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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| Vydané v: | Advances in operator theory Ročník 6; číslo 2 |
|---|---|
| Hlavný autor: | |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Cham
Springer International Publishing
01.04.2021
|
| Predmet: | |
| ISSN: | 2662-2009, 2538-225X |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. |
| ArticleNumber | 33 |
| Author | Dragomir, Silvestru Sever |
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| Cites_doi | 10.7153/jmi-09-04 10.15352/afa/1391614576 10.15352/afa/1396833504 10.1016/0024-3795(95)00201-2 10.1016/0024-3795(94)90449-9 10.1080/03081087.2017.1295432 10.1007/BF01170633 10.1007/s002200050279 10.1007/BF02054965 10.2183/pjab1945.49.205 10.1007/BF01941801 10.1073/pnas.0807965106 10.1002/mana.201200194 10.1007/BF01371042 |
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| Issue | 2 |
| Keywords | Operator monotone functions 26D1 Noncommutative perspectives 15A60 47A63 47A30 26D15 |
| Language | English |
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| PublicationTitleAbbrev | Adv. Oper. Theory |
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| Publisher | Springer International Publishing |
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Z doi: 10.1007/BF01170633 – volume: 191 start-page: 603 issue: 3 year: 1998 ident: 130_CR5 publication-title: Commun. Math. Phys. doi: 10.1007/s002200050279 – volume: 81 start-page: 89 year: 1981 ident: 130_CR1 publication-title: Commun. Math. Phys. doi: 10.1007/BF01941801 – volume: 66 start-page: 243 issue: 2 year: 2018 ident: 130_CR24 publication-title: Linear Multilinear Algebras doi: 10.1080/03081087.2017.1295432 – volume: 208 start-page: 367 issue: 209 year: 1994 ident: 130_CR2 publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(94)90449-9 – volume: 286 start-page: 1514 issue: 14–15 year: 2013 ident: 130_CR21 publication-title: Math. Nachr. doi: 10.1002/mana.201200194 – volume: 123 start-page: 415 year: 1951 ident: 130_CR17 publication-title: Math. Ann. doi: 10.1007/BF02054965 – ident: 130_CR7 – volume: 20 start-page: 224 issue: 3 year: 2008 ident: 130_CR12 publication-title: Algebra i Analiz – volume: 56 start-page: 143 year: 1976 ident: 130_CR11 publication-title: Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. – volume: 9 start-page: 47 issue: 1 year: 2015 ident: 130_CR16 publication-title: J. Math. Inequal. doi: 10.7153/jmi-09-04 – volume: 5 start-page: 121 issue: 1 year: 2014 ident: 130_CR22 publication-title: Ann. Funct. Anal. doi: 10.15352/afa/1391614576 – volume: 5 start-page: 74 issue: 2 year: 2014 ident: 130_CR9 publication-title: Ann. Funct. Anal. doi: 10.15352/afa/1396833504 – volume: 49 start-page: 143 year: 1973 ident: 130_CR18 publication-title: Proc. Jpn. Acad. doi: 10.2183/pjab1945.49.205 – volume: 34 start-page: 541 issue: 4 year: 1989 ident: 130_CR13 publication-title: Math. Jpn. – volume: 1 start-page: 301 year: 1998 ident: 130_CR15 publication-title: Sci. Math. – volume: 34 start-page: 341 issue: 3 year: 1989 ident: 130_CR14 publication-title: Math. Jpn. – volume-title: Calculus on Normed Vector Spaces year: 2010 ident: 130_CR6 – volume: 226 start-page: 639 issue: 228 year: 1995 ident: 130_CR3 publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(95)00201-2 – start-page: xii+347 volume-title: Graduate Texts in Mathematics year: 1997 ident: 130_CR4 – ident: 130_CR19 doi: 10.1007/BF01371042 – volume: 37 start-page: 149 year: 1961 ident: 130_CR23 publication-title: Proc. Jpn. Acad. – volume: 106 start-page: 1006 year: 2009 ident: 130_CR8 publication-title: Proc. Natl. Acad. Sci. USA doi: 10.1073/pnas.0807965106 |
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| Snippet | 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
/... |
| SourceID | crossref springer |
| SourceType | Index Database Publisher |
| SubjectTerms | Mathematics Mathematics and Statistics Operator Theory Original Paper |
| Title | Lipschitz type inequalities for noncommutative perspectives of operator monotone functions in Hilbert spaces |
| URI | https://link.springer.com/article/10.1007/s43036-021-00130-9 |
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