A Note on the Range of the Operator 𝑋↦𝑇𝑋−𝑋𝑇 Defined on 𝒞₂(ℋ)

Abstract

We show how a proof of J. Stampfli can be extended to prove that the operator 𝑋↦𝑇𝑋−𝑋𝑇 defined on the Hilbert-Schmidt class, when 𝑇 is an 𝑀-hyponormal, 𝑝-hyponormal, or log-hyponormal operator, has a closed range if and only if 𝜎(𝑇) is finite.

1. Introduction

Let ℋ be a complex, separable, infinite dimensional Hilbert space, and let ℒ(ℋ) denote the algebra of all linear bounded operators on ℋ. The Hilbert-Schmidt class, denoted by 𝒞₂(ℋ), is a Hilbert space with the ||⋅||₂-norm that arises from the inner product ⟨𝑋,𝑌⟩=tr(𝑋𝑌^∗), where tr is the scalar-valued trace. For 𝑇∈ℒ(ℋ), define Δ_𝑇∶ℒ(ℋ)→ℒ(ℋ) by Δ_𝑇(𝑋)=𝑇𝑋−𝑋𝑇, and let 𝜎(𝑇) denote the spectrum of 𝑇. Let the range of a linear operator 𝑆 be denoted by ℛ(𝑆). For a normal operator 𝑁∈ℒ(ℋ), Anderson and Foiaş [1] proved that ℛ(Δ_𝑁) is norm closed if and only if 𝜎(𝑁) is a finite set. In [2], Stampfli extended this result to the class of hyponormal operators.

Theorem 1 A ([2]). Let 𝑇∈ℒ(ℋ) be a hyponormal operator. Then ℛ(Δ_𝑇) is norm closed if and only if𝜎(𝑇) is finite.

In fact, Stampfli provided a proof of the “only if" implication which extends to a larger class of operators than the class of hyponormal operators (see Proposition 2.2). For an operator 𝑇∈ℒ(ℋ), let 𝜎_nap(𝑇) denote its normal approximate point spectrum , that is, the set of those 𝜆∈ℂ for which there exists an orthonormal sequence {𝜙_𝑛}_𝑛 in ℋ such that ‖(𝑇−𝜆)𝜙_𝑛‖+‖(𝑇−𝜆)^∗𝜙_𝑛‖→0. (1.1) Define the class 𝒢(ℋ) as follows: 𝒢(ℋ)∶={𝑇∈ℒ(ℋ)∣𝜎_nap(𝑇)isaninfiniteset}. (1.2) Some classes of hyponormal related operators, such as 𝑀-hyponormal operators, that is, 𝑚⋅‖(𝑇−𝜆)^∗𝜙‖≤‖(𝑇−𝜆)𝜙‖,(∀)𝜙∈ℋ,(∀)𝜆∈ℂ,forsome𝑚>0, (1.3)𝑝-hyponormal operators, that is, (𝑇^∗𝑇)^𝑝≥(𝑇𝑇^∗)^𝑝 for some 𝑝>0, or log-hyponormal operators, that is, invertible operators such that log(𝑇^∗𝑇)≥log(𝑇𝑇^∗), have spectrum that is finite or they belong to 𝒢(ℋ). Particularly, the hyponormal operators (i.e., 1-hyponormal) have this property.

In [3] Stampfli proved the following lemma which will be used in Section 2.

Lemma 1 B. Let 𝑇∈𝒢(ℋ) and let {𝜆_𝑛}^∞_𝑛=1 be a sequence of distinct points of 𝜎_𝑛𝑎𝑝(𝑇).Then for any sequence {𝜀_𝑛}^∞_𝑛=1 of positive numbers converging to zero, there exists an orthonormal sequence {𝜙_𝑛}^∞_𝑛=1 of vectors in ℋ such that ‖(𝑇−𝜆_𝑛)𝜙_𝑛‖+‖(𝑇−𝜆_𝑛)^∗𝜙_𝑛‖<𝜀_𝑛𝑓𝑜𝑟𝑛=1,2,…,⟨𝜙_𝑛,𝑇𝜙_𝑘⟩=0𝑓𝑜𝑟𝑘=1,…,𝑛−1. (1.4)(1.5)

2. The Closedness of the Range of Δ_𝐓⁽²⁾

The operator Δ_𝑇 defined on the Hilbert-Schmidt class will be denoted in the remainder of this note by Δ_𝑇⁽²⁾, that is, Δ_𝑇⁽²⁾∶𝒞₂(ℋ)→𝒞₂(ℋ), Δ_𝑇⁽²⁾(𝑋)=𝑇𝑋−𝑋𝑇. Let 𝐻^𝑀(ℋ) denote the set of 𝑀-hyponormal operators.

Proposition 2.1. Let 𝑇∈𝐻^𝑀(ℋ). If 𝜎(𝑇) is finite, then ℛ(Δ_𝑇⁽²⁾) is closed.

Proof. It is well known that an operator 𝑇∈𝐻^𝑀(ℋ) with finite spectrum is normal. Indeed, for such an operator, the restriction to an invariant subspace ℳ belongs to 𝐻^𝑀(ℳ). On the other hand, if 𝑇∈𝐻^𝑀(ℋ) with 𝜎(𝑇)={𝜆}, then 𝑇=𝜆𝐼, (cf. [4]). Thus, we can write 𝑇=∑^𝑛0_𝑖=1𝜆_𝑖𝐸_𝑖, where 𝐸_𝑖's are the spectral projections.
Let 𝑋_𝑛 and 𝐶 be in 𝒞₂(ℋ) such that‖Δ_𝑇⁽²⁾(𝑋_𝑛)−𝐶‖₂→0. Therefore Δ_𝑇(𝑋_𝑛)−𝐶→0 in the ℒ(ℋ) norm, and according to Theorem A, there exists 𝑋⁰∈ℒ(ℋ) such that 𝐶=𝑇𝑋⁰−𝑋⁰𝑇. For an arbitrary 𝑋∈ℒ(ℋ), let [𝑋_𝑖𝑗] be the block-matrix representation of 𝑋 relative to the decomposition ℋ=∑^𝑛0_𝑖=1⊕𝐸_𝑖ℋ. Thus 𝐶_𝑖𝑗=(𝜆_𝑖−𝜆_𝑗)𝑋⁰_𝑖𝑗, (2.1)
for all 𝑖,𝑗=1,…,𝑛₀. This implies that each 𝑋⁰_𝑖𝑗=(1/(𝜆_𝑖−𝜆_𝑗))𝐶_𝑖𝑗 is a Hilbert-Schmidt operator. Moreover 𝑋⁰_𝑖𝑖 can be chosen 0, and thus 𝑋⁰∈𝒞₂(ℋ).

Proposition 2.2. Let 𝑇∈𝒢(ℋ).Then ℛ(Δ_𝑇⁽²⁾) is not closed.

Proof. We will use same notation and circle of ideas as in [2]. Let {𝜆_𝑛}_𝑛≥1 be sequence of distinct points of 𝜎_nap(𝑇) so that 𝜆_𝑛→𝜆₀. Let 𝜂_𝑛=max{|𝜆_𝑗+1−𝜆_𝑗|^−1/2∣𝑗=1,…,𝑛}, (2.2) and choose a nonincreasing sequence {𝜀_𝑛}_𝑛≥1 so that 0<𝜀_𝑛≤|𝜆_𝑛+1−𝜆_𝑛|²,𝑛≥1, and ∑_𝑛≥1𝜀²_𝑛𝜂²_𝑛<∞. According to Lemma B, there exists an orthonormal sequence {𝜙_𝑛}_𝑛≥1 that satisfies (1.4) and(1.5). Let ℋ₁=∨{𝜙_𝑛∣𝑛≥1},ℋ₂=ℋ^⟂₁, and let 𝛿_𝑛 such that 𝑇𝜙_𝑛=𝜇_𝑛𝜙_𝑛+𝛿_𝑛,𝛿_𝑛⟂𝜙_𝑛,𝑛≥1. (2.3) It results that |𝜇_𝑛−𝜆_𝑛|<𝜀_𝑛,‖𝛿_𝑛‖<2𝜀_𝑛,𝑛≥1. (2.4) Define 𝑉∶ℋ→ℋ by 𝑉𝜙_𝑛=|𝜆_𝑗+1−𝜆_𝑗|^−1/2𝜙_𝑛+1,𝑛≥1, and 𝑉𝑔=0,𝑔∈ℋ₂. Let ℳ_𝑛=∨{𝜙_𝑗∣𝑗=1,…,𝑛} and let 𝑃_𝑛 be the orthogonal projection onto ℳ_𝑛, and define 𝑉_𝑛=𝑉𝑃_𝑛. A tedious calculation shows that Δ_𝑇(𝑉_𝑛)𝜙_𝑗=𝑣_𝑗(𝜇_𝑗+1−𝜇_𝑗)𝜙_𝑗+1+𝑣_𝑗𝛿_𝑗+1−𝑉_𝑛𝛿_𝑗,−𝑉_𝑛𝛿_𝑗,𝑗≤𝑛,𝑗>𝑛, (2.5) where 𝑣_𝑗=|𝜆_𝑗+1−𝜆_𝑗|^−1/2. Denoting Δ_𝑇(𝑉_𝑛)−Δ_𝑇(𝑉_𝑚) by Δ_𝑇^𝑛,𝑚, then for 𝑛<𝑚,Δ_𝑇^𝑛,𝑚𝜙_𝑗=0,−𝑣_𝑗(𝜇_𝑗+1−𝜇_𝑗)𝜙_𝑗+1+𝑣_𝑗𝛿_𝑗+1+(𝑉_𝑚−𝑉_𝑛)𝛿_𝑗,(𝑉_𝑚−𝑉_𝑛)𝛿_𝑗,𝑗≤𝑛,𝑛<𝑗≤𝑚,𝑗>𝑚. (2.6) Furthermore, from (2.3) it results that 𝛿_𝑗⟂𝜙_𝑗,𝜙_𝑗+1,𝜙_𝑗+2,… (2.7) and from (2.4) ‖𝑉_𝑛𝛿_𝑗‖≤2𝜂_𝑗𝜀_𝑗,∀𝑗,𝑛≥1. (2.8)
We will show next that ‖Δ_𝑇^𝑛,𝑚‖₂→0 when 𝑚,𝑛→∞, thus there exists 𝐶∈𝒞₂(ℋ) such that ‖Δ_𝑇(𝑉_𝑛)−𝐶‖₂→0, that is, 𝐶∈ℛ(Δ_𝑇⁽²⁾).
First, we will show that ‖Δ_𝑇^𝑛,𝑚∣_ℋ1‖²₂→0, when 𝑚,𝑛→∞. Indeed, ‖Δ_𝑇^𝑛,𝑚∣_ℋ1‖²₂=^∞_𝑗=1‖Δ_𝑇^𝑛,𝑚𝜙_𝑗‖=^2(2.6)=^𝑚_𝑗=𝑛+1‖−𝑣_𝑗(𝜇_𝑗+1−𝜇_𝑗)𝜙_𝑗+1+𝑣_𝑗𝛿_𝑗+1+(𝑉_𝑚−𝑉_𝑛)𝛿_𝑗‖²+^∞_𝑗=𝑚+1‖(𝑉_𝑚−𝑉_𝑛)𝛿_𝑗‖². (2.9)
The first sum of the right-hand side of the above can be majorized by 2⋅^𝑚_𝑗=𝑛+1‖−𝑣_𝑗(𝜇_𝑗+1−𝜇_𝑗)𝜙_𝑗+1+𝑣_𝑗𝛿_𝑗+1‖²+2⋅^𝑚_𝑗=𝑛+1‖(𝑉_𝑚−𝑉_𝑛)𝛿_𝑗‖². (2.10) Since 𝜙_𝑗+1⟂𝛿_𝑗+1, we have ‖Δ_𝑇^𝑛,𝑚∣_ℋ1‖²₂≤2[^𝑚_𝑗=𝑛+1(𝑣²_𝑗|𝜇_𝑗+1−𝜇_𝑗|²+𝑣²_𝑗‖𝛿_𝑗+1‖²)+^∞_𝑗=𝑛+1‖(𝑉_𝑚−𝑉_𝑛)𝛿_𝑗‖²]. (2.11) According to (2.8), ‖(𝑉_𝑚−𝑉_𝑛)𝛿_𝑗‖²≤16𝜂²_𝑗𝜀²_𝑗, (2.12) and according to (2.4), 𝑣²_𝑗‖𝛿_𝑗+1‖²≤4𝜂²_𝑗𝜀²_𝑗+1≤4𝜂²_𝑗𝜀²_𝑗,|𝜇_𝑗+1−𝜇_𝑗|²≤(2𝜀_𝑗+|𝜆_𝑗+1−𝜆_𝑗|)²≤8𝜀²_𝑗+2|𝜆_𝑗+1−𝜆_𝑗|², (2.13) which implies 𝑣²_𝑗|𝜇_𝑗+1−𝜇_𝑗|²≤8𝜂²_𝑗𝜀²_𝑗+2|𝜆_𝑗+1−𝜆_𝑗|. (2.14) Therefore ‖Δ_𝑇^𝑛,𝑚∣_ℋ1‖²₂≤𝑐₁⋅^∞_𝑗=𝑛+1𝜂²_𝑗𝜀²_𝑗+𝑐₂⋅^𝑚_𝑗=𝑛+1|𝜆_𝑗+1−𝜆_𝑗|, (2.15) where 𝑐₁ and 𝑐₂ are some constants. After a careful review of the proof, one can see that the sequence {𝜆_𝑛} can be assumed to converge fast enough (otherwise choose a subsequence of it), more precisely ∑^𝑚_𝑗=𝑛+1|𝜆_𝑗+1−𝜆_𝑗|→0 when 𝑛,𝑚→∞.
We show next that ‖Δ_𝑇^𝑛,𝑚∣_ℋ2‖²₂→0, when 𝑚,𝑛→∞. Indeed, we can write 𝑇^∗𝜙_𝑛=𝜇_𝑛𝜙_𝑛+𝛾_𝑛with⟨𝛾_𝑛,𝜙_𝑛⟩=0,‖𝛾_𝑛‖≤2𝜀_𝑛,𝑛≥1. (2.16)
Obviously, we can write 𝑇^∗𝜙_𝑛=𝜃_𝑛𝜙_𝑛+𝛾_𝑛 with ⟨𝛾_𝑛,𝜙_𝑛⟩=0, which implies 𝜃_𝑛=⟨𝜃_𝑛𝜙_𝑛+𝛾_𝑛,𝜙_𝑛⟩=⟨𝑇^∗𝜙_𝑛,𝜙_𝑛⟩=⟨𝜙_𝑛,𝑇𝜙_𝑛⟩=⟨𝜙_𝑛,𝜇_𝑛𝜙_𝑛+𝛿_𝑛⟩=𝜇_𝑛,‖𝛾_𝑛‖=‖(𝑇^∗−𝜇_𝑛)𝜙_𝑛‖≤‖(𝑇−𝜆_𝑛)^∗𝜙_𝑛‖+|𝜆_𝑛−𝜇_𝑛|≤^(1.4),(2.4)2𝜀_𝑛. (2.17) For an orthonormal basis {𝜓_𝑖}_𝑖≥1 of ℋ₂, we will show that ^∞_𝑖=1‖Δ_𝑇^𝑛,𝑚𝜓_𝑖‖²→0when𝑛,𝑚→∞. (2.18) For each 𝑖, write 𝑇𝜓_𝑖=∑^∞_𝑘=1𝑎_𝑘^(𝑖)𝜙_𝑘+𝑤_𝑖 with 𝑤_𝑖∈ℋ₂. Thus 𝑉_𝑚𝑇𝜓_𝑖=^𝑚_𝑘=1𝑎_𝑘^(𝑖)𝑉_𝑚𝜙_𝑘+𝑉_𝑚𝑤_𝑖=^𝑚_𝑘=1𝑎_𝑘^(𝑖)𝑣_𝑘𝜙_𝑘+1. (2.19) Since 𝑉_𝑚𝜓_𝑖=0, we have Δ_𝑇(𝑉_𝑚)𝜓_𝑖=−𝑉_𝑚𝑇𝜓_𝑖, and consequently, for 𝑛<𝑚,Δ_𝑇^𝑛,𝑚𝜓_𝑖=^𝑚_𝑘=𝑛+1𝑎_𝑘^(𝑖)𝑣_𝑘𝜙_𝑘+1. (2.20) Since the sequence {𝜙_𝑘} is orthonormal, we have ‖Δ_𝑇^𝑛,𝑚𝜓_𝑖‖²=^𝑚_𝑘=𝑛+1|𝑎_𝑘^(𝑖)|²⋅𝑣²_𝑘. (2.21) Therefore ^∞_𝑖=1‖Δ_𝑇^𝑛,𝑚𝜓_𝑖‖²=^∞𝑚_{𝑖=1𝑘=𝑛+1}|𝑎_𝑘^(𝑖)|²⋅𝑣²_𝑘=^𝑚_𝑘=𝑛+1𝑣²_𝑘(^∞_𝑖=1|𝑎_𝑘^(𝑖)|²). (2.22) For a fixed 𝑘,^∞_𝑖=1|𝑎_𝑘^(𝑖)|²=^∞_𝑖=1|⟨𝑇𝜓_𝑖,𝜙_𝑘⟩|²=^∞_𝑖=1|⟨𝜓_𝑖,𝑇^∗𝜙_𝑘⟩|²_(2.16)=^∞_𝑖=1|⟨𝜓_𝑖,𝜇_𝑘𝜙_𝑘+𝛾_𝑘⟩|²=^∞_𝑖=1|⟨𝜓_𝑖,𝛾_𝑘⟩|²≤‖𝛾_𝑘‖²_(2.16)≤4𝜀²_𝑘. (2.23) Consequently, ∑^∞_𝑖=1‖Δ_𝑇^𝑛,𝑚𝜓_𝑖‖²≤4∑^𝑚_𝑘=𝑛+1𝑣²_𝑘⋅𝜀²_𝑘→0 for 𝑛,𝑚→∞.
The operator 𝐶 is not in ℛ(Δ_𝑇⁽²⁾) since, according to the proof of Theorem A in [2], 𝐶∉ℛ(Δ_𝑇).

Theorem 2.3. Let 𝑇∈𝐻^𝑀(ℋ). Then ℛ(Δ_𝑇⁽²⁾) is closed if and only if 𝜎(𝑇) is finite.

Proof. If 𝑇∈𝐻^𝑀(ℋ) and 𝜎(𝑇) are finite, then according to Proposition 2.1, ℛ(Δ_𝑇⁽²⁾)is closed. Conversely, if 𝑇∈𝐻^𝑀(ℋ) has an infinite spectrum, then there are infinitely many distinct points {𝜆_𝑛}_𝑛 that are either isolated points of the spectrum, in which case they are eigenvalues, or accumulation points of the spectrum, in which case they are in 𝜎_ap(𝑇). Since 𝑇∈𝐻^𝑀(ℋ), we have 𝜎_𝑝(𝑇),𝜎_ap(𝑇)⊆𝜎_nap(𝑇). Thus 𝑇∈𝒢(ℋ) and according to Proposition 2.2, ℛ(Δ_𝑇⁽²⁾) is not closed.