The weighted Davis-Wielandt Berezin number for reproducing kernel Hilbert space operators
A functional Hilbert space is the Hilbert space of complex-valued functions on some set Θ ⊆ C that the evaluation functionals φ λ ( f ) = f ( λ ) , λ ∈ Θ are continuous on H . Then, by the Riesz representation theorem, there is a unique element k λ ∈ H such that f ( λ ) = 〈 f , k λ 〉 for all f ∈ H a...
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| Published in | Journal of inequalities and applications Vol. 2025; no. 1; pp. 24 - 13 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
27.02.2025
Springer Nature B.V SpringerOpen |
| Subjects | |
| Online Access | Get full text |
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| Summary: | A functional Hilbert space is the Hilbert space of complex-valued functions on some set
Θ
⊆
C
that the evaluation functionals
φ
λ
(
f
)
=
f
(
λ
)
,
λ
∈
Θ
are continuous on
H
. Then, by the Riesz representation theorem, there is a unique element
k
λ
∈
H
such that
f
(
λ
)
=
〈
f
,
k
λ
〉
for all
f
∈
H
and every
λ
∈
Θ
. The function
k
on
Θ
×
Θ
defined by
k
(
z
,
λ
)
=
k
λ
(
z
)
is called the reproducing kernel of
H
. In this study, we defined the weighted Davis-Wielandt Berezin number, and then we obtained some related inequalities. It is shown, among other inequalities, that if
X
∈
L
(
H
)
and
ν
∈
[
0
,
1
]
, then
1
2
(
ber
2
(
X
ν
+
|
X
ν
|
2
)
+
c
ber
2
(
X
ν
−
|
X
ν
|
2
)
)
≤
d
w
ber
ν
2
(
X
)
≤
1
2
(
ber
2
(
X
ν
+
|
X
ν
|
2
)
+
ber
2
(
X
ν
−
|
X
ν
|
2
)
)
,
where
X
ν
=
(
1
−
2
ν
)
X
∗
+
X
. Some bounds for the weighted Davis-Wielandt Berezin number are also established. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1029-242X 1025-5834 1029-242X |
| DOI: | 10.1186/s13660-025-03255-0 |