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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2008Β (2008), Article IDΒ 362409, 6 pages
http://dx.doi.org/10.1155/2008/362409
Research Article

Some Estimates of Certain Subnormal and Hyponormal Derivations

Department of Mathematics, Florida A&M University, Tallahassee, FL 32307, USA

Received 23 October 2007; Accepted 9 January 2008

Academic Editor: NilsΒ Ackermann

Copyright Β© 2008 Vasile Lauric. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove that if and are subnormal operators and is a bounded linear operator such that is a Hilbert-Schmidt operator, then is also a Hilbert-Schmidt operator and for belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that , are hyponormal operators and is such that belongs to a norm ideal , and we prove that and for being in a certain class of functions.

1. Introduction

Let be a separable, infinite dimensional, complex Hilbert space, and denote by the algebra of all bounded linear operators on and by the Hilbert-Schmidt class. For denotes the spectrum of and for a compact subset denotes the set of Lipschitz functions on Furthermore, denotes the algebra of rational functions with poles of and denotes the closure of in the supremum norm over

For operators the mapping is called a (generalized) derivation. If are normal (subnormal or co-subnormal, hyponormal or co-hyponormal) operators, then 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 are subnormal operators and to the case in which 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 are normal operators and satisfies , then and

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

Theorem 1.2 (See [4]). If are subnormal operators and satisfies then and

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 be normal operators and such that and let Then is also a Hilbert-Schmidt operator andwhere is the Lipschitz constant of the function .

2. Subnormal Derivations

In this section, we investigate the validity of this inequality in the case that are subnormal operators, but with a drawback concerning the extent of the class of functions in which 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 are subnormal operators, then there exists a Hilbert space and normal operators that are extensions of respectively, and

For a subnormal operator and a function one can associate an operator as follows. Let such thatand let where be the minimal normal extension of Since we haveand in the operator norm of Therefore, converges in the operator norm of to an operator that will be denoted by It is obvious that this operator does not depend on the sequence In a similar way, for one can define when is a subnormal operator.

Theorem 2.2. Let be subnormal operators and such that and let and Then is also a Hilbert-Schmidt operator andwhere is the Lipschitz constant of the function .

Proof. For subnormal operators according to Lemma 2.1, there exists a Hilbert space and there are some normal operators such that relative to the decomposition of we haveand
If we put on then we have and therefore
For where a simple calculation shows thatThus, if using a limiting argument, one can see that and have similar matrix representation as in (2.5), but with replacing According to Theorem 1.3,Since the proof is finished.

Corollary 2.3. Let be subnormal operators and such that and let and Thenand thuswhere is the Lipschitz constant of the function

Proof. The first inequality is a consequence of Theorem 1.2 after observing that and 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 are hyponormal operators and the Hilbert-Schmidt class is replaced with an arbitrary norm ideal.

For a hyponormal operator the analytic functional calculus can be extended to a class of β€œpseudo-analytic” functions on 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 be a perfect compact set of the complex plane and let be a positive noninteger with its integer part, The class is defined as the set of tuples of continuous functions on that are related byfor and Since is a perfect set,and thus the tuple depends only on . The space , endowed with the maximum of the smallest constants that satisfy (3.1) plus the supremum norm on of , becomes a unital Banach algebra and is a closed subalgebra of the algebra of Lipschitz functions of order

Theorem 3.1 (See [8]). Let be a perfect compact set, and a positive noninteger. The following are equivalent:
(a)(b) has an extension with (c)there exists such that and where is planar Lebesgue measure and is a constant that does not depend on .

If is a hyponormal operator, then where denotes the spectral radius of that is It is well known that if then is also hyponormal and thusThus, for a hyponormal operator whose spectrum is a perfect set and for a function with one can associate an operator defined bythat will be denoted by The above integral does not depend on that is, the definition of is not ambiguous, and the mapping acting from into is a continuous, unital morphism of Banach algebras, and which extends the Riesz-Dunford calculus.

Let be a norm ideal, that is, a proper two-sided ideal of with a norm that satisfies th following: is a Banach space and for all and any In particular, the Shatten-von Neumann -classes, for are instances of norm ideals.

Theorem 3.2. Let be a norm ideal, let be hyponormal operators for which both and are perfect sets, let belong to with and and let such that Then andwhere is a constant that depends on but it does not depend on .

Proof. For according to Theorem 3.1, there exists such thatTherefore,The domain of integration is which is a compact set that has in common with only possibly boundary points of For and, according to (3.4),where is a constant that depends on Therefore, the integrant in (3.8) belongs to the norm ideal andfor After integration one obtains the desired conclusion of the theorem.

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