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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.