Abstract
In this paper, we introduce new classes of operators related to the class of polynomially normal operators which are described as follows: (i)
m
-quasi polynomially normal operators includes polynomially normal operators recently studied in [7, 6]. A bounded linear operator
S
on a complex Hilbert space
H
is said to be
m
-quasi polynomially normal operator if there exists a nontrivial polynomial
P
=
∑
0
≤
k
≤
n
b
k
z
k
∈
C
[
z
]
for which
S
∗
m
(
P
(
S
)
S
∗
-
S
∗
P
(
S
)
)
S
m
=
0
(
⇔
∑
0
≤
k
≤
n
b
k
S
∗
m
(
S
k
S
∗
-
S
∗
S
k
)
S
m
=
0
)
,
where
m
is a natural number. (ii) Polynomially
C
-normal operators includes
C
-normal operators studied in [13, 23, 25]. An operator
S
is called polynomially
C
-normal operator if there exists a nontrivial polynomial
P
∈
C
[
z
]
and a conjugation operator
C
on
H
for which
C
P
(
S
)
S
∗
-
S
∗
P
(
S
)
C
=
0
(
⇔
∑
0
≤
k
≤
n
b
k
(
C
S
k
S
∗
-
S
∗
S
k
C
)
=
0
)
.
A detailed study of certain properties of some members of the first class has been presented. However, an initiation to the study of the second class has been given.