ⊆ σ(T ). If T ∈ B(H1 , H2 ), then T is said to be compact if for every bounded set S of H1 , the set T (S) is pre-compact in H2 . Equivalently, for every bounded sequence (xn ) of H1 , (T xn ) has a convergent subsequence in H2 . We denote the set of all compact operators between H1 and H2 by K(H1 , H2 ). In case if H1 = H2 = H, then K(H1 , H2 ) is denoted by K(H). A bounded linear operator T : H1 → H2 is called finite rank if R(T ) is finite dimensional. The space of all finite rank operators between H1 and H2 is denoted by F(H1 , H2 ) and we write F(H, H) = F(H). All the above mentioned basics of operator theory can be found in [12, 4, 2, 11]. For T ∈ B(H1 , H2 ), the quantity m(T ) := inf {T x : x ∈ SH1 } is called the minimum modulus of T . If H1 = H2 = H and T −1 ∈ B(H), 1 (see [1, Theorem 1] for details). then m(T ) = T −1 The following definition is available in [9] for densely defined closed operators (not necessarily bounded) in a Hilbert space, and this holds true automatically for bounded operators. Definition 1.1. [9, Definition 8.3 page 178] Let T = T ∗ ∈ B(H). The discrete spectrum σd (T ) of T is defined as the set of all eigenvalues of T with finite multiplicities which are isolated points of the spectrum σ(T ) of T . The complement set σess (T ) = σ(T ) \ σd (T ) is called the essential spectrum of T . By the Weyl’s theorem we can assert that if T = T ∗ and K = K ∗ ∈ K(H), then σess (T + K) = σess (T ) (see [9, Corollary 8.16, page 182] for details). If H is a separable Hilbert space, the essential minimum modulus of T is
4
G. RAMESH
defined to be me (T ) := inf {λ : λ ∈ σess (|T |)} (see [1] for details). The same result in the general case is dealt in [8, Proposition 2.1]. Remark 1.2. Let T ∈ B(H) be self-adjoint. Since σess (T ) ⊆ σ(T ), then by [6, Proposition 2.1] we have m(T ) = inf {|λ| : λ ∈ σ(T )} ≤ inf {|μ| : μ ∈ σess (T )} = me (T ). Let H = H1 ⊕ H2 and T ∈ B(H). LetPj : H → H be an orthogonal T11 T12 , where Tij : Hj → projection onto Hj for j = 1, 2. Then T = T21 T22 Hi is the operator given by Tij = Pi T Pj |Hj . In particular, T (H1 ) ⊆ H1 if and only if T12 = 0. Also, H1 reduces T if and only if T12 = 0 = T21 (for details see [11, 4]. 2. Positive AN -operators In this section we describe the structure of operators which are positive and satisfy the AN -property. First, we recall a few results which are necessary for proving our results. An operator T ∈ B(H1 , H2 ) is said to be norm attaining if there exists a x ∈ SH1 such that T x = T . We denote the class of norm attaining operators by N (H1 , H2 ). It is known that N (H1 , H2 ) is dense in B(H1 , H2 ) with respect to the operator norm of B(H1 , H2 ). We refer [5] for a simple proof this fact. Recall that T ∈ B(H1 , H2 ) is said to be absolutely norm attaining or AN -operator (shortly), if T |M , the restriction of T to M , is norm attaining for every non zero closed subspace M of H1 . That is T |M ∈ N (M, H2 ) for every non zero closed subspace M of H1 . This class contains K(H1 , H2 ), and the class of partial isometries with finite dimensional null space or finite dimensional range space (See [3] for more details). We have the following characterization of norm attaining operators: Proposition 2.1. [3, Proposition 2.4] Let T ∈ B(H) be self-adjoint. Then (1) T ∈ N (H) if and only if either T ∈ σp (T ) or −T ∈ σp (T ) (2) if T ≥ 0, then T ∈ N (H) if and only if T ∈ σp (T ). We recall a characterization of positive AN -operators that we need later. Theorem 2.2. [10, Theorem 5.1] Let H be a complex Hilbert space of arbitrary dimension and let P be a positive operator on H. Then P is an AN operator iff P is of the form P = αI + K + F , where α ≥ 0, K is a positive compact operator and F is self-adjoint finite rank operator. A slight improved version of the above result is proved in [7]. Theorem 2.3. [7] Let H be an infinite dimensional Hilbert space and T ∈ AN (H). Then the following are equivalent: (1) T ≥ 0 and T ∈ AN (H)
ON AN -OPERATORS
5
(2) there exists a unique triple (K, F, α) where K ∈ K(H) is positive, F ∈ F(H) is positive with KF = 0, and α ≥ 0, K ≤ αI such that T = K − F + αI. Next, we prove a new characterization of positive AN -operators in terms of the essential spectrum of the operator. Theorem 2.4. Let H be an infinite dimensional Hilbert space H and T ∈ B(H) be positive. Then T ∈ AN (H) if and only σess (T ) is a singleton set and [m(T ), me (T )) contains only finitely many points. Proof. If T ∈ AN (H), then there exists a unique triple (K, F, α) as in Theorem 2.3. Observe that σess (T ) = {α} and me (T ) = α. If [m(T ), me (T )) contains infinitely many points say (βn ), then (βn ) has monotone, subsequence, say (βnk ). If (βnk ) is decreasing, then (βnk ) converges, say βnk → β. This shows that β ∈ σess (T ), which means me (T ) = β. This is a contradiction. If (βnk ) increases to β, then β = me (T ). But this contradicts the AN property of T , which can be seen by [10, Proposition 3.4]. To prove the converse, assume that σess (T ) is a singleton set and [m(T ), me (T )) contains only finitely many points. If σess (T ) = {α}, then me (T ) = α. Thus σ(T ) \ {me (T )} = σd (T ). As σd (T ) contains only isolated points of σ(T ) with finite multiplicities, it must be countable. First, let us consider the case when σd (T ) is finite. Suppose σd (T ) = {λ1 , λ2 , . . . .λn }. Then clearly in this case me (T ) cannot be the limit of σd (T ), it can be an eigenvalue with infinite multiplicity. Let H1 = nj=1 (T − λj I) and H2 = N (T − me (T )I). Then H = H1 ⊕ H2 . Define T1 on H1 by T1 = diag(λ1 , λ2 , . . . , λn ), the diagonal operator with diagonal entries λ1 , λ2 , . . . , λn . Define T2 on H2 by T2 = me (T )IH2 . Then clearly, T = T1 ⊕ T2 . Thus T = F + me (T )I,
where F = diag λ1 − me (T ), λ2 − me (T ), . . . , λn − me (T ) ⊕ 0 is a finite rank, self-adjoint operator. Now by Theorem 2.2, we can conclude that T ∈ AN (H). Next, we consider the case ∞when σd (T ) is countably infinite. Let σd (T ) = {λ1 , λ2 , λ3 , . . .} and H1 = n=1 Hn , where Hn := N (T − λn I). Since σd (T ) is countable, it follows that σ(T ) is countable, if the multiplicity of α is ignored (this case occurs when α is an eigenvalue with infinite multiplicity). Without loss of generality assume that λ1 = sup {λ : λ ∈ σd (T )}. Then λ1 = T . Let T1 = diag(λ1 , λ2 , . . . ) be the diagonal operator. Then T1 ∈ B(H1 ) and σ(T1 ) = σd (T ). Let {β , β , . . . , β } be the set of points in [m(T ), me (T )). Define H2 = m ˜ 1 2 ˜ m k=1 Hk , with Hk = N (T − βk I) and T2 := diag(β1 , β2 , . . . , βm ). Then T = T1 ⊕ T2 with σ(T1 ) = {λ1 , λ2 , . . .} ∪ {me (T )} and σ(T2 ) = {β1 , β2 , . . . , βm }. Note that T1 − me (T )IH1 =: K1 , a positive compact operator. Also, T2 −
6
G. RAMESH
me (T )IH2 = −F1 , where F1 is positive m × m diagonal matrix with diagonal me (T ) − β1 , . . . , me (T ) − βm . Thus T can be written as T =
T1 0 0 T2
K + me (T )IH1 = 0 −F + me (T )IH2 I H1 K1 0 0 0 0 + me (T ) . = − 0 0 0 F1 0 IH2 0 0 K1 0 , F := Write K := . Then T can be written as 0 0 0 F1 T = K − F + me (T )I, which is an AN -operator, by Theorem 2.3. In general if T ∈ AN (H1 , H2 ) it need not imply that T ∗ ∈ AN (H2 , H1 ). An example supporting this fact can be found in [10, Example 6.3]. Here we give a sufficient condition for such a result to hold true. The following lemma will be used in our next result. Lemma 2.5. [18, Lemma 3] Let T ∈ B(H). Then λ ∈ σess (|T |) if and only if λ2 ∈ σess (T ∗ T ). Theorem 2.6. [7, Corollary 2.10] Let T ∈ B(H1 , H2 ). Then T ∈ AN (H1 , H2 ) if and only if T ∗ T ∈ AN (H1 ). Theorem 2.7. Let T ∈ B(H1 , H2 ). Suppose that σess (T ∗ T ) = σess (T T ∗ ). Then the following holds. (1) T ∈ AN (H1 , H2 ) if and only T ∗ ∈ AN (H2 , H1 ) (2) T ∗ T ∈ AN (H1 ) if and only T T ∗ ∈ AN (H2 ). Proof. Proof of (1): Suppose T ∈ AN (H1 , H2 ). Then T ∗ T ∈ AN (H1 ), by Theorem 2.6. Thus by Theorem 2.3, we have T ∗ T = K − F + αI, where K ∈ K(H1 ) is positive, F ∈ F(H1 ) is positive and α ≥ 0 has the property that F ≤ αI. It is clear that σess (T ∗ T ) = {α} and me (T ∗ T ) = α. Note that if α = 0, then clearly T ∈ K(H1 , H2 ) and hence T ∗ ∈ K(H2 , H1 ). So T ∗ ∈ AN (H2 , H1 ). Thus in this case the result is proved. Now, assume that α > 0. By the hypothesis, we have σess (T T ∗ ) = {α} = σess (T T ∗ ). Since σ(T ∗ T ) \ {0} = σ(T T ∗ ) \ {0}, [α, T ∗ T ] and [α, T T ∗ ] contains same points. Also, [m(T T ∗ ), α) contains finitely many points as the non zero points of [m(T T ∗ ), α) and [m(T ∗ T ), α) are the same. We can conclude by Theorem 2.4 that T T ∗ ∈ AN (H2 ). Thus T ∗ ∈ AN (H2 , H1 ). The other implication follows by replacing T by T ∗ in the above argument. Proof of (2): If T ∗ T ∈ AN (H1 ), then T ∈ AN (H1 , H2 ) by Theorem 2.6. Now by (1), it is clear that T ∗ ∈ AN (H2 , H1 ). Now by Theorem 2.6 again, we can conclude that T T ∗ ∈ AN (H2 ). Applying these arguments for T ∗ , we get the reverse argument.
ON AN -OPERATORS
7
Remark 2.8. Let T ∈ B(H1 , H2 ) such that σess (T ∗ T ) = σess (T T ∗ ). Then by Lemma 2.5, we can easily show that me (T ) = me (T ∗ ). Theorem 2.9. Let T ∈ B(H). Suppose that (1) me (T ) = me (T ∗ ) (2) both T and T ∗ ∈ AN (H) Then σess (T ∗ T ) = σess (T T ∗ ). Proof. Since both T and T ∗ are AN -operators, we have σess (T ∗ T ) = {α} and σess (T T ∗ ) = {β}. Clearly, me (T ∗ T ) = α and me (T T ∗ ) = β. Note that by Lemma 2.5 and the hypothesis, we have me (|T |) = me (T ) = me (T ∗ ) = me (|T ∗ |), it follows that α = β. Thus σess (T ∗ T ) = σess (T T ∗ ). 3. Paranormal AN -operators In this section we describe the structure of paranormal AN -operators. We give sufficient conditions under which a paranormal operator will be AN -operator. The structure of self-adjoint and normal AN -operators is discussed in [7]. First, we prove a few elementary results which will be useful in proving main theorems. The following Lemma is proved in [16, Lemma 1]. We write the same in a format that is useful to us. Lemma 3.1. Let T ∈ B(H) and M = {x ∈ H : T x = T x}. Then M = N (T 2 I − T ∗ T ) = N (|T | − T I). Proof. If x ∈ M , then T x2 = T 2 x2 . This is equivalent to the condition; (T 2 I − T ∗ T )x, x = 0. Since T 2 I − T ∗ T ≥ 0, it follows that T ∗ T x = T 2 x. Thus x ∈ N (T 2 I − T ∗ T ). On the other hand if, x ∈ N (T 2 I − T ∗ T ), then T 2 I − |T |2 x = 0. Hence (T I + |T |)(T I − |T |)x = 0. Since (T I + |T |)−1 ∈ B(H), we can conclude that |T |x = T x. That is x ∈ N (|T | − T I). Moreover, T x = |T |x, we obtain that x ∈ M . Hence M = N (T 2 I − T ∗ T ). If x ∈ N (|T | − T I), then |T |x = T x and |T |2 x = T 2 x. That is ∗ T T x = T 2 x, concluding N (|T | − T I) ⊆ N (T 2 − T ∗ T ). Corollary 3.2. Let T ∈ B(H). Then T ∈ N (H) if and only if M = {0}. Definition 3.3. Let T ∈ B(H). Then T is said to be (1) Hyponormal if T ∗ T − T T ∗ ≥ 0 or T x2 ≥ T ∗ x2 for all x ∈ H. (2) Paranormal if T 2 x ≥ T x2 for all x ∈ SH . In other words, T is paranormal if T u2 ≤ T 2 uu for all u ∈ H. A hyponormal operator is paranormal. We refer [13, 14, 15] and [17] for more details on these class of operators. Lemma 3.4. [16, Theorem 3] Suppose T ∈ B(H) be paranormal, norm attaining. Then N (T 2 I − T ∗ T ) is invariant under T . Moreover, TT is isometry on M .
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G. RAMESH
Lemma 3.5. [13, Lemma 2.3] Let T ∈ B(H) be paranormal with T = 1. Then MT ∗ = {x ∈ H : T T ∗ x = x} = N (I − T T ∗ ) is invariant under T . Remark 3.6. Let T ∈ B(H). Then MT ∗ = N (T 2 I − T T ∗ ) is invariant under T . By combining the above two lemmas we can prove the following result which is a key point in the structure theorem of paranormal AN -operators. Lemma 3.7. Let T ∈ B(H) be paranormal. Then MT = N (T 2 I − T ∗ T ) is a reducing subspace for T . In addition if T ∈ N (H), then T ∈ σp (|T |). Proof. By Lemma 3.4, MT is invariant under T . To prove the Lemma it is enough to prove that MT is invariant under T ∗ . This follows by applying Lemma 3.5 to T ∗ and the fact that T ∗∗ = T . If T ∈ N (H), then MT = {0} and by Lemma 3.1, we have T ∈ σp (|T |). Lemma 3.8. Let T ∈ B(H) be paranormal. If M is a non zero invariant subspace, then T |M is also paranormal. Proof. Let TM := T |M . For x ∈ M , we have 2 xx. TM x2 = T x2 ≤ T 2 xx = TM (T x)x = TM
Thus TM is paranormal.
Theorem 3.9. Let T ∈ AN (H) be paranormal and Λ = σ(|T |). Then there exists Hβ , Uβ , where β∈Λ
(1) Hβ is a reducing subspace for T (2) Uβ ∈ B(Hβ ) is an isometry such that (a) H = Hβ β∈Λ (b) T = βUβ β∈Λ
(c) σ(T ) ⊆
βT, where T := {z ∈ C : |z| = 1}.
β∈Λ
Proof. Since T ∈ AN (H), H1 := N (T ∗ T − T 2 I) = N (|T | − T I) is a reducing subspace for T . Let T1 := T |H1 . Then T1 ∈ AN (H1 ). Note that for all x ∈ H1 , we have T ∗ T x = T 2 x. In fact, T1∗ = T ∗ |H1 , since H1 reduces T . If H1 = H, then T ∗ T = T 2 I. Write α1 = T . Then 1 α1 ∈ σp (|T |) and T ∗ T = α2 I. That is T is an isometric operator, call it α U1 . Then T1 = α1 U1 .
ON AN -OPERATORS
9
If H1 ⊂ H, then H = H1 ⊕⊥ H1⊥ . We can write T = T1 ⊕ T2 , where T2 := T |H ⊥ . In fact, T has the following representation; 1 T1 0 α1 U1 0 (3.1) T = = . 0 T2 0 T2 Write H2 = H1⊥ . Note that T2 ∈ AN (H2 ). As T2 is paranormal by Lemma 3.8, by proceeding as above, either T2 = α2 U2 or T2 = α2 U2 ⊕ T3 , where α2 = T2 ∈ σ(|T |). Thus T has the following representations; α1 U 1 0 (3.2) T = 0 α 2 U2 or ⎛ ⎞ α1 U 1 0 0 α2 U 2 0 ⎠ . (3.3) T =⎝ 0 0 0 T3 Proceeding as above we end up with the following two cases: Case 1: The process stops after n number of steps; In this case, we have that H = ⊕nk=1 Hk and T = α1 U1 ⊕ α2 U2 ⊕ · · · ⊕ αn Un . Note that α1 < α2 < · · · < αn and each αi ∈ σp (|T |) for i = 1, 2 . . . , n. Clearly, |T | = nk=1 Ik , where Ik : Hk → Hk is the identity operator. If all the eigenvalue αk has finite multiplicity, then each Hk is finite dimensional and each Uk is a unitary. In this case H = ⊕nk=1 Hk
(3.4)
and
T =
n
αi U i .
i=1
Clearly H is a finite dimensional Hilbert space and each of Uj is a unitary. Since |T | ∈ AN (H) it can have atmost one eigenvalue with infinite multiplicity, say αk , (1 ≤ k ≤ n). That is Hk is infinite dimensional. Thus
(3.5)
H=
n j=1,j=k
⎛ Hj ⊕ H k
and
T =⎝
n
⎞ αj U j ⎠ ⊕ α k U k .
j=1, j=k
Note that each Uj (j = k) is an unitary and Uk is an isometry. Clearly, we have σ(T ) ⊂ ⊕nj=1 αj T. Case 2: The process does not stop after finite steps; By proceeding as above we get a sequence (αn ) of positive numbers such that αn < αn+1 , subspaces Hn such that Tn = T |Hn = αn Un for each n ∈ N. Since αn ∈ σp (|T |) for each n ∈ N and αn ≥ m(T ), αn converges to α, say. Then by [7, Remark 2.5] we can conclude that α = me (T ), the essential minimum modulus of T . Also, there may exists atmost finitely many spectral values, β1 , β2 , . . . , βm between m(T ) and me (T ), by the structure of positive AN -operators (Theorem 2.4). Let H˜j , (j = 1, 2, . . . , m) be the eigenspaces and U˜j , (j = 1, 2, . . . , m) be the corresponding unitaries associated with
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G. RAMESH
β1 , β2 , . . . , βm , respectively. Since by [10, Theorem 3.1], the eigenvectors of m ˜ ∞ |T | spans a dense subspace of H, we must have H = k=1 Hk j=1 Hk . ∞ m Also note that σ(|T |) = {αn }n=1 ∪ {α} ∪ {βj }j=1 . Now the representation of T can be written as; ⎞
⎛ m ∞ αn U n ⊕ ⎝ βj U˜j ⎠ = βUβ . (3.6) T = n=1
j=1
β∈Λ
Now it is easy to compute the spectrum. Clearly, ∞ n ∞ n (3.7) σ(T ) = σ(αn Un ) ⊕ σ(βk Uk ) ⊆ αn T ⊕ βk T = βT,
n=1
n=1
k=1
k=1
β∈Λ
where denote the disjoint union. If β is an eigenvalue of |T | with infinite multiplicity, then by [10, Theorem 3.8(iv)] α = β. That is α is both an eigenvalue with infinite multplicity as well as the unique limit point of σ(|T |). In this case the representation of T can be written as; ⎛ (3.8)
T =⎝
⎞ βUβ ⎠ ⊕ αUα .
β∈Λ, β=α
Corollary 3.10. If T ∈ AN (H) paranormal, then T has infinitely many finite dimensional reducing subspaces. Proof. The subspaces {Hα }α∈Λ is a reducing subspace of T . Since Λ is countable, the conclusion holds. Remark 3.11. Unlike compact operators, a paranormal AN -operator need not be normal. For example, the right shift operator on 2 is paranormal, AN -operator, but it is not normal. Hence we have the following question. Question 3.12. When can a paranormal AN -operator is normal? Here we answer the above question. Theorem 3.13. Let T ∈ B(H) be paranormal and T ∈ AN (H). We have the following; (1) if |T | has no eigenvalue with infinite multiplicity, then T is normal. (2) If |T | has an eigenvalue with infinite multiplicity, say β, and Hβ = N (|T | − βI), then T is normal if and only if T |Hβ is normal. Proof. Proof of (1): If |T | has no eigenvalue with infinite multiplicity, then T is represented as in Equation 3.4 or Equation 3.6. In both cases, since each Uj is finite dimensional isometry, it must be unitary. Thus T must be normal. Proof of (2): We have Hβ = N (T ∗ T − β 2 I). If T is normal, as Hβ is a reducing subspace for T , it is clear that T |Hβ is normal.
ON AN -OPERATORS
11
1 T |Hβ is unitary. Now β the normality of T follows by Equations 3.5 and 3.8, depending on the case. On the other hand, if T |Hβ is normal, then Uβ =
Remark 3.14. If T ∈ K(H) is paranormal, then by (1) of Theorem 3.13, T must be normal. References [1] R. Bouldin, The essential minimum modulus, Indiana Univ. Math. J. 30 (1981), no. 4, 513–517. MR0620264 (82i:47001) [2] Paul Richard Halmos, A Hilbert space problem book, volume 19 of Graduate Texts in Mathematics, second edition, Encyclopedia of Mathematics and its Applications, 17 (Springer-Verlag, New York, 1982). [3] X. Carvajal and W. Neves, Operators that achieve the norm, Integral Equations Operator Theory 72 (2012), no. 2, 179–195. MR2872473 (2012k:47044) [4] John B. Conway, A course in functional analysis, volume 96 of Graduate Texts in Mathematics, second edition (Springer-Verlag, New York, 1990). [5] P. Enflo, J. Kover and L. Smithies, Denseness for norm attaining operator-valued functions, Linear Algebra Appl. 338 (2001), 139–144. MR1861118 (2002g:47148) [6] G. Ramesh, Structure theorem for AN -operators, J. Aust. Math. Soc. 96 (2014), no. 3, 386–395. MR3217722 [7] G. Ramesh and D. Venkunaidu, On Absolutely norm attaining operators, Preprint, 2016(https://arxiv.org/abs/1801.02432). [8] I. S. Feshchenko, On the essential spectrum of the sum of self-adjoint operators and the closedness of the sum of operator ranges, Banach J. Math. Anal. 8 (2014), no. 1, 55–63. MR3161682 [9] Schm¨ udgen, Konrad, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, 265, Springer, Dordrecht, 2012, xx+432, MR2953553 [10] S. K. Pandey and V. I. Paulsen, A spectral characterization of AN operators, J. Aust. Math. Soc. 102 (2017), no. 3, 369–391. MR3650963 [11] A. E. Taylor and D. C. Lay, Introduction to functional analysis, second edition, John Wiley & Sons, New York, 1980. MR0564653 (81b:46001) [12] W. Rudin, Functional analysis, McGraw-Hill, New York, 1973. MR0365062 (51 #1315) [13] V. Istr˘ a¸tescu, On some hyponormal operators, Pacific J. Math. 22 (1967), 413–417. MR0213893 [14] S. K. Berberian, A note on hyponormal operators, Pacific J. Math. 12 (1962), 1171– 1175. MR0149281 [15] J. G. Stampfli, Hyponormal operators, Pacific J. Math. 12 (1962), 1453–1458. MR0149282 [16] Lee, Jun Ik, On the norm attaining operators, The Korean Journal of Mathematics 20(2012) [17] T. Furuta, Invitation to linear operators, Taylor & Francis (2001). MR1978629 [18] J. Kover, Compact perturbations and norm attaining operators, Quaest. Math. 28 (2005)no. 4. MR2182451 Department of Mathematics, I. I. T. Hyderabad,, E-Block, 305, Kandi (V), Sangareddy, Telangana, India-502 285. E-mail address: [email protected]
G. Ramesh
PII: DOI: Reference:
S0022-247X(18)30417-7 https://doi.org/10.1016/j.jmaa.2018.05.024 YJMAA 22256
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Journal of Mathematical Analysis and Applications
Received date:
14 February 2017
Please cite this article in press as: G. Ramesh, Absolutely Norm attaining Paranormal operators, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.05.024
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ABSOLUTELY NORM ATTAINING PARANORMAL OPERATORS G. RAMESH Abstract. A bounded linear operator T : H1 → H2 , where H1 , H2 are Hilbert spaces is said to be norm attaining if there exists a unit vector x ∈ H1 such that T x = T . If for any closed subspace M of H1 , the restriction T |M : M → H2 of T to M is norm attaining, then T is called an absolutely norm attaining operator or AN -operator. We prove the following characterization theorem: A positive operator T defined on an infinite dimensional Hilbert space H is an AN -operator if and only if the essential spectrum of T is a single point and [m(T ), me (T )) contains atmost finitely many points. Here m(T ) and me (T ) are the minimum modulus and essential minimum modulus of T . As a consequence we obtain a sufficient condition under which the AN -property of an operator implies AN -property of its adjoint. We also study the structure of paranormal AN -operators and give a necessary and sufficient condition under which a paranormal AN operator is normal.
1. Introduction In this article we continue the study of absolutely norm attaining operators of the earlier work from [7]. The class of absolutely norm attaining operators is introduced in [3] and further the detailed study of these operators is appeared in [6, 10, 7]. In the present article first we prove a characterization theorem for positive AN -operators. In general if T is an AN -operator, it may not be true that T ∗ is also an AN -operator (See [10, Example 6.3] for more details). We give a sufficient condition under which this result holds true. Next, we study the structure of absolutely norm attaining paranormal operators. Specifically, we show that if T is a paranormal AN -operator, then there exists pairs (Hα , Uα ), where Hα is a reducing subspace of T and Uα is an isometry on Hα such that Date: 08:09 Wednesday 9th May, 2018. 1991 Mathematics Subject Classification. Primary 47A15; Secondary 47B07, 47B20, 47B40. Key words and phrases. Compact operator, norm attaining operator, AN -operator, Weyl’s theorem, paranormal operator, reducing subspace. 1
2
G. RAMESH
H=
β∈σ(|T |)
Hβ
and
T =
βUβ .
β∈σ(|T |)
Here σ(|T |) is the spectrum of |T |, the modulus of T . Among all Hα ’s atmost one can have infinite dimension. That means atmost of the Uβ ’s can be an isometry and the remaining must be unitary. We describe a necessary and sufficient condition which ensure the normality of a paranormal AN -operator. We organize the article as follows: In the remaining part of this section we describe basic definitions and notations needed to prove our main results. In the second section, we prove a characterization of positive AN -operators and as a consequence we give a sufficient condition which guarantees the AN -property of the adjoint operator when the operator has AN -property. In the final section, we discuss the structure of paranormal AN -operators. As a consequence we show that every such operator has countably many finite dimensional reducing subspaces. Another important consequence is that a paranormal compact operator is normal. Throughout we consider complex Hilbert spaces which will be denoted by H, H1 , H2 etc. The inner product and the induced norm on H are designated by ·, · and · . If M is a subspace of H, then SM := {x ∈ M : x = 1} is the unit sphere of M and the orthogonal complement of M in H is denoted by M ⊥ . If M is a closed subspace of M , then the orthogonal projection onto M is denoted by PM . We denote the space of all bounded linear operators between H1 and H2 by B(H1 , H2 ). In case if H1 = H2 = H, then we denote B(H1 , H2 ) by B(H). For T ∈ B(H1 , H2 ), there exists a unique operator denoted by T ∗ : H2 → H1 satisfying T x, y = x, T ∗ y for all x ∈ H1 and for all y ∈ H2 . This operator T ∗ is called the adjoint of T . The null space and the range spaces of T are denoted by N (T ) and R(T ), respectively. Let T ∈ B(H). Then T is said to be normal if T ∗ T = T T ∗ , self-adjoint if T = T ∗ . If T is self-adjoint and T x, x ≥ 0 for all x ∈ H, then T is called positive. It is well known that for a positive operator T , there exists a unique positive operator S ∈ B(H) such that S 2 = T . The operator S is 1 called as the positive square root of T and is denoted by T 2 . If S, T ∈ B(H) are self-adjoint and S − T ≥ 0, then we write this by S ≥ T. If P ∈ B(H) is such that P 2 = P , then P is called a projection. If N (P ) and R(P ) are orthogonal to each other, then P is called an orthogonal projection. A projection P is an orthogonal projection if and only if it is self-adjoint if and only if it is normal. We call an operator V ∈ B(H1 , H2 ) to be an isometry if V x = x for each x ∈ H1 . An operator V ∈ B(H1 , H2 ) is said to be a partial isometry
ON AN -OPERATORS
3
if V |N (V )⊥ is an isometry. That is V x = x for all x ∈ N (V )⊥ . If V ∈ B(H) is isometry and onto, then V is said to be a unitary operator. 1 If T ∈ B(H1 , H2 ), then T ∗ T ∈ B(H1 ) is positive and |T | := (T ∗ T ) 2 is called the modulus of T . In fact, there exists a unique partial isometry V ∈ B(H1 , H2 ) such that T = V |T | and N (V ) = N (T ). This factorization is called the polar decomposition of T . A closed subspace M of H is said to be invariant under T ∈ B(H) if T M ⊆ M and reducing if both M and M ⊥ are invariant under T . For T ∈ B(H), the set ρ(T ) := {λ ∈ C : T − λI : H → H is invertible and (T − λI)−1 ∈ B(H)} is called the resolvent set and the complement σ(T ) = C \ ρ(T ) is called the spectrum of T . It is well known that σ(T ) is a non empty compact subset of C. The point spectrum of T is defined by σp (T ) = {λ ∈ C : T − λI is not one-to-one}. Note that σp (T ) ⊆ σ(T ). If T ∈ B(H1 , H2 ), then T is said to be compact if for every bounded set S of H1 , the set T (S) is pre-compact in H2 . Equivalently, for every bounded sequence (xn ) of H1 , (T xn ) has a convergent subsequence in H2 . We denote the set of all compact operators between H1 and H2 by K(H1 , H2 ). In case if H1 = H2 = H, then K(H1 , H2 ) is denoted by K(H). A bounded linear operator T : H1 → H2 is called finite rank if R(T ) is finite dimensional. The space of all finite rank operators between H1 and H2 is denoted by F(H1 , H2 ) and we write F(H, H) = F(H). All the above mentioned basics of operator theory can be found in [12, 4, 2, 11]. For T ∈ B(H1 , H2 ), the quantity m(T ) := inf {T x : x ∈ SH1 } is called the minimum modulus of T . If H1 = H2 = H and T −1 ∈ B(H), 1 (see [1, Theorem 1] for details). then m(T ) = T −1 The following definition is available in [9] for densely defined closed operators (not necessarily bounded) in a Hilbert space, and this holds true automatically for bounded operators. Definition 1.1. [9, Definition 8.3 page 178] Let T = T ∗ ∈ B(H). The discrete spectrum σd (T ) of T is defined as the set of all eigenvalues of T with finite multiplicities which are isolated points of the spectrum σ(T ) of T . The complement set σess (T ) = σ(T ) \ σd (T ) is called the essential spectrum of T . By the Weyl’s theorem we can assert that if T = T ∗ and K = K ∗ ∈ K(H), then σess (T + K) = σess (T ) (see [9, Corollary 8.16, page 182] for details). If H is a separable Hilbert space, the essential minimum modulus of T is
4
G. RAMESH
defined to be me (T ) := inf {λ : λ ∈ σess (|T |)} (see [1] for details). The same result in the general case is dealt in [8, Proposition 2.1]. Remark 1.2. Let T ∈ B(H) be self-adjoint. Since σess (T ) ⊆ σ(T ), then by [6, Proposition 2.1] we have m(T ) = inf {|λ| : λ ∈ σ(T )} ≤ inf {|μ| : μ ∈ σess (T )} = me (T ). Let H = H1 ⊕ H2 and T ∈ B(H). LetPj : H → H be an orthogonal T11 T12 , where Tij : Hj → projection onto Hj for j = 1, 2. Then T = T21 T22 Hi is the operator given by Tij = Pi T Pj |Hj . In particular, T (H1 ) ⊆ H1 if and only if T12 = 0. Also, H1 reduces T if and only if T12 = 0 = T21 (for details see [11, 4]. 2. Positive AN -operators In this section we describe the structure of operators which are positive and satisfy the AN -property. First, we recall a few results which are necessary for proving our results. An operator T ∈ B(H1 , H2 ) is said to be norm attaining if there exists a x ∈ SH1 such that T x = T . We denote the class of norm attaining operators by N (H1 , H2 ). It is known that N (H1 , H2 ) is dense in B(H1 , H2 ) with respect to the operator norm of B(H1 , H2 ). We refer [5] for a simple proof this fact. Recall that T ∈ B(H1 , H2 ) is said to be absolutely norm attaining or AN -operator (shortly), if T |M , the restriction of T to M , is norm attaining for every non zero closed subspace M of H1 . That is T |M ∈ N (M, H2 ) for every non zero closed subspace M of H1 . This class contains K(H1 , H2 ), and the class of partial isometries with finite dimensional null space or finite dimensional range space (See [3] for more details). We have the following characterization of norm attaining operators: Proposition 2.1. [3, Proposition 2.4] Let T ∈ B(H) be self-adjoint. Then (1) T ∈ N (H) if and only if either T ∈ σp (T ) or −T ∈ σp (T ) (2) if T ≥ 0, then T ∈ N (H) if and only if T ∈ σp (T ). We recall a characterization of positive AN -operators that we need later. Theorem 2.2. [10, Theorem 5.1] Let H be a complex Hilbert space of arbitrary dimension and let P be a positive operator on H. Then P is an AN operator iff P is of the form P = αI + K + F , where α ≥ 0, K is a positive compact operator and F is self-adjoint finite rank operator. A slight improved version of the above result is proved in [7]. Theorem 2.3. [7] Let H be an infinite dimensional Hilbert space and T ∈ AN (H). Then the following are equivalent: (1) T ≥ 0 and T ∈ AN (H)
ON AN -OPERATORS
5
(2) there exists a unique triple (K, F, α) where K ∈ K(H) is positive, F ∈ F(H) is positive with KF = 0, and α ≥ 0, K ≤ αI such that T = K − F + αI. Next, we prove a new characterization of positive AN -operators in terms of the essential spectrum of the operator. Theorem 2.4. Let H be an infinite dimensional Hilbert space H and T ∈ B(H) be positive. Then T ∈ AN (H) if and only σess (T ) is a singleton set and [m(T ), me (T )) contains only finitely many points. Proof. If T ∈ AN (H), then there exists a unique triple (K, F, α) as in Theorem 2.3. Observe that σess (T ) = {α} and me (T ) = α. If [m(T ), me (T )) contains infinitely many points say (βn ), then (βn ) has monotone, subsequence, say (βnk ). If (βnk ) is decreasing, then (βnk ) converges, say βnk → β. This shows that β ∈ σess (T ), which means me (T ) = β. This is a contradiction. If (βnk ) increases to β, then β = me (T ). But this contradicts the AN property of T , which can be seen by [10, Proposition 3.4]. To prove the converse, assume that σess (T ) is a singleton set and [m(T ), me (T )) contains only finitely many points. If σess (T ) = {α}, then me (T ) = α. Thus σ(T ) \ {me (T )} = σd (T ). As σd (T ) contains only isolated points of σ(T ) with finite multiplicities, it must be countable. First, let us consider the case when σd (T ) is finite. Suppose σd (T ) = {λ1 , λ2 , . . . .λn }. Then clearly in this case me (T ) cannot be the limit of σd (T ), it can be an eigenvalue with infinite multiplicity. Let H1 = nj=1 (T − λj I) and H2 = N (T − me (T )I). Then H = H1 ⊕ H2 . Define T1 on H1 by T1 = diag(λ1 , λ2 , . . . , λn ), the diagonal operator with diagonal entries λ1 , λ2 , . . . , λn . Define T2 on H2 by T2 = me (T )IH2 . Then clearly, T = T1 ⊕ T2 . Thus T = F + me (T )I,
where F = diag λ1 − me (T ), λ2 − me (T ), . . . , λn − me (T ) ⊕ 0 is a finite rank, self-adjoint operator. Now by Theorem 2.2, we can conclude that T ∈ AN (H). Next, we consider the case ∞when σd (T ) is countably infinite. Let σd (T ) = {λ1 , λ2 , λ3 , . . .} and H1 = n=1 Hn , where Hn := N (T − λn I). Since σd (T ) is countable, it follows that σ(T ) is countable, if the multiplicity of α is ignored (this case occurs when α is an eigenvalue with infinite multiplicity). Without loss of generality assume that λ1 = sup {λ : λ ∈ σd (T )}. Then λ1 = T . Let T1 = diag(λ1 , λ2 , . . . ) be the diagonal operator. Then T1 ∈ B(H1 ) and σ(T1 ) = σd (T ). Let {β , β , . . . , β } be the set of points in [m(T ), me (T )). Define H2 = m ˜ 1 2 ˜ m k=1 Hk , with Hk = N (T − βk I) and T2 := diag(β1 , β2 , . . . , βm ). Then T = T1 ⊕ T2 with σ(T1 ) = {λ1 , λ2 , . . .} ∪ {me (T )} and σ(T2 ) = {β1 , β2 , . . . , βm }. Note that T1 − me (T )IH1 =: K1 , a positive compact operator. Also, T2 −
6
G. RAMESH
me (T )IH2 = −F1 , where F1 is positive m × m diagonal matrix with diagonal me (T ) − β1 , . . . , me (T ) − βm . Thus T can be written as T =
T1 0 0 T2
K + me (T )IH1 = 0 −F + me (T )IH2 I H1 K1 0 0 0 0 + me (T ) . = − 0 0 0 F1 0 IH2 0 0 K1 0 , F := Write K := . Then T can be written as 0 0 0 F1 T = K − F + me (T )I, which is an AN -operator, by Theorem 2.3. In general if T ∈ AN (H1 , H2 ) it need not imply that T ∗ ∈ AN (H2 , H1 ). An example supporting this fact can be found in [10, Example 6.3]. Here we give a sufficient condition for such a result to hold true. The following lemma will be used in our next result. Lemma 2.5. [18, Lemma 3] Let T ∈ B(H). Then λ ∈ σess (|T |) if and only if λ2 ∈ σess (T ∗ T ). Theorem 2.6. [7, Corollary 2.10] Let T ∈ B(H1 , H2 ). Then T ∈ AN (H1 , H2 ) if and only if T ∗ T ∈ AN (H1 ). Theorem 2.7. Let T ∈ B(H1 , H2 ). Suppose that σess (T ∗ T ) = σess (T T ∗ ). Then the following holds. (1) T ∈ AN (H1 , H2 ) if and only T ∗ ∈ AN (H2 , H1 ) (2) T ∗ T ∈ AN (H1 ) if and only T T ∗ ∈ AN (H2 ). Proof. Proof of (1): Suppose T ∈ AN (H1 , H2 ). Then T ∗ T ∈ AN (H1 ), by Theorem 2.6. Thus by Theorem 2.3, we have T ∗ T = K − F + αI, where K ∈ K(H1 ) is positive, F ∈ F(H1 ) is positive and α ≥ 0 has the property that F ≤ αI. It is clear that σess (T ∗ T ) = {α} and me (T ∗ T ) = α. Note that if α = 0, then clearly T ∈ K(H1 , H2 ) and hence T ∗ ∈ K(H2 , H1 ). So T ∗ ∈ AN (H2 , H1 ). Thus in this case the result is proved. Now, assume that α > 0. By the hypothesis, we have σess (T T ∗ ) = {α} = σess (T T ∗ ). Since σ(T ∗ T ) \ {0} = σ(T T ∗ ) \ {0}, [α, T ∗ T ] and [α, T T ∗ ] contains same points. Also, [m(T T ∗ ), α) contains finitely many points as the non zero points of [m(T T ∗ ), α) and [m(T ∗ T ), α) are the same. We can conclude by Theorem 2.4 that T T ∗ ∈ AN (H2 ). Thus T ∗ ∈ AN (H2 , H1 ). The other implication follows by replacing T by T ∗ in the above argument. Proof of (2): If T ∗ T ∈ AN (H1 ), then T ∈ AN (H1 , H2 ) by Theorem 2.6. Now by (1), it is clear that T ∗ ∈ AN (H2 , H1 ). Now by Theorem 2.6 again, we can conclude that T T ∗ ∈ AN (H2 ). Applying these arguments for T ∗ , we get the reverse argument.
ON AN -OPERATORS
7
Remark 2.8. Let T ∈ B(H1 , H2 ) such that σess (T ∗ T ) = σess (T T ∗ ). Then by Lemma 2.5, we can easily show that me (T ) = me (T ∗ ). Theorem 2.9. Let T ∈ B(H). Suppose that (1) me (T ) = me (T ∗ ) (2) both T and T ∗ ∈ AN (H) Then σess (T ∗ T ) = σess (T T ∗ ). Proof. Since both T and T ∗ are AN -operators, we have σess (T ∗ T ) = {α} and σess (T T ∗ ) = {β}. Clearly, me (T ∗ T ) = α and me (T T ∗ ) = β. Note that by Lemma 2.5 and the hypothesis, we have me (|T |) = me (T ) = me (T ∗ ) = me (|T ∗ |), it follows that α = β. Thus σess (T ∗ T ) = σess (T T ∗ ). 3. Paranormal AN -operators In this section we describe the structure of paranormal AN -operators. We give sufficient conditions under which a paranormal operator will be AN -operator. The structure of self-adjoint and normal AN -operators is discussed in [7]. First, we prove a few elementary results which will be useful in proving main theorems. The following Lemma is proved in [16, Lemma 1]. We write the same in a format that is useful to us. Lemma 3.1. Let T ∈ B(H) and M = {x ∈ H : T x = T x}. Then M = N (T 2 I − T ∗ T ) = N (|T | − T I). Proof. If x ∈ M , then T x2 = T 2 x2 . This is equivalent to the condition; (T 2 I − T ∗ T )x, x = 0. Since T 2 I − T ∗ T ≥ 0, it follows that T ∗ T x = T 2 x. Thus x ∈ N (T 2 I − T ∗ T ). On the other hand if, x ∈ N (T 2 I − T ∗ T ), then T 2 I − |T |2 x = 0. Hence (T I + |T |)(T I − |T |)x = 0. Since (T I + |T |)−1 ∈ B(H), we can conclude that |T |x = T x. That is x ∈ N (|T | − T I). Moreover, T x = |T |x, we obtain that x ∈ M . Hence M = N (T 2 I − T ∗ T ). If x ∈ N (|T | − T I), then |T |x = T x and |T |2 x = T 2 x. That is ∗ T T x = T 2 x, concluding N (|T | − T I) ⊆ N (T 2 − T ∗ T ). Corollary 3.2. Let T ∈ B(H). Then T ∈ N (H) if and only if M = {0}. Definition 3.3. Let T ∈ B(H). Then T is said to be (1) Hyponormal if T ∗ T − T T ∗ ≥ 0 or T x2 ≥ T ∗ x2 for all x ∈ H. (2) Paranormal if T 2 x ≥ T x2 for all x ∈ SH . In other words, T is paranormal if T u2 ≤ T 2 uu for all u ∈ H. A hyponormal operator is paranormal. We refer [13, 14, 15] and [17] for more details on these class of operators. Lemma 3.4. [16, Theorem 3] Suppose T ∈ B(H) be paranormal, norm attaining. Then N (T 2 I − T ∗ T ) is invariant under T . Moreover, TT is isometry on M .
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G. RAMESH
Lemma 3.5. [13, Lemma 2.3] Let T ∈ B(H) be paranormal with T = 1. Then MT ∗ = {x ∈ H : T T ∗ x = x} = N (I − T T ∗ ) is invariant under T . Remark 3.6. Let T ∈ B(H). Then MT ∗ = N (T 2 I − T T ∗ ) is invariant under T . By combining the above two lemmas we can prove the following result which is a key point in the structure theorem of paranormal AN -operators. Lemma 3.7. Let T ∈ B(H) be paranormal. Then MT = N (T 2 I − T ∗ T ) is a reducing subspace for T . In addition if T ∈ N (H), then T ∈ σp (|T |). Proof. By Lemma 3.4, MT is invariant under T . To prove the Lemma it is enough to prove that MT is invariant under T ∗ . This follows by applying Lemma 3.5 to T ∗ and the fact that T ∗∗ = T . If T ∈ N (H), then MT = {0} and by Lemma 3.1, we have T ∈ σp (|T |). Lemma 3.8. Let T ∈ B(H) be paranormal. If M is a non zero invariant subspace, then T |M is also paranormal. Proof. Let TM := T |M . For x ∈ M , we have 2 xx. TM x2 = T x2 ≤ T 2 xx = TM (T x)x = TM
Thus TM is paranormal.
Theorem 3.9. Let T ∈ AN (H) be paranormal and Λ = σ(|T |). Then there exists Hβ , Uβ , where β∈Λ
(1) Hβ is a reducing subspace for T (2) Uβ ∈ B(Hβ ) is an isometry such that (a) H = Hβ β∈Λ (b) T = βUβ β∈Λ
(c) σ(T ) ⊆
βT, where T := {z ∈ C : |z| = 1}.
β∈Λ
Proof. Since T ∈ AN (H), H1 := N (T ∗ T − T 2 I) = N (|T | − T I) is a reducing subspace for T . Let T1 := T |H1 . Then T1 ∈ AN (H1 ). Note that for all x ∈ H1 , we have T ∗ T x = T 2 x. In fact, T1∗ = T ∗ |H1 , since H1 reduces T . If H1 = H, then T ∗ T = T 2 I. Write α1 = T . Then 1 α1 ∈ σp (|T |) and T ∗ T = α2 I. That is T is an isometric operator, call it α U1 . Then T1 = α1 U1 .
ON AN -OPERATORS
9
If H1 ⊂ H, then H = H1 ⊕⊥ H1⊥ . We can write T = T1 ⊕ T2 , where T2 := T |H ⊥ . In fact, T has the following representation; 1 T1 0 α1 U1 0 (3.1) T = = . 0 T2 0 T2 Write H2 = H1⊥ . Note that T2 ∈ AN (H2 ). As T2 is paranormal by Lemma 3.8, by proceeding as above, either T2 = α2 U2 or T2 = α2 U2 ⊕ T3 , where α2 = T2 ∈ σ(|T |). Thus T has the following representations; α1 U 1 0 (3.2) T = 0 α 2 U2 or ⎛ ⎞ α1 U 1 0 0 α2 U 2 0 ⎠ . (3.3) T =⎝ 0 0 0 T3 Proceeding as above we end up with the following two cases: Case 1: The process stops after n number of steps; In this case, we have that H = ⊕nk=1 Hk and T = α1 U1 ⊕ α2 U2 ⊕ · · · ⊕ αn Un . Note that α1 < α2 < · · · < αn and each αi ∈ σp (|T |) for i = 1, 2 . . . , n. Clearly, |T | = nk=1 Ik , where Ik : Hk → Hk is the identity operator. If all the eigenvalue αk has finite multiplicity, then each Hk is finite dimensional and each Uk is a unitary. In this case H = ⊕nk=1 Hk
(3.4)
and
T =
n
αi U i .
i=1
Clearly H is a finite dimensional Hilbert space and each of Uj is a unitary. Since |T | ∈ AN (H) it can have atmost one eigenvalue with infinite multiplicity, say αk , (1 ≤ k ≤ n). That is Hk is infinite dimensional. Thus
(3.5)
H=
n j=1,j=k
⎛ Hj ⊕ H k
and
T =⎝
n
⎞ αj U j ⎠ ⊕ α k U k .
j=1, j=k
Note that each Uj (j = k) is an unitary and Uk is an isometry. Clearly, we have σ(T ) ⊂ ⊕nj=1 αj T. Case 2: The process does not stop after finite steps; By proceeding as above we get a sequence (αn ) of positive numbers such that αn < αn+1 , subspaces Hn such that Tn = T |Hn = αn Un for each n ∈ N. Since αn ∈ σp (|T |) for each n ∈ N and αn ≥ m(T ), αn converges to α, say. Then by [7, Remark 2.5] we can conclude that α = me (T ), the essential minimum modulus of T . Also, there may exists atmost finitely many spectral values, β1 , β2 , . . . , βm between m(T ) and me (T ), by the structure of positive AN -operators (Theorem 2.4). Let H˜j , (j = 1, 2, . . . , m) be the eigenspaces and U˜j , (j = 1, 2, . . . , m) be the corresponding unitaries associated with
10
G. RAMESH
β1 , β2 , . . . , βm , respectively. Since by [10, Theorem 3.1], the eigenvectors of m ˜ ∞ |T | spans a dense subspace of H, we must have H = k=1 Hk j=1 Hk . ∞ m Also note that σ(|T |) = {αn }n=1 ∪ {α} ∪ {βj }j=1 . Now the representation of T can be written as; ⎞
⎛ m ∞ αn U n ⊕ ⎝ βj U˜j ⎠ = βUβ . (3.6) T = n=1
j=1
β∈Λ
Now it is easy to compute the spectrum. Clearly, ∞ n ∞ n (3.7) σ(T ) = σ(αn Un ) ⊕ σ(βk Uk ) ⊆ αn T ⊕ βk T = βT,
n=1
n=1
k=1
k=1
β∈Λ
where denote the disjoint union. If β is an eigenvalue of |T | with infinite multiplicity, then by [10, Theorem 3.8(iv)] α = β. That is α is both an eigenvalue with infinite multplicity as well as the unique limit point of σ(|T |). In this case the representation of T can be written as; ⎛ (3.8)
T =⎝
⎞ βUβ ⎠ ⊕ αUα .
β∈Λ, β=α
Corollary 3.10. If T ∈ AN (H) paranormal, then T has infinitely many finite dimensional reducing subspaces. Proof. The subspaces {Hα }α∈Λ is a reducing subspace of T . Since Λ is countable, the conclusion holds. Remark 3.11. Unlike compact operators, a paranormal AN -operator need not be normal. For example, the right shift operator on 2 is paranormal, AN -operator, but it is not normal. Hence we have the following question. Question 3.12. When can a paranormal AN -operator is normal? Here we answer the above question. Theorem 3.13. Let T ∈ B(H) be paranormal and T ∈ AN (H). We have the following; (1) if |T | has no eigenvalue with infinite multiplicity, then T is normal. (2) If |T | has an eigenvalue with infinite multiplicity, say β, and Hβ = N (|T | − βI), then T is normal if and only if T |Hβ is normal. Proof. Proof of (1): If |T | has no eigenvalue with infinite multiplicity, then T is represented as in Equation 3.4 or Equation 3.6. In both cases, since each Uj is finite dimensional isometry, it must be unitary. Thus T must be normal. Proof of (2): We have Hβ = N (T ∗ T − β 2 I). If T is normal, as Hβ is a reducing subspace for T , it is clear that T |Hβ is normal.
ON AN -OPERATORS
11
1 T |Hβ is unitary. Now β the normality of T follows by Equations 3.5 and 3.8, depending on the case. On the other hand, if T |Hβ is normal, then Uβ =
Remark 3.14. If T ∈ K(H) is paranormal, then by (1) of Theorem 3.13, T must be normal. References [1] R. Bouldin, The essential minimum modulus, Indiana Univ. Math. J. 30 (1981), no. 4, 513–517. MR0620264 (82i:47001) [2] Paul Richard Halmos, A Hilbert space problem book, volume 19 of Graduate Texts in Mathematics, second edition, Encyclopedia of Mathematics and its Applications, 17 (Springer-Verlag, New York, 1982). [3] X. Carvajal and W. Neves, Operators that achieve the norm, Integral Equations Operator Theory 72 (2012), no. 2, 179–195. MR2872473 (2012k:47044) [4] John B. Conway, A course in functional analysis, volume 96 of Graduate Texts in Mathematics, second edition (Springer-Verlag, New York, 1990). [5] P. Enflo, J. Kover and L. Smithies, Denseness for norm attaining operator-valued functions, Linear Algebra Appl. 338 (2001), 139–144. MR1861118 (2002g:47148) [6] G. Ramesh, Structure theorem for AN -operators, J. Aust. Math. Soc. 96 (2014), no. 3, 386–395. MR3217722 [7] G. Ramesh and D. Venkunaidu, On Absolutely norm attaining operators, Preprint, 2016(https://arxiv.org/abs/1801.02432). [8] I. S. Feshchenko, On the essential spectrum of the sum of self-adjoint operators and the closedness of the sum of operator ranges, Banach J. Math. Anal. 8 (2014), no. 1, 55–63. MR3161682 [9] Schm¨ udgen, Konrad, Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, 265, Springer, Dordrecht, 2012, xx+432, MR2953553 [10] S. K. Pandey and V. I. Paulsen, A spectral characterization of AN operators, J. Aust. Math. Soc. 102 (2017), no. 3, 369–391. MR3650963 [11] A. E. Taylor and D. C. Lay, Introduction to functional analysis, second edition, John Wiley & Sons, New York, 1980. MR0564653 (81b:46001) [12] W. Rudin, Functional analysis, McGraw-Hill, New York, 1973. MR0365062 (51 #1315) [13] V. Istr˘ a¸tescu, On some hyponormal operators, Pacific J. Math. 22 (1967), 413–417. MR0213893 [14] S. K. Berberian, A note on hyponormal operators, Pacific J. Math. 12 (1962), 1171– 1175. MR0149281 [15] J. G. Stampfli, Hyponormal operators, Pacific J. Math. 12 (1962), 1453–1458. MR0149282 [16] Lee, Jun Ik, On the norm attaining operators, The Korean Journal of Mathematics 20(2012) [17] T. Furuta, Invitation to linear operators, Taylor & Francis (2001). MR1978629 [18] J. Kover, Compact perturbations and norm attaining operators, Quaest. Math. 28 (2005)no. 4. MR2182451 Department of Mathematics, I. I. T. Hyderabad,, E-Block, 305, Kandi (V), Sangareddy, Telangana, India-502 285. E-mail address: [email protected]