Abstract

In this paper, we aim to develop formulas of spectral radius for an operator in terms of generalized Aluthge transform, numerical radius, iterated generalized Aluthge transform, and asymptotic behavior of powers of . These formulas generalize some of the formulas of spectral radius existing in literature. As an application, these formulas are used to obtain several characterizations of normaloid operators.

1. Introduction

Generally, in mathematical analysis and particularly in functional analysis, the spectral analysis of operators is an essential research topic. It is useful to study the properties of operators, including spectrum and the spectral radius of operators (see [1]). The spectrum of an operator is connected with an invariant subspace problem on a complex Hilbert space (see [2]), and the important property of spectrum is the expression of spectral radius in various formulas (see [3–5]). These formulas help to obtain several characterizations of operators, including normaloid and spectraloid operators (see [6]). Since the advent of various transformations of bounded linear operators, including Aluthge transform and its generalizations, the study of spectral properties of operators has become the center point for many researchers (see [7–9]).

An operator can be decomposed into two Hermitian operators being its real and imaginary parts, and this decomposition is known as Cartesian decomposition. Clearly, Hermitian operators are self-adjoint and hence symmetric operators. The symmetric operators involved in Cartesian decomposition are helpful to develop the spectral radius formulas and numerical radius inequalities involving Aluthge transform [10–12].

This paper is aimed at studying the generalization of spectral radius formulas involving generalized Aluthge transform. Henceforward, we will give the notions to proceed with the results of this paper.

Let be the algebra of all bounded linear operators on complex Hilbert space . Let be the polar decomposition of , where is the square root of an operator defined as and is a partial isometry.

In [13], Aluthge introduced a transform to study the properties of hyponormal operators that were connected with the invariant subspace problem in operator theory. This transform is called Aluthge transform, which is defined as and its th iterated Aluthge transform is defined as

Yamazaki, in [3], gave the formula of spectral radius for bounded linear operator involving iterated Aluthge transform, i.e.,

In [14], a generalization of Aluthge transform was introduced that is called -Aluthge transform which is defined as

Tam [4] gave a formula of spectral radius involving iterated -Aluthge transform for invertible operators using unitarily invariant norm, i.e.,

Chabbabi and Mbekhta [12] gave various expressions for spectral radius formulas involving -Aluthge transform, iterated -Aluthge transform, asymptotic behavior of powers of an operator, and numerical radius. The expression of spectral radius involving -Aluthge transform is given by

and the expressions of spectral radius involving iterated -Aluthge transform and the asymptotic behavior of powers of are given by

for each .

The expressions of spectral radius involving iterated -Aluthge transform, numerical radius, and the asymptotic behavior of powers of are given by

for each . With the help of the above formulas, the author [14] gave a characterization of normaloid operators.

In [15], Shebrawi and Bakherad introduced a new generalization of Aluthge transform, called generalized Aluthge transform. This transform is defined as

where and both are continuous functions such that . The iterated generalized Aluthge transform is defined as

In this paper, we establish the formulas of spectral radius for operator by assuming that These formulas generalize the spectral radius formulas (6)–(10).

The paper is organized as follows. In Section 2, we give the properties of the generalized Aluthge transform. In Section 3, spectral radius formulas involving generalized Aluthge transform and asymptotic behavior of powers of the bounded operator are given. In Section 4, we develop spectral radius formulas of bounded linear operators involving numerical radius of generalized Aluthge transform. Furthermore, some characterizations of normaloid operators are established.

2. Preliminaries and Some Auxiliary Results

We start this section with some basic definitions and properties of generalized Aluthge transform which will be useful in establishing the main results of this paper. An operator is similar to if there exists an invertible operator such that (see [16]). If , then the operator is said to be normaloid. An operator is said to be a contraction if . The spectral radius of an operator is defined as where is the spectrum of the operator .

To prove spectral radius formulas, we recall some properties of generalized Aluthge transform.

Proposition 1 [7]. Let . Then, we have (i)(ii)

Proposition 2. Let . If is similar to , then (i)(ii)(iii)

Proof. The proofs of parts (i) and (iii) are trivial. The proof of part (ii) follows from part (i) and Proposition 1 (i).

Proposition 3. Let and be any continuous function on . Then, for any unitary .

Proof. Since , we have for each , which implies for any polynomial . Since is a continuous, so there exist a sequence of polynomial such that for each , and converges uniformly to on the interval . Then, from Equation (16), we have as required.

Proposition 4. Let such that is unitary. Then, we have

Proof. Let be the polar decomposition of Then, we have Now by using Proposition 3, we have The polar decomposition of operator is as follows: where is partial isometry. Therefore, The second equality holds by Proposition 3 and by the fact that .

Proposition 5. Let . Then, the sequence is nonincreasing.

Proof. The proof follows from the repeated application of the inequality

3. Formulas of Spectral Radius Involving Generalized Aluthge Transform

In this section, we give formulas of the spectral radius by using Rota’s theorem [16] and the properties of generalized Aluthge transform.

Theorem 6. Let . Then, we have

Proof. From Propositions 1 and 2, we have It follows that Hence, Let be the polar decomposition of Since is an invertible operator, then is unitary and invertible. Therefore, there exists such that . Consequently, exists and self-adjoint; then, we have Therefore, The second equality holds by Proposition 4. Hence, To prove above inequality in other direction, for an arbitrary , we define an operator For operator , we have From [16], Theorem 2, the spectrum of operator lies in the unit disk; thus, the operator is similar to contraction for which there exists an invertible operator such that and this implies that For , we obtain Since is arbitrary, therefore

The next Corollary is the direct result of Theorem 6 involving iterated generalized Aluthge transform.

Corollary 7. Let . Then, for each , we have

Proof. From Propositions 1 and 2, we can easily obtain From above equality and by using Proposition 5, we have for all invertible Therefore, The third inequality holds by Proposition 5, and the last equality holds by Theorem 6, which completes the proof.

The next Corollary is the direct result of Corollary 7 that is the characterization of normaloid operators.

Corollary 8. Let . Then, the following assertions are equivalent (i) is normaloid(ii), for invertible

Proof. Assume that is normaloid. Then, for all invertible The first equality holds by Proposition 2. The first inequality holds because the spectral radius is less than the operator norm, and the second inequality holds by Proposition 5.
Assume that assertion (ii) holds. Then, we have for all invertible The last inequality holds by inequality (33) in Theorem 6. Since is arbitrary, hence is normaloid.

Corollary 9. Let . Then the following assertions are equivalent. (i) is normaloid;(ii) for invertible (iii) for invertible and every

Proof. (i)⇒(iii)⇒(ii). Since is normaloid, therefore for all invertible The first inequality holds because the spectral radius is less than the operator norm, and the second inequality holds by Proposition 5. Hence, (ii)⇒(i)
Since spectral radius is less than operator norm and by assertion (ii), we have for all invertible The third inequality holds by inequality (34) of Theorem 6. Since is arbitrary, therefore is normaloid.

Now, we will give a formula of spectral radius involving iterated generalized Aluthge transform and asymptotic behavior of powers of .

Theorem 10. Let . Then, we have

Proof. The first equality holds by Proposition 1, second inequality holds by , and third inequality holds by Proposition 5. Thus, for th power of an operator, we have Since Thus, which completes the proof.