In this survey, we shall present the characterizations of some distinguished classes of bounded linear operators acting on a complex separable Hilbert space in terms of operator inequalities related to the arithmetic–geometric mean inequality.

In this note, we present several characterizations for some distinguished
classes of bounded Hilbert space operators (self-adjoint operators, normal
operators, unitary operators, and isometry operators) in terms of operator
inequalities.

In this note, we present several characterizations for some distinguished
classes of bounded Hilbert space operators (self-adjoint operators, normal
operators, unitary operators, and isometry operators) in terms of operator
inequalities.

We correct a mistake which affect our main results, namely the proof of Lema 1.
The main results of the article remain unchanged.

In this note, we shall give complete characterizations of the class of all normal
operators with closed range, and the class of all selfadjoint operators with closed range
multiplied by scalars in terms of some operator inequalities.

We prove some refinements of an inequality due to X. Zhan in an
arbitrary complex Hilbert space by using some results on the Heinz
inequality. We present several related inequalities as well as new
variants of the Corach–Porta–Recht inequality.We also characterize
the class of operators satisfying
SXS
−1 + S
−1XS + kX

(k +
2) X under certain conditions.

In the present paper, taking some advantages oered by the con-
text of nite dimensional Hilbert spaces, we shall give a complete characteriza-
tions of certain distinguished classes of operators (self-adjoint, unitary re ec-
tion, normal) in terms of operator inequalities. These results extend previous
characterizations obtained by the second author.

In [3, 4] , we have given some characterizations of some subclasses of complex Hilbert
space normal operators by inequalities. In this note, we shall reformulate these results using a
concept of closed inequalities. Also, we shall give some new characterizations. Some open
problems concerning closed inequalities are posed at the end of this note.

Let B(H) denotes the C∗-algebra of all bounded linear operators acting on the
complex Hilbert space H . In this note, we shall give some lower estimates for the injective norm
of the element
n
i=1
Ai⊗Bi in the tensor product B(H)⊗B(H), where A = (A1, ..., An) and B =
(B1, ..., Bn) are two n-tuples of elements in B(H) ; and we shall characterize the normaloid
operators in B(H) using the injective norm.
1.

Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuples
A = (A1, . . . , An) and B = (B1, . . . , Bn) of elements in A, we define the elementary operator RA,B on
A by the relation RA,B(X) = n
i=1 AiXBi for all X in A. For a single operator A ∈ A, we define the
two particular elementary operators LA and RA onAby LA(X) = AX and RA(X) = XA, for every X in
A. We denote by d(RA,B) the supremum of the norm of RA,B(X) over all unit rank one operators on E.
In this note, we shall characterize: (i) the supremun d(RA,B), (ii) the relation d(RA,B) = n
i=1
Ai
Bi
,
(iii) the relation d(LA
− RB) = A + B, (iv) the relation d(LARB
+ LBRA) = 2AB. Moreover,
we shall showthe lower estimate d(LA
− RB) max{supλ∈V (B)
A − λI, supλ∈V (A)
B − λI} (where
V (X) is the algebraic numerical range of X inA).

Let BðEÞ be the complex Banach algebra of all bounded linear operators on a complex Banach space E: For
n-tuples A ¼ ðA1, . . . ,AnÞ and B ¼ ðB1, . . . ,BnÞ of operators on E, let RA, B denote the operator on BðEÞ
defined by RA, BðXÞ ¼
Pn
i¼1AiXBi :
For A, B 2 BðEÞ, we put UA, B ¼ RðA, BÞ, ðB,AÞ:
In this note, we prove that
co
Xn
i¼1
ii : ð1, . . . , nÞ 2 VðAÞ, ð1, . . . , nÞ 2 VðBÞ
( )
W0ðRA, BjJÞ
where VðÞ is the joint spatial numerical range, W0ðÞ is the algebraic numerical range and J is a norm ideal of
BðEÞ: We shall show that this inclusion becomes an equality when RA, B is taken to be a derivation. Also, we
deduce that wðUA, BjJÞ 2ð
ffiffiffi2
p
1ÞwðAÞwðBÞ, for A,B 2 BðEÞ and J is a norm ideal of BðEÞ, where wðÞ is the
numerical radius.
On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate
bound kUA, BjJk 2ð
ffiffiffi2
p
1ÞkAkkBk holds, if one of the following two conditions is satisfied:
(i) J is a standard operator algebra of BðEÞ and A,B 2 J:
(ii) J is a norm ideal of BðEÞ and A, B 2 BðEÞ:

Let A be a complex normed algebra. For A,B ∈ A, define a basic
elementary operator MA,B : A → A by MA,B(X) = AXB.
Given a standard operator algebra A acting on a complex normed space
and A,B ∈ A we have:
(i) The lower estimate kMA,B +MB,Ak ≥ 2(√2 − 1)kAkkBk holds.
(ii) The lower estimate kMA,B +MB,Ak ≥ kAkkBk holds if
inf
∈C kA + Bk = kAk or inf
∈C kB + Ak = kBk.
(iii) The equality kMA,B +MB,Ak = 2kAkkBk holds if
kA + Bk = kAk + kBk for some unit scalar .
These results extend analogous estimates established earlier for standard
operator subalgebras of Hilbert space operators.

Let B(H) be the C
∗-algebra of all bounded linear operators on a complex Hilbert space
H, S be an invertible and selfadjoint operator in B(H) and let (I, .
I ) denote a norm ideal
of B(H). In this note, we shall show the following inequality:
∀X ∈ I : SXS
−1 − S
−1XS
I (SS
−1 − 1)SXS
−1 + S
−1XS
I .

For n-tuples A = (At, ..., AN) and B = (BI, ..., Bn) of operators on aHilber! space
H, let RA,B denote the operator on L(H) defined by RA,B(X) = ~i=1AiX Bi. In
this paper we prove that
co c~i13i : (a,,..., an) C W(A), (131, ..., 1~) e W(B) C Wo(RA,B)
where W is the joint spatial numerical range and W0 is the numerical range. We
will show also that this inclusion becomes an equality when I~A,B is taken to be
a generalized derivation, and it is strict when RA,B is taken to be an elementary
multiplication operator induced by non scalar self-adjoints operators.

For A,B ∈ L(H) (the algebra of all bounded linear operators on the Hilbert space H),
it is proved that: (i) the generalized derivation δA,B is convexoid if and only if A and B are
convexoid; (ii) the operators δA,B and δA,B
|Cp (where p 1) have the same numerical range
and are equal to W0(A) − W0(B) (where Cp is the Banach space of the p-Schatten class
operators on H).

Let A E_Y(H~), B E_F(H.J ( w h ere Hi, H, are Hilbert spaces), and let S,, H
denote the operator on _Y( H,, H,) given by
6,,,(X) = AX - XB, X =.Y( H,, H,)
J. P. Williams asked: For which A is R(i%,)-n {A*) = {O}?(w here S,, A = 6,) We
obtain some operators in this class. The case of 6, s, A f B, is interesting in itself;
moreover it is useful if we have to use a decomposition of the Hilbert space in a direct
some for the consideration of 6,. In this note we describe some classes of operators
A, B for which we have R(6,, H) - f? ker S,,, se = (0). 0 1998 Elsevier Science

For A E/:(H) ( the algebra of all operators on the complex Hilbert space H)~ let 5A
denote the operator on/~(H) defined by : (~A(X) = AX - XA.
We show here that for all Jordan operators A : R(SA) N {A*)' = {0}, where R(6A) is
the range of 6A and (A*}; is the commntant of the adjoint of A.