# Some Estimates of Certain Subnormal and Hyponormal Derivations

Authors: Vasile Lauric
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

We prove that ifA 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.


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## Body

## 1. Introduction

Letℋ 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 operatorsA,B∈ℒ(ℋ), the mapping ΔA,B(X)=AX−XB is called a (generalized) derivation. If A,B are normal (subnormal or co-subnormal,
hyponormal or co-hyponormal) operators, then ΔA,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 whichA,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]).
IfA,B∈ℒ(ℋ) are normal operators and X∈ℒ(ℋ) satisfies AX−XB∈𝒞2(ℋ),
then A*X−XB*∈𝒞2(ℋ) and(1.1)∥AX−XB∥2=∥A*X−XB*∥2.In [4], Furuta extended the above result to subnormal
operators.Theorem 1.2 (See [4]).
IfA,B*∈ℒ(ℋ) are subnormal operators and X∈ℒ(ℋ) satisfies AX−XB∈𝒞2(ℋ), then A*X−XB*∈𝒞2(ℋ) and(1.2)∥AX−XB∥2≥∥A*X−XB*∥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]).
LetA,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(1.3)∥f(A)X−Xf(B)∥2≤L∥AX−XB∥2,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 thatA,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 theminimal
normal extension of a subnormal operator. Its proof is left for the reader.Lemma 2.1.
IfS1,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(Ni)⊆σ(Si),i=1,2For a subnormal operatorS∈ℒ(ℋ) and a function f∈R(σ(S)), one can associate an operator f(S)∈ℒ(ℋ) as follows. Let rn∈Rat(σ(S)),n∈ℕ, such that(2.1)∥f−rn∥σ(S),∞→0,asn→∞,and let NS∈ℒ(𝒦), where 𝒦⊃ℋ, be the minimal normal extension of S. Since σ(NS)⊆σ(S), we have(2.2)rn(NS)=(rn(S)S12′0S22′),and rn(NS)→f(NS) in the operator norm of ℒ(𝒦). Therefore, rn(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 {rn}. 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.
LetA,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(2.3)∥f(A)X−Xf(B)∥2≤L∥AX−XB∥2,where L is the Lipschitz constant of the function f.Proof.
For subnormal operatorsA,B*∈ℒ(ℋ), according to Lemma 2.1, there exists a
Hilbert space 𝒦⊃ℋ and there are some normal operators NA,NB*∈ℒ(𝒦) such that relative to the decomposition of 𝒦=ℋ⊕ℋ⊥, we have(2.4)NA=(AA120A22),NB*=(B*B120B22),and σ(NA)⊆σ(A),σ(NB*)⊆σ(B*).
If we putX˜=X⊕0 on ℋ⊕ℋ⊥, then we have NAX˜−X˜NB**=(AX−XB)⊕0, and therefore NAX˜−X˜NB**∈𝒞2(𝒦).
Forr∈Rat(Σ), where Σ=σ(A)∪σ(B), a simple calculation shows that(2.5)r(NA)=(r(A)A12′0A22′),r(NB**)=(r(B)0B21′B22′).Thus, if f∈Lip(Σ)∩R(Σ), using a limiting argument, one can see that f(NA) and f(NB**) have similar matrix representation as in (2.5),
but with f replacing r. According to Theorem 1.3,(2.6)f(NA)X˜−X˜f(NB**)∈𝒞2(𝒦),∥f(NA)X˜−X˜f(NB**)∥2≤L∥NAX˜−X˜NB**∥2.Since f(NA)X˜−X˜f(NB**)=(f(A)X−Xf(B))⊕0, the proof is finished.Corollary 2.3.
LetA,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(2.7)∥f(A)*X−Xf(B)*∥2≤∥f(A)X−Xf(B)∥2,and thus(2.8)||f(A)*X−Xf(B)*||2≤L||AX−XB||2,where L is the Lipschitz constant of the function f.Proof.
The first inequality is a consequence of Theorem1.2 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 whichA=S,B=T are hyponormal operators and the
Hilbert-Schmidt class is replaced with an arbitrary norm ideal.For a hyponormal operatorT∈ℒ(ℋ), the analytic functional calculus can be
extended to a class Aα(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. LetΣ be a perfect compact set of the complex plane
and let α be a positive noninteger with k its integer part, [α]. The class Aα(Σ) is defined as the set of (k+1) tuples of continuous functions on Σ,(f0,…,fk):Σ→ℂk+1 that are related by(3.1)fj(z)=fj(z0)+fj+1(z0)1!(z−z0)+…+fk(z0)(k−j)!(z−z0)k−j+Rj(z0,z),|Rj(z0,z)|≤Cj|z−z0|α−jfor j=0,…,k and z,z0∈Σ. Since Σ is a perfect set,(3.2)fj(z0)=limz→z0fj−1(z)−fj−1(z0)z−z0,j=0,…,k−1,and thus the (k+1) tuple depends only on f0. The space Aα(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]).
LetΣ be a perfect compact set, f∈C(Σ), and α a positive noninteger. The following are
equivalent:
(a)
f∈Aα(Σ);(b)
f has an extension F∈C1(ℂ∖Σ) with |∂¯F(z)|≤C⋅dist(z,Σ)α−1,z∉Σ;(c)
there existsϕ∈C0(ℂ) such that(3.3)f(z)=∫ϕ(w)w−zdμ(w),z∈Σ, and |ϕ(w)|≤C0⋅∥f∥Aα(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.IfT∈ℒ(ℋ) is a hyponormal operator, then ||T||=||T||σ, where ||T||σ denotes the spectral radius of T, that is supz∈σ(T)|z|. It is well known that if z∉σ(T), then (z−T)−1 is also hyponormal and thus(3.4)∥(z−T)−1∥=1dist(z,σ(T)).Thus, for a hyponormal operator T whose spectrum σ(T) is a perfect set and for a function f∈Aα(σ(T)) with α>2, one can associate an operator defined by(3.5)∫ϕ(w)(w−T)−1dμ(w)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 Aα(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, 𝒞p(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 Aα(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(3.6)||f(S)X−Xf(T)||J≤C1⋅||f||Aα(Σ)⋅||SX−XT||J,where C1 is a constant that depends on Σ but it does not depend on f.Proof.
Forf∈Aα(Σ), according to Theorem 3.1, there exists ϕ∈C0(ℂ) such that(3.7)f(z)=∫ϕ(w)w−zdμ(w),z∈Σ,|ϕ(w)|≤C0⋅∥f∥Aα(Σ)⋅dist(w,Σ)α−1,w∈ℂ.Therefore,(3.8)f(S)X−Xf(T)=∫ϕ(w)[(w−S)−1X−X(w−T)−1]dμ(w).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),(3.9)(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,and, according to (3.4),(3.10)∥(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,where C′ is a constant that depends on Σ. Therefore, the integrant in (3.8) belongs to the
norm ideal J and(3.11)∥ϕ(w)(w−S)−1X−X(w−T)−1∥J≤C0⋅C′⋅∥f∥Aα(Σ)⋅dist(w,Σ)α−3∥SX−XT∥J,for w∈supp(ϕ)∩(ℂ∖Σ). After integration one obtains the desired
conclusion of the theorem.



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*Source: 362409-2008-04-13.xml*