ON OPERATORS SATISFYING 1. Introduction Let H

Mohammad H.M. Rashid

r in

ON OPERATORS SATISFYING T (T ∗ |T |2k T )1/(k+1) T m ≥ T ∗m |T |2 T m ∗m

t

Commun. Korean Math. Soc. 0 (0), No. 0, pp. 1–0 https://doi.org/10.4134/CKMS.c160191 pISSN: 1225-1763 / eISSN: 2234-3024

fP

Abstract. Let T be a bounded linear operator acting on a complex Hilbert space H . In this paper we introduce the class, denoted Q(A(k), m), of operators satisfying T m∗ (T ∗ |T |2k T )1/(k+1) T m ≥ T ∗m |T |2 T m , where m is a positive integer and k is a positive real number and we prove basic structural properties of these operators. Using these results, we prove that if P is the Riesz idempotent for isolated point λ of the spectrum of T ∈ Q(A(k), m), then P is self-adjoint, and we give a necessary and sufficient condition for T ⊗ S to be in Q(A(k), m) when T and S are both non-zero operators. Moreover, we characterize the quasinilpotent part H0 (T − λ) of class A(k) operator.

1. Introduction

Ah ea do

Let H be a complex Hilbert space and let L (H ) be the algebra of all bounded linear operators acting on H . An operator T ∈ L (H ) has a unique polar decomposition T = U |T |, where |T | = (T ∗ T )1/2 and U is partial isometry satisfying ker(U ) = ker(T ) = ker(|T |) and ker(U ) = ker(T ∗ ). An operator T is said to be positive (denoted by T ≥ 0) if hT x, xi ≥ 0 for all x ∈ H and also T is said to be strictly positive (denoted by T > 0) if T is positive and invertible. An operator T is called p-hyponormal if |T |2p ≥ |T ∗ |2p for every 0 < p ≤ 1 and log-hyponormal

if T is2 invertible and log(T ∗ T ) ≥ log(T T ∗ ), T is called paranormal if T 2 x ≥ kT xk for every unit vector x ∈ H , and T is called normaloid if kT k = r(T ), the spectral radius of T . Following [9, 10], we say that T ∈ L (H ) belongs to class A if |T 2 | ≥ |T |2 and class A(k) for k > 0 (abbreviation T ∈ A(k)) if (T ∗ |T |2k T )1/(k+1) ≥ |T |2 , we note that T is class A if and only if T is class A(1). According to [3], an operator T ∈ L (H ) is said to be w-hyponormal if |Te| ≥ |T | ≥ |Tf∗ |, where Te is the Aluthge transformation Te = |T |1/2 U |T |1/2 . As a generalization of whyponormal and class A(k), Ito [10] introduced class wA(s, t) as follows. An 2t/(s+t) g operator T is called class wA(s, t) for s > 0 and t > 0 if |T ≥ |T |2t s,t | Received September 17, 2016. 2010 Mathematics Subject Classification. 47A55, 47A10, 47A11. Key words and phrases. Riesz idempotent, tensor product, class A(k), m-quasi-class A(k). c

2017 Korean Mathematical Society

1

2

M. H. M. RASHID

r in

t

∗ 2s/(s+t) , where T g g and |T |2s ≥ |T s,t is generalized Aluthge transformation, i.e., s,t | s t g Ts,t = |T | U |T | . An operator T ∈ L (H ) is called k-paranormal for positive

k+1 integer k, if T k+1 x ≥ kT xk for every unit vector x ∈ H .

Definition 1.1. We say that an operator T ∈ L (H ) is of m-quasi class Ak (abbreviate Q(A(k), m)), if T ∗m (T ∗ |T |2k T )1/(k+1) T m ≥ T m∗ |T |2 T m ,

where m is a positive integers and k > 0. If m = 1, then T is called a quasi-class A(k) and k = m = 1, then Q(A(k), m) coincides with quasi-class A operator. Example 1.2. Let H =

∞ M

C2 and define an operator T on H by

where A =

1 2 1 2

1 2 1 2



⊕ x1 ⊕ · · · ) = · · · ⊕ Ax−2 ⊕ Ax−1 ⊕ Bx0 ⊕ Bx1 ⊕ · · · ,

and B = ( 10 00 ). Then T is of m-quasi-class A(k) for each

In fact, for each k ≥ 14 , D   E T ∗m (T ∗ |T |2k T )1/(k+1) − |T |2 T m x, x D   E = Am (ABA)1/(k+1) − A2 Am x−1 , x−1

2  m ( 1/(k+1)  )  1 1 

1 1 1 2 2

− x−1 = 1 1

≥0 16 32 16 2 2

Ah ea do

k≥

1 4.

1 4

fP

n=0 (0) T (· · · ⊕ x−2 ⊕ x−1 ⊕ x0

for each x ∈ H . 1 Let 0 < α < 1 and A = α 12 2

1 2 1 2



. Then T ∈ Q(A(k), m) with k ≥

− log 2 2 log α .

log 2 Since − 2 log α → 0 as α → 0 for any k > 0. Then T ∈ Q(A(k), m) for each k > 0 and m is a positive integer.

Since T ≥ 0 implies R∗ T R ≥ 0, we have:

Proposition 1.3. Let T ∈ L (H ). If T ∈ A(k), then T ∈ Q(A(k), m)).

Throughout this paper, we shall denote the spectrum, the point spectrum and the isolated points of the spectrum of T ∈ L (H ) by σ(T ), σp (T ) and isoσ(T ), respectively. The range and the kernel of T ∈ L (H ) will be denoted by