Some Estimates of Certain Subnormal and Hyponormal Derivations

Abstract

We prove that if 𝐴 and 𝐵^∗ are subnormal operators and 𝑋 is a bounded linear operator such that 𝐴𝑋−𝑋𝐵 is a Hilbert-Schmidt operator, then 𝑓(𝐴)𝑋−𝑋𝑓(𝐵) is also a Hilbert-Schmidt operator and ‖𝑓(𝐴)𝑋−𝑋𝑓(𝐵)‖₂≤𝐿‖𝐴𝑋−𝑋𝐵‖₂ for 𝑓 belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that 𝑆, 𝑇 are hyponormal operators and 𝑋∈ℒ(ℋ) is such that 𝑆𝑋−𝑋𝑇 belongs to a norm ideal (𝐽,‖⋅‖_𝐽), and we prove that 𝑓(𝑆)𝑋−𝑋𝑓(𝑇)∈𝐽 and ‖𝑓(𝑆)𝑋−𝑋𝑓(𝑇)‖_𝐽≤𝐶‖𝑆𝑋−𝑋𝑇‖_𝐽 for 𝑓 being in a certain class of functions.

1. Introduction

Let ℋ be a separable, infinite dimensional, complex Hilbert space, and denote by ℒ(ℋ) the algebra of all bounded linear operators on ℋ and by 𝒞₂(ℋ) the Hilbert-Schmidt class. For 𝑇∈ℒ(ℋ),𝜎(𝑇) denotes the spectrum of 𝑇, and for a compact subset Σ⊂ℂ,𝐿𝑖𝑝(Σ) denotes the set of Lipschitz functions on Σ. Furthermore, 𝑅𝑎𝑡(Σ) denotes the algebra of rational functions with poles of Σ, and 𝑅(Σ) denotes the closure of 𝑅𝑎𝑡(Σ) in the supremum norm over Σ.

For operators 𝐴,𝐵∈ℒ(ℋ), the mapping Δ_𝐴,𝐵(𝑋)=𝐴𝑋−𝑋𝐵 is called a (generalized) derivation. If 𝐴,𝐵 are normal (subnormal or co-subnormal, hyponormal or co-hyponormal) operators, then Δ_𝐴,𝐵 will be called a normal (subnormal, hyponormal) derivation, respectively.

Next, we recall some theorems that involve normal derivations, and then we extend some of these theorems to the case in which 𝐴,𝐵^∗ are subnormal operators and to the case in which 𝐴=𝑆,𝐵=𝑇 are hyponormal operators.

In [1], a generalization of Fuglede-Putnam theorem for normal operators was proved. For further results concerning normal derivations, the reader can see [2, 3].

Theorem 1.1 (See [1]). If 𝐴,𝐵∈ℒ(ℋ) are normal operators and 𝑋∈ℒ(ℋ) satisfies 𝐴𝑋−𝑋𝐵∈𝒞₂(ℋ), then 𝐴^∗𝑋−𝑋𝐵^∗∈𝒞₂(ℋ) and‖𝐴𝑋−𝑋𝐵‖₂=‖𝐴^∗𝑋−𝑋𝐵^∗‖₂. (1.1)

In [4], Furuta extended the above result to subnormal operators.

Theorem 1.2 (See [4]). If 𝐴,𝐵^∗∈ℒ(ℋ) are subnormal operators and 𝑋∈ℒ(ℋ) satisfies 𝐴𝑋−𝑋𝐵∈𝒞₂(ℋ), then 𝐴^∗𝑋−𝑋𝐵^∗∈𝒞₂(ℋ) and‖𝐴𝑋−𝑋𝐵‖₂≥‖𝐴^∗𝑋−𝑋𝐵^∗‖₂. (1.2)

In his paper [5], Kittaneh proved the following theorem using a famous result of Voiculescu [6] according to which every normal operator can be written as the sum of a diagonal operator and a Hilbert-Schmidt operator of an arbitrarily small Hilbert-Schmidt norm.

Theorem 1.3 (See [5]). Let 𝐴,𝐵∈ℒ(ℋ) be normal operators and 𝑋∈ℒ(ℋ) such that 𝐴𝑋−𝑋𝐵∈𝒞₂(ℋ), and let 𝑓∈𝐿𝑖𝑝(𝜎(𝐴)∪𝜎(𝐵)). Then 𝑓(𝐴)𝑋−𝑋𝑓(𝐵) is also a Hilbert-Schmidt operator and‖𝑓(𝐴)𝑋−𝑋𝑓(𝐵)‖₂≤𝐿‖𝐴𝑋−𝑋𝐵‖₂, (1.3)where 𝐿 is the Lipschitz constant of the function 𝑓.

2. Subnormal Derivations

In this section, we investigate the validity of this inequality in the case that 𝐴,𝐵^∗ are subnormal operators, but with a drawback concerning the extent of the class of functions in which 𝑓 can run.

The following lemma is elementary and can be easily established making use of the minimal normal extension of a subnormal operator. Its proof is left for the reader.

Lemma 2.1. If 𝑆₁,𝑆₂∈ℒ(ℋ) are subnormal operators, then there exists a Hilbert space 𝒦⊃ℋ and normal operators 𝑁₁,𝑁₂∈ℒ(𝒦) that are extensions of 𝑆₁,𝑆₂, respectively, and 𝜎(𝑁_𝑖)⊆𝜎(𝑆_𝑖),𝑖=1,2

For a subnormal operator 𝑆∈ℒ(ℋ) and a function 𝑓∈𝑅(𝜎(𝑆)), one can associate an operator 𝑓(𝑆)∈ℒ(ℋ) as follows. Let 𝑟_𝑛∈𝑅𝑎𝑡(𝜎(𝑆)),𝑛∈ℕ, such that‖𝑓−𝑟_𝑛‖_{𝜎(𝑆),∞}→0,as𝑛→∞, (2.1)and let 𝑁_𝑆∈ℒ(𝒦), where 𝒦⊃ℋ, be the minimal normal extension of 𝑆. Since 𝜎(𝑁_𝑆)⊆𝜎(𝑆), we have𝑟_𝑛(𝑁_𝑆𝑟)=^𝑛(𝑆)𝑆^′₁₂0𝑆^′₂₂, (2.2)and 𝑟_𝑛(𝑁_𝑆)→𝑓(𝑁_𝑆) in the operator norm of ℒ(𝒦). Therefore, 𝑟_𝑛(𝑆) converges in the operator norm of ℒ(ℋ) to an operator that will be denoted by 𝑓(𝑆). It is obvious that this operator does not depend on the sequence {𝑟_𝑛}. In a similar way, for 𝑓∈𝑅(𝜎(𝑇)), one can define 𝑓(𝑇), when 𝑇^∗∈ℒ(ℋ) is a subnormal operator.

Theorem 2.2. Let 𝐴,𝐵^∗∈ℒ(ℋ) be subnormal operators and 𝑋∈ℒ(ℋ) such that 𝐴𝑋−𝑋𝐵∈𝒞₂(ℋ), and let Σ=𝜎(𝐴)∪𝜎(𝐵) and 𝑓∈𝐿𝑖𝑝(Σ)∩𝑅(Σ). Then 𝑓(𝐴)𝑋−𝑋𝑓(𝐵) is also a Hilbert-Schmidt operator and‖𝑓(𝐴)𝑋−𝑋𝑓(𝐵)‖₂≤𝐿‖𝐴𝑋−𝑋𝐵‖₂, (2.3)where 𝐿 is the Lipschitz constant of the function 𝑓.

Proof. For subnormal operators 𝐴,𝐵^∗∈ℒ(ℋ), according to Lemma 2.1, there exists a Hilbert space 𝒦⊃ℋ and there are some normal operators 𝑁_𝐴,𝑁_𝐵^∗∈ℒ(𝒦) such that relative to the decomposition of 𝒦=ℋ⊕ℋ^⟂, we have𝑁_𝐴=𝐴𝐴₁₂0𝐴₂₂,𝑁_𝐵^∗=𝐵^∗𝐵₁₂0𝐵₂₂, (2.4)and 𝜎(𝑁_𝐴)⊆𝜎(𝐴),𝜎(𝑁_𝐵^∗)⊆𝜎(𝐵^∗).
If we put ^0𝑥0𝑋^02𝑑𝑐=𝑋⊕0 on ℋ⊕ℋ^⟂, then we have 𝑁𝑋_𝐴^{0𝑥002𝑑𝑐}−𝑋^{0𝑥002𝑑𝑐}𝑁^∗_𝐵^∗=(𝐴𝑋−𝑋𝐵)⊕0, and therefore 𝑁𝑋_𝐴^{0𝑥002𝑑𝑐}−𝑋^{0𝑥002𝑑𝑐}𝑁^∗_𝐵^∗∈𝒞₂(𝒦).
For 𝑟∈𝑅𝑎𝑡(Σ), where Σ=𝜎(𝐴)∪𝜎(𝐵), a simple calculation shows that𝑟(𝑁_𝐴)=𝑟(𝐴)𝐴^′₁₂0𝐴^′₂₂,𝑟(𝑁^∗_𝐵^∗𝐵)=𝑟(𝐵)0_′21𝐵^′₂₂. (2.5)Thus, if 𝑓∈𝐿𝑖𝑝(Σ)∩𝑅(Σ), using a limiting argument, one can see that 𝑓(𝑁_𝐴) and 𝑓(𝑁^∗_𝐵^∗) have similar matrix representation as in (2.5), but with 𝑓 replacing 𝑟. According to Theorem 1.3,𝑓(𝑁_𝐴)𝑋^{0𝑥002𝑑𝑐}−𝑋^{0𝑥002𝑑𝑐}𝑓(𝑁^∗_𝐵^∗)∈𝒞₂(𝒦),‖𝑓(𝑁_𝐴)𝑋^{0𝑥002𝑑𝑐}−𝑋^{0𝑥002𝑑𝑐}𝑓(𝑁^∗_𝐵^∗)‖₂≤𝐿‖𝑁𝑋_𝐴^{0𝑥002𝑑𝑐}−𝑋^{0𝑥002𝑑𝑐}𝑁^∗_𝐵^∗‖₂. (2.6)Since 𝑓(𝑁_𝐴)𝑋^{0𝑥002𝑑𝑐}−𝑋^{0𝑥002𝑑𝑐}𝑓(𝑁^∗_𝐵^∗)=(𝑓(𝐴)𝑋−𝑋𝑓(𝐵))⊕0, the proof is finished.

Corollary 2.3. Let 𝐴,𝐵^∗∈ℒ(ℋ) be subnormal operators and 𝑋∈ℒ(ℋ) such that 𝐴𝑋−𝑋𝐵∈𝒞₂(ℋ), and let Σ=𝜎(𝐴)∪𝜎(𝐵) and 𝑓∈𝐿𝑖𝑝(Σ)∩𝑅(Σ). Then‖𝑓(𝐴)^∗𝑋−𝑋𝑓(𝐵)^∗‖₂≤‖𝑓(𝐴)𝑋−𝑋𝑓(𝐵)‖₂, (2.7)and thus||𝑓(𝐴)^∗𝑋−𝑋𝑓(𝐵)^∗||₂≤𝐿||𝐴𝑋−𝑋𝐵||₂, (2.8)where 𝐿 is the Lipschitz constant of the function 𝑓.

Proof. The first inequality is a consequence of Theorem 1.2 after observing that 𝑓(𝐴) and 𝑓(𝐵)^∗ are subnormal operators. The second inequality follows from Theorem 2.2.

3. Hyponormal Derivations

In this section, we approach the same problem, but in the case in which 𝐴=𝑆,𝐵=𝑇 are hyponormal operators and the Hilbert-Schmidt class is replaced with an arbitrary norm ideal.

For a hyponormal operator 𝑇∈ℒ(ℋ), the analytic functional calculus can be extended to a class 𝐴^𝛼(𝜎(𝑇)) of “pseudo-analytic” functions on 𝜎(𝑇) that satisfy a certain growth condition at the boundary.

The extension of the analytic functional calculus for a hyponormal operator was introduced by Dyn'kin (cf. [7, 8]) and it also can be found in [9].

We briefly review the definition and the main tools that are necessary. Let Σ be a perfect compact set of the complex plane and let 𝛼 be a positive noninteger with 𝑘 its integer part, [𝛼]. The class 𝐴^𝛼(Σ) is defined as the set of (𝑘+1) tuples of continuous functions on Σ,(𝑓₀,…,𝑓_𝑘)∶Σ→ℂ^𝑘+1 that are related by𝑓_𝑗(𝑧)=𝑓_𝑗(𝑧₀𝑓)+^𝑗+1(𝑧₀)1!(𝑧−𝑧₀𝑓)+…+^𝑘(𝑧₀)(𝑘−𝑗)!(𝑧−𝑧₀)^𝑘−𝑗+𝑅_𝑗(𝑧₀,𝑧),|𝑅_𝑗(𝑧₀,𝑧)|≤𝐶_𝑗|𝑧−𝑧₀|^𝛼−𝑗 (3.1)for 𝑗=0,…,𝑘 and 𝑧,𝑧₀∈Σ. Since Σ is a perfect set,𝑓_𝑗(𝑧₀)=lim_𝑧→𝑧0𝑓_𝑗−1(𝑧)−𝑓_𝑗−1(𝑧₀)𝑧−𝑧₀,𝑗=0,…,𝑘−1, (3.2)and thus the (𝑘+1) tuple depends only on 𝑓₀. The space 𝐴^𝛼(Σ), endowed with the maximum of the smallest constants that satisfy (3.1) plus the supremum norm on Σ of 𝑓₀, becomes a unital Banach algebra and is a closed subalgebra of 𝐿𝑖𝑝(𝛼,Σ), the algebra of Lipschitz functions of order 𝛼.

Theorem 3.1 (See [8]). Let Σ be a perfect compact set, 𝑓∈𝐶(Σ), and 𝛼 a positive noninteger. The following are equivalent:
(a)𝑓∈𝐴^𝛼(Σ);(b)𝑓 has an extension 𝐹∈𝐶¹(ℂ⧵Σ) with |𝜕𝐹(𝑧)|≤𝐶⋅dist(𝑧,Σ)^𝛼−1,𝑧∉Σ;(c)there exists 𝜙∈𝐶₀(ℂ) such that𝑓(𝑧)=∫𝜙(𝑤)𝑤−𝑧𝑑𝜇(𝑤),𝑧∈Σ, (3.3) and |𝜙(𝑤)|≤𝐶₀⋅‖𝑓‖_𝐴^𝛼_(Σ)⋅dist(𝑤,Σ)^𝛼−1,𝑤∈ℂ, where 𝜇 is planar Lebesgue measure and 𝐶₀ is a constant that does not depend on 𝑓.

If 𝑇∈ℒ(ℋ) is a hyponormal operator, then ||𝑇||=||𝑇||_𝜎, where ||𝑇||_𝜎 denotes the spectral radius of 𝑇, that is sup_{𝑧∈𝜎(𝑇)}|𝑧|. It is well known that if 𝑧∉𝜎(𝑇), then (𝑧−𝑇)⁻¹ is also hyponormal and thus‖(𝑧−𝑇)⁻¹1‖=dist(𝑧,𝜎(𝑇)). (3.4)Thus, for a hyponormal operator 𝑇 whose spectrum 𝜎(𝑇) is a perfect set and for a function 𝑓∈𝐴^𝛼(𝜎(𝑇)) with 𝛼>2, one can associate an operator defined by𝜙(𝑤)(𝑤−𝑇)⁻¹𝑑𝜇(𝑤) (3.5)that will be denoted by 𝑓(𝑇). The above integral does not depend on 𝜙, that is, the definition of 𝑓(𝑇) is not ambiguous, and the mapping 𝜙↦𝑓(𝑇) acting from 𝐴^𝛼(𝜎(𝑇)) into ℒ(ℋ) is a continuous, unital morphism of Banach algebras, and which extends the Riesz-Dunford calculus.

Let (𝐽,||⋅||_𝐽) be a norm ideal, that is, a proper two-sided ideal 𝐽 of ℒ(ℋ) with a norm ||⋅||_𝐽 that satisfies th following: (𝐽,||⋅||_𝐽) is a Banach space and ‖𝐴𝑋𝐵‖_𝐽≤‖𝐴‖‖𝐵‖‖𝑋‖_𝐽, for all 𝑋∈𝐽 and any 𝐴,𝐵∈ℒ(ℋ). In particular, the Shatten-von Neumann 𝑝-classes, 𝒞_𝑝(ℋ), for 𝑝≥1, are instances of norm ideals.

Theorem 3.2. Let (𝐽,||⋅||_𝐽) be a norm ideal, let 𝑆,𝑇∈ℒ(ℋ) be hyponormal operators for which both 𝜎(𝑆) and 𝜎(𝑇) are perfect sets, let 𝑓 belong to 𝐴^𝛼(Σ) with 𝛼>3 and Σ=𝜎(𝑆)∪𝜎(𝑇), and let 𝑋∈ℒ(ℋ) such that 𝑆𝑋−𝑋𝑇∈𝐽. Then 𝑓(𝑆)𝑋−𝑋𝑓(𝑇)∈𝐽 and||𝑓(𝑆)𝑋−𝑋𝑓(𝑇)||_𝐽≤𝐶₁⋅||𝑓||_𝐴^𝛼_(Σ)⋅||𝑆𝑋−𝑋𝑇||_𝐽, (3.6)where 𝐶₁ is a constant that depends on Σ but it does not depend on 𝑓.

Proof. For 𝑓∈𝐴^𝛼(Σ), according to Theorem 3.1, there exists 𝜙∈𝐶₀(ℂ) such that𝑓(𝑧)=∫𝜙(𝑤)𝑤−𝑧𝑑𝜇(𝑤),𝑧∈Σ,|𝜙(𝑤)|≤𝐶₀⋅‖𝑓‖_𝐴^𝛼_(Σ)⋅dist(𝑤,Σ)^𝛼−1,𝑤∈ℂ. (3.7)Therefore,𝑓(𝑆)𝑋−𝑋𝑓(𝑇)=𝜙(𝑤)[(𝑤−𝑆)⁻¹𝑋−𝑋(𝑤−𝑇)⁻¹]𝑑𝜇(𝑤). (3.8)The domain of integration is supp(𝜙), which is a compact set that has in common with Σ only possibly boundary points of Σ. For 𝑤∈supp(𝜙)∩(ℂ⧵Σ),(𝑤−𝑆)⁻¹𝑋−𝑋(𝑤−𝑇)⁻¹=(𝑤−𝑆)⁻¹[𝑋(𝑤−𝑇)−(𝑤−𝑆)𝑋](𝑤−𝑇)⁻¹=(𝑤−𝑆)⁻¹[𝑆𝑋−𝑋𝑇](𝑤−𝑇)⁻¹∈𝐽, (3.9)and, according to (3.4),‖(𝑤−𝑆)⁻¹𝑋−𝑋(𝑤−𝑇)⁻¹‖_𝐽≤dist(𝑤,𝜎(𝑆))⁻¹⋅dist(𝑤,𝜎(𝑇))⁻¹⋅‖𝑆𝑋−𝑋𝑇‖_𝐽≤𝐶^′⋅dist(𝑤,Σ)⁻²⋅‖𝑆𝑋−𝑋𝑇‖_𝐽, (3.10)where 𝐶^′ is a constant that depends on Σ. Therefore, the integrant in (3.8) belongs to the norm ideal 𝐽 and‖𝜙(𝑤)(𝑤−𝑆)⁻¹𝑋−𝑋(𝑤−𝑇)⁻¹‖_𝐽≤𝐶₀⋅𝐶^′⋅‖𝑓‖_𝐴^𝛼_(Σ)⋅dist(𝑤,Σ)^𝛼−3‖𝑆𝑋−𝑋𝑇‖_𝐽, (3.11)for 𝑤∈supp(𝜙)∩(ℂ⧵Σ). After integration one obtains the desired conclusion of the theorem.