Spectral Radius Formulas Involving Generalized Aluthge Transform

In this paper, we aim to develop formulas of spectral radius for an operator S in terms of generalized Aluthge transform, numerical radius, iterated generalized Aluthge transform, and asymptotic behavior of powers of S. 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.


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][4][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][8][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 for-mulas and numerical radius inequalities involving Aluthge transform [10][11][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 BðH Þ be the algebra of all bounded linear operators on complex Hilbert space H. Let S = UjSj be the polar decomposition of S ∈ BðH Þ, where jSj is the square root of an operator defined as jSj = ffiffiffiffiffiffiffi S * S p and U 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 nth 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 λ-Aluthge transform which is defined as Tam [4] gave a formula of spectral radius involving iterated λ-Aluthge transform for invertible operators using unitarily invariant norm, i.e., Chabbabi and Mbekhta [12] gave various expressions for spectral radius formulas involving λ-Aluthge transform, iterated λ-Aluthge transform, asymptotic behavior of powers of an operator, and numerical radius. The expression of spectral radius involving λ-Aluthge transform is given by and the expressions of spectral radius involving iterated λ-Aluthge transform and the asymptotic behavior of powers of S are given by for each n ≥ 0. The expressions of spectral radius involving iterated λ-Aluthge transform, numerical radius, and the asymptotic behavior of powers of S are given by for each n ≥ 0. 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 f and g both are continuous functions such that gðxÞf ðxÞ = x, x ≥ 0. The iterated generalized Aluthge transform is defined as In this paper, we establish the formulas of spectral radius for operator S by assuming that kΔ f ,g ðSÞk ≤ kSk: 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 S 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.

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 T is similar to S if there exists an invertible operator Y such that S = Y −1 TY (see [16]). If rðSÞ = kSk, then the operator is said to be normaloid. An operator S is said to be a contraction if kSk ≤ 1. The spectral radius of an operator S is defined as where σðSÞ is the spectrum of the operator S.
To prove spectral radius formulas, we recall some properties of generalized Aluthge transform.
Proposition 1 [7]. Let S ∈ BðH Þ. Then, we have 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 S ∈ BðH Þ and f be any continuous function on σðSÞ. Then, for any unitary U ∈ BðH Þ.
for each n ∈ ℕ, which implies for any polynomial PðtÞ. Since f is a continuous, so there exist a sequence of polynomial fP n ðtÞg ∞ n=1 such that P n ð0Þ = 0 for each n ∈ ℕ, and fP n ðtÞg ∞ n=1 converges uniformly to f ðtÞ on the interval ½0, kjTjk. Then, from Equation (16), we have as required.
Proposition 4. Let S, U ∈ BðH Þ such that U is unitary. Then, we have Proof. Let S = V | S | be the polar decomposition of S: Then, we have Now by using Proposition 3, we have The polar decomposition of operator USU * is as follows: where UVU * is partial isometry. Therefore, The second equality holds by Proposition 3 and by the fact that U * U = I. Proposition 5. Let S ∈ BðH Þ. Then, the sequence fkΔ n f ,g ðSÞkg ∞ n=1 is nonincreasing.
Proof. The proof follows from the repeated application of the inequality

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 S ∈ BðH Þ. Then, we have Proof. From Propositions 1 and 2, we have It follows that Hence, Let Y = U | Y | be the polar decomposition of Y: Since Y is an invertible operator, then U is unitary and |Y | invertible. Therefore, there exists β > 0 such that σð|Y | Þ ⊆ ½β,∞Þ . Consequently, A = ln ð|Y | Þ exists and self-adjoint; then, we have Therefore, 3

Journal of Function Spaces
The second equality holds by Proposition 4. Hence, To prove above inequality in other direction, for an arbitrary ε > 0, we define an operator For operator S ε , we have From [16], Theorem 2, the spectrum of operator S ε lies in the unit disk; thus, the operator S ε is similar to contraction for which there exists an invertible operator Y ε ∈ Bð H Þ such that and this implies that For ε > 0, we obtain Since ε > 0 is arbitrary, therefore The next Corollary is the direct result of Theorem 6 involving iterated generalized Aluthge transform.

Corollary 7.
Let S ∈ BðH Þ. Then, for each n ∈ ℕ, 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 Y ∈ BðH Þ: 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 S ∈ BðH Þ. Then, the following assertions are equivalent for all invertible Y ∈ BðH Þ: 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 Y ∈ BðH Þ: The last inequality holds by inequality (33) in Theorem 6. Since ε > 0 is arbitrary, hence S is normaloid. Corollary 9. Let S ∈ BðH Þ. Then the following assertions are equivalent.
(i) S is normaloid; (ii) kSk ≤ kΔ f ,g ðYSY −1 Þk for invertible Y ∈ BðH Þ ; (iii) kSk ≤ kΔ n f ,g ðYSY −1 Þk for invertible Y ∈ BðH Þ and every n ∈ ℕ: 4 Journal of Function Spaces Proof. (i)⇒(iii)⇒(ii). Since S is normaloid, therefore for all invertible Y ∈ BðH Þ: 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 Y ∈ BðH Þ: The third inequality holds by inequality (34) of Theorem 6. Since ε > 0 is arbitrary, therefore S is normaloid. Now, we will give a formula of spectral radius involving iterated generalized Aluthge transform and asymptotic behavior of powers of S. Theorem 10. Let S ∈ BðH Þ. Then, we have Proof.
The first equality holds by Proposition 1, second inequality holds by rðSÞ ≤ kSk, and third inequality holds by Proposition 5. Thus, for kth power of an operator, we have which completes the proof.
The next Corollary is obtain in the consequence of Theorem 10.
The first equality holds by assertion (i) and Theorem 10.
The first equality holds by assertion (i) and Theorem 10.
The last equality holds by Theorem 10.