Lipschitz type inequalities for noncommutative perspectives of operator monotone functions in Hilbert spaces
Assume that f : [ 0 , ∞ ) → R is a continuous function. We can define the perspective P f B , A by setting P f B , A : = A 1 / 2 f A - 1 / 2 B A - 1 / 2 A 1 / 2 , where A , B > 0 . We show in this paper among others that P f B , P - P f A , P ≤ P 2 B - A p 2 P f m 2 , p - P f m 1 , p m 2 - m 1 i...
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| Published in: | Advances in operator theory Vol. 6; no. 2 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01.04.2021
|
| Subjects: | |
| ISSN: | 2662-2009, 2538-225X |
| Online Access: | Get full text |
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| Summary: | Assume that
f
:
[
0
,
∞
)
→
R
is a continuous function. We can define the
perspective
P
f
B
,
A
by setting
P
f
B
,
A
:
=
A
1
/
2
f
A
-
1
/
2
B
A
-
1
/
2
A
1
/
2
,
where
A
,
B
>
0
.
We show in this paper among others that
P
f
B
,
P
-
P
f
A
,
P
≤
P
2
B
-
A
p
2
P
f
m
2
,
p
-
P
f
m
1
,
p
m
2
-
m
1
if
m
1
≠
m
2
,
f
′
m
p
if
m
1
=
m
2
=
m
for all
A
≥
m
1
>
0
,
B
≥
m
2
>
0
and
P
≥
p
>
0
. If
f
is operator monotone on
[
0
,
∞
)
, then for all
C
≥
n
1
>
0
,
D
≥
n
2
>
0
,
Q
>
q
>
0
we also have
P
f
Q
,
D
-
P
f
Q
,
C
≤
Q
2
D
-
C
q
2
P
f
q
,
n
2
-
P
f
q
,
n
1
n
2
-
n
1
if
n
2
≠
n
1
,
f
q
n
-
q
n
f
′
q
n
if
n
2
=
n
1
=
n
.
Some applications for
weighted operator geometric mean
and
relative operator entropy
are also given. |
|---|---|
| ISSN: | 2662-2009 2538-225X |
| DOI: | 10.1007/s43036-021-00130-9 |