Remarks on Essential Maximal Numerical Range of Aluthge Transform

Abstract

This paper focuses on the properties of the essential maximal numerical range of Aluthge transform T̃ . For instance, among other results, we show that the essential maximal numerical range of Aluthge transform is nonempty and convex. Further, we prove that the essential maximal numerical range of Aluthge transform T̃ is contained in the essential maximal numerical range of T . This study is therefore an extention of the research on Aluthge transform which was begun by Aluthge in his study of p−hyponormal operators. 2010 Subject Classification Numbers: 46CXX, 46EXX, 47LXX. Introduction LetB(X) denote the algebra of bounded linear operators acting on a complex Hilbert spaceX . Let us recall that the Aluthge transform T̃ of T is the operator |T | 12U |T | 12 . Here, we denote by T the bounded linear operator on a complex Hilbert space X and let T = U |T | be any polar decomposition of T with U a partial isometry and |T | = (T ∗T ) 12 . Recall also that if kerT is the kernel of a bounded linear operator T then a bounded linear operator T ∈ B(X) is said to be an isometry if ∥Tx∥ = ∥x∥ ∀x ∈ X . We say that T is a partial isometry if it is an isometry on the orthogonal complement of its kernel, that is, for every x ∈ ker(T )⊥, ∥Tx∥ = ∥x∥. After its conception in 1900 by Aluthge [1], the notion of Aluthge transform and the study of its properties with their generalizations has attracted the attention of many authors such as in [5], [6] among others. This extensive research is because Aluthge transform is a very useful tool for studying some operator classes. Especially, it is used by many researchers in the study of p−hyponormal and semi\xadhyponormal operators. In 2007, Guoxing Ji, Ni Liu and Ze Li [6] together showed that the es\xad sential numerical range of Aluthge transform is contained in the essential numerical range of T . It is also known that spectrum of the normal operator T coincides with the spectrum of Aluthge transform T̃ , that is, σ(T ) = σ(T̃ ). See [5] for this and more. Aluthge transform T̃ of an m−tuple operator T = (T1, ..., Tm) ∈ B(X) was studied in [2], [3] and [4] and interesting results established. Essential Maximal Numerical Range of Aluthge Transform This section establishes some of the properties of the essential maximal numerical range of Aluthge transform. If T = U |T | is any polar decomposition of an operator T ∈ B(X)withU a partial isometry and T̃ = |T | 1 2U |T | 12 then we denote the essential maximal numerical range of Aluthge transform as MaxWe(T̃ ) and define it as MaxWe(T̃ ) = {r ∈ C : ⟨T̃ xn, xn⟩ → r, xn → 0 weakly and ∥T̃ xn∥ → ∥T̃∥e}. Theorem1. Let T = U |T | be any polar decomposition of an operator T ∈ B(X). If r ∈MaxWe(T̃ ) for any r ∈ C, then there exists an orthonormal sequence {xn} ∈ X such that International Journal of Pure Mathematical Sciences Submitted: 2019-01-22 ISSN: 2297-6205, Vol. 20, pp 1-5 Revised: 2020-01-02 doi:10.18052/www.scipress.com/IJPMS.20.1 Accepted: 2020-01-02 2019 SciPress Ltd, Switzerland Online: 2019-12-31 SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ ⟨T̃ xn, xn⟩ → r and ∥T̃ xn∥ → ∥T̃∥e. Proof. Suppose r ∈MaxWe(T̃ ). Then there is a sequence {xn} of vectors such that ⟨T̃ xn, xn⟩ → r, ∥xn∥ = 1, xn → 0 weakly and ∥T̃ xn∥ → ∥T̃∥e. Choosing the set {x1, ..., xn} which satisfy |⟨T̃ xn, xn⟩ − r| < 1i ∀i and letting M be the subspace spanned by x1, ..., xn and P be the projection onto M then we have ∥Pxn∥ → 0 as n → ∞. Let zn = ∥(I − P )xn∥((I − P )xn).We obtain T̃ zn = ∥(I − P )xn∥(T̃ (I − P )xn). This gives ⟨T̃ zn, zn⟩ = ⟨∥(I − P )xn∥(T̃ (I − P )xn), ∥(I − P )xn∥(T̃ (I − P )xn)⟩ = ∥(I − P )xn∥{⟨T̃ xn, xn⟩ − ⟨T̃ xn, Pxn⟩ − ⟨T̃Pxn, xn⟩+ ⟨T̃Pxn, Pxn⟩} → r as n → ∞. We then choose n large enough such that |⟨T̃ zn, zn⟩ − r| < 1 n+1 . If we let zn = xn+1 we get |⟨T̃ xn+1, xn+1⟩ − r| < 1 n+1 which completes the proof. Lemma2. Suppose T ∈ B(X), ∥T∥ = 1, ∥xn∥ = 1 and T = U |T | any polar decomposition of an operator T ∈ B(X). If ∥T̃ xn∥ ≥ (1− ε), then ∥ ( T̃ ∗T̃ − I ) xn∥ ≤ 2ε. Proof. Since T̃ ∗T̃ − I ≥ 0 it follows that, ∥ ( T̃ ∗T̃ − I ) xn∥ = ∥T̃ ∗T̃ xn∥ − 2∥T̃ xn∥ + ∥xn∥ ≤ 2 ( 1− ∥T̃ xn∥ ) ≤ 2ε. Theorem3. Let T = U |T | be any polar decomposition of an operator T ∈ B(X). Suppose that for a point r ∈ C there exists an orthonormal sequence {xn} ∈ X such that ⟨T̃ xn, xn⟩ → r and ∥T̃ xn∥ → ∥T̃∥e. Then r ∈MaxWe(T̃ ) Proof. Assume without loss of generality that for a point r ∈ C there exists an orthonormal sequence {xn} ∈ X such that ⟨T̃ xn, xn⟩ → r and ∥T̃ xn∥ → ∥T̃∥e. Since ∥xn∥ = 1 and every orthonormal sequence {xn} converges weakly to zero, it implies that r ∈ MaxWe(T̃ ). Theorem4. The set MaxWe(T̃ ) is nonempty and convex. Proof. Weprove thatMaxWe(T̃ ) is nonempty. To do this, fromTheorem 1, there exists an orthonormal sequence {xn} ∈ X such that ⟨T̃ xn, xn⟩ → r and ∥T̃ xn∥ → ∥T̃∥e. Thus the sequence {⟨T̃ xn, xn⟩} is bounded. Choose a subsequence and assume that ⟨T̃ xn, xn⟩ converges. Then MaxWe(T̃ ) is nonempty To show convexity, let r, μ ∈MaxWe(T̃ ). Since r, μ ∈MaxWe(T̃ ), it implies that there exist orthonor\xad mal sequences xn, yn ∈ X such that ∥xn∥ = 1 = ∥yn∥, ⟨ T̃ xn, xn⟩ → r and ∥T̃ xn∥ → ∥T̃∥e. Also ⟨T̃ yn, yn⟩ → μ and ∥T̃ yn∥ → ∥T̃∥e. Let Mn be a subspace spanned by xn and yn and Pn be a pro\xad jection ofX ontoMn. Suppose T̃n = PnT̃Pn, then ⟨T̃ xn, xn⟩ = ⟨T̃ yn, yn⟩ are in the numerical range of PnT̃Pn. By Toeplitz\xadHausdorff Theorem, W (PnT̃Pn) is convex and so for each n we can choose αn, βn with νn = αnxn + βnyn = 1 (where νn is a sequence in X). If η is a point on the line segment joining r and μ then ⟨T̃ νn, νn⟩ → η and ∥νn∥l = 1. Note that |⟨xn, yn⟩| ≤ θ < 1 for n sufficiently large. This implies that the angle between xn and yn is bounded away from 0. Therefore, there exists a constant M such that |αn| ≤ M and |βn| ≤ M for n sufficiently large, where ∥αnxn + βnyn∥ = 1. By Lemma 2, ∥T̃ νn∥ = ⟨T̃ ∗T̃ νn, νn⟩ = ∥νn∥ − 2Mε where ε → 0. That is, ∥ ( T̃ ∗T̃ − I ) xn∥ → 0 and ∥ ( T̃ ∗T̃ − I ) yn∥ → 0 as n → ∞. Thus ∥T̃ νn∥ → 1 as n → ∞ implying that ∥T̃ νn∥ → ∥T̃∥ as n → ∞. 2 IJPMS Volume 20