Abstract
A new class of operators, larger than
∗
\ast
-finite operators, named generalized
∗
\ast
-finite operators and noted by
Gℱ
∗
(
ℋ
)
{{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})
is introduced, where:
Gℱ
∗
(
ℋ
)
=
{
(
A
,
B
)
∈
ℬ
(
ℋ
)
×
ℬ
(
ℋ
)
:
∥
T
A
−
B
T
∗
−
λ
I
∥
≥
∣
λ
∣
,
∀
λ
∈
C
,
∀
T
∈
ℬ
(
ℋ
)
}
.
{{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})=\{(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}):\parallel TA-B{T}^{\ast }-\lambda I\parallel \ge | \lambda | ,\hspace{0.33em}\forall \lambda \in {\mathbb{C}},\hspace{0.33em}\forall T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\}.
Basic properties are given. Some examples are also presented.