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Acta Scientiarum Mathematicarum 84:3–4 (2018) c

Bolyai Institute, University of Szeged

doi: 10.14232/actasm-017- 773-6 Acta Sci. Math. (Szeged)

84 (2018), 611–627

On the characterizations of some distinguished

subclasses of Hilbert space operators

C. Bou raya and A. Seddik

Communicated by L. Molnár

Abstract.

In this note, we present several characterizations for some distin-

guished classes of bounded Hilbert space operators (self-adjoint operators, nor-

mal operators, unitary operators, and isometry operators) in terms of operator

inequalities.

1. Introduction and preliminaries

Let

B

(

H

)be the

C∗

-algebra of all bounded linear operators acting on a complex

Hilbert space

H,

and let

S

(

H

),

N

(

H

),

U

(

H

), and

V

(

H

)denote the class of all self-

adjoint operators, the class of all normal operators, the class of all unitary operators,

and the class of all isometry operators in B(H), respectively.

We use the following notations:

•I(H)is the group of all invertible elements in B(H),

•S0

(

H

)=

S

(

H

)

∩I

(

H

)is the set of all invertible self-adjoint operators in

B

(

H

),

• N0

(

H

) =

N

(

H

)

∩I

(

H

)is the set of all invertible normal operators in

B

(

H

),

• R(H)is the set of all operators with closed ranges in B(H),

•Scr

(

H

) =

S

(

H

)

∩ R

(

H

)is the set of all self-adjoint operators with closed

ranges in B(H),

• Ncr

(

H

) =

N

(

H

)

∩ R

(

H

)is the set of all normal operators with closed range

in B(H),

•(M)1

=

M∩ {x∈X:kxk= 1}

, where

X

is any normed space and

M

is a

subset of X,

Received April 8, 2017, and in ﬁnal form November 25, 2017.

AMS Subject Classiﬁcation (2010): 47A30, 47A05, 47B15.

Key words and phrases: closed range operator, Moore-Penrose inverse, group inverse, self-adjoint

operator, unitary operator, normal operator, partial isometry operator, isometry operator, operator

inequality.