Some classical inequalities and their applications

Abstract

In this paper, we define analogies of classical H?lder-McCarthy and Young type inequalities in terms of the Berezin symbols of operators on a reproducing kernel Hilbert space H = H (?). These inequalities are applied in proving of some new inequalities for the Berezin number of operators. We also define quasi-paranormal and absolute-k-quasi paranormal operators and study their properties by using the Berezin symbols.

Full text

Filomat 35:7 (2021), 2165–2173
https://doi.org/10.2298/FIL2107165H
Published by Faculty of Sciences and Mathematics,
University of Niˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Some Classical Inequalities and their Applications
Mualla Birg ¨ul Hubana, Mehmet G ¨urdalb, Havva Tilkib
aIsparta University of Applied Sciences, Isparta, Turkey
bSuleyman Demirel University, Department of Mathematics, 32260, Isparta, Turkey
Abstract. In this paper, we define analogies of classical H¨
older-McCarthy and Young type inequalities
in terms of the Berezin symbols of operators on a reproducing kernel Hilbert space H=H(Ω).These
inequalities are applied in proving of some new inequalities for the Berezin number of operators. We also
define quasi-paranormal and absolute-k-quasi paranormal operators and study their properties by using
the Berezin symbols.
1. Introduction
Let H=H(Ω) be a Hilbert space of complex-valued functions on some set Ωsuch that f→f(λ)is a
continuous functional (evaluation functional) for any λin Ω. Then, according to the Riesz’s representation
theorem there exists uniquely kλ∈ H such that
f(λ)=f,kλ
for all f∈ H . The function kλ(z), λ ∈Ω,is called the reproducing kernel of the space H,and b
kλ:=kλ
kkλk
is called the normalized reproducing kernel in H(see [2]). The space Hwith the reproducing kernels
kλ, λ ∈Ω,is called reproducing kernel Hilbert space (RKHS). For a bounded linear operator A(i.e., for
A∈ B(H), the Banach algebra of all bounded linear operators on H) its Berezin symbol e
Ais defined by
(Berezin [6, 7])
e
A(λ) :=DA
b
kλ,b
kλE, λ ∈Ω.
The Berezin number ber (A)of operator Ais the following number:
ber (A):=sup
λ∈Ωe
A(λ).
2010 Mathematics Subject Classification. Primary 47A63; Secondary 26D15, 47B10
Keywords. Reproducing kernel Hilbert space, Berezin symbol, Berezin number, quasi-paranormal operator, H ¨
older-McCarthy
type inequality, Young type inequality
Received: 07 July 2020; Accepted: 20 December 2020
Communicated by Fuad Kittaneh
This paper was supported by T ¨
UBA through Young Scientist Award Program (T ¨
UBA-GEBIP/2015).
Email addresses: muallahuban@isparta.edu.tr (Mualla Birg ¨
ul Huban), gurdalmehmet@sdu.edu.tr (Mehmet G ¨
urdal),
havvatilki32@gmail.com (Havva Tilki)

M. B. Huban et al. /Filomat 35:7 (2021), 2165–2173 2166
Since e
A(λ)≤kAk(by the Cauchy-Schwarz inequality) for all λ∈Ω,the Berezin number is a finite number
and ber (A)≤kAk.Recall that
W(A):=hAx,xi:x∈ H and kxk=1
is the numerical range of operator Aand
w(A):=sup |hAx,xi| :x∈ H and kxk=1
=sup nµ:µ∈W(A)o
is the numerical radius of A(for more information, see [1, 20–22]). It is well known that
Ber (A)⊂W(A)and ber (A)≤w(A)
for any A∈ B(H).More information about ber (A)and relations between ber (A),w(A)and kAkcan be
found in Karaev [16, 18], and also in [3–5, 9–15, 17, 19, 23–25].
In this paper, we will use some known operator inequalities to prove some new inequalities for the
Berezin number of operators acting on the RKHS H=H(Ω).Some other related questions also will be
studied. In general, the present paper is motivated by the paper of Garayev [16], where the McCarthy,
H¨
older-McCarthy and Kantorovich operator inequalities were extensively used to get some new inequalities
for the Berezin number of operators and their powers. Recall that for any positive operator A(i.e., hAx,xi≥0
for any x∈ H, shortly A≥0), there exists a unique positive operator Rsuch that R2=A(denoted by R=A1
2).
An operator T∈ B(H) can be decomposed into T=UP,where Uis a partial isometry and P=|T|:=(T∗T)1
2
(moduli of operator T) with ker (T)=ker (P)and the last condition uniquely determines Uand Pof the
polar decomposition T=UP (see Furuta [8]). In general, we will refer to the book of Furuta [8] for main
definitions and notations.
2. H ¨older-McCarthy Type Inequalities and Berezin number
In this section, by using the H¨
older-McCarthy inequality, we prove some inequalities for the Berezin
number of some operators on the RKHS H.
Theorem 2.1. Let A ∈ B(H)be a positive operator. Then :
1) ber (Aµ)≥ber (A)µfor any µ > 1.
2) ber (Aµ)≤ber (A)µfor any µ∈[0,1].
3) If A is invertible, then ber (Aµ)≥ber (A)µfor any µ < 0.
Proof. First we prove 2). Indeed, assume that 2) holds for some α,β∈[0,1].Then we only have to prove 2) holds for
α+β
2∈[0,1]by continuity of an operator. In fact, we have for any λ∈Ωthat
Aα+β
2b
kλ,b
kλ
2
=Aα
2b
kλ,Aβ
2b
kλ
2
(by Cauchy-Schwarz inequality)
≤DAα
b
kλ,b
kλEDAβ
b
kλ,b
kλE(by assumption)
≤DA
b
kλ,b
kλEα+β,
so that g
Aα+β
2(λ)≤e
A(λ)α+β
2holds for α+β
2∈[0,1].This implies the desired inequality ber (Aµ)≤ber (A)µfor any
µ∈[0,1].

M. B. Huban et al. /Filomat 35:7 (2021), 2165–2173 2167
1) Let µ > 1.Then 1
µ∈[0,1].For any λ∈Ω
DA
b
kλ,b
kλE=DAµ1
µb
kλ,b
kλE
≤DAµ
b
kλ,b
kλE1
µby 2),
hence DAµ
b
kλ,b
kλE>DA
b
kλ,b
kλEµfor any µ > 1,which shows that ber (Aµ)>ber (A)µfor any µ > 1, as desired.
3) Since A is invertible, we have the following for any λ∈Ωthat
1=


b
kλ



4=DA1
2b
kλ,A−1
2b
kλE
2
≤


A1
2b
kλ



2


A−1
2b
kλ



2
=DA
b
kλ,b
kλEDA−1
b
kλ,b
kλE
=e
A(λ)g
A−1(λ),
and hence
1≤e
A(λ)g
A−1(λ)for any λ∈Ω,(1)
which gives us
ber (A)ber A−1>1,
or equivalently
ber A−1>ber (A)−1.
Case: µ∈(−∞,−1).Then we have the following for any λ∈Ωthat
DAµ
b
kλ,b
kλE=DA−|µ|b
kλ,b
kλE
>DA−1
b
kλ,b
kλEµ(by 1) since µ>1)
>DA
b
kλ,b
kλE−|µ|(by (1))
=DA
b
kλ,b
kλEµ
which implies that ber (Aµ)>ber (A)µ, as desired.
Case: µ∈[−1,0).For every λ∈Ωwe have
e
Aµ(λ)=DAµ
b
kλ,b
kλE=DA−|µ|b
kλ,b
kλE
>DA|µ|b
kλ,b
kλE−1(by (1))
>DA
b
kλ,b
kλE−|µ|=DA
b
kλ,b
kλEµ=e
A(λ)µ,
and the last inequality follows by 2) since µ∈[0,1]and taking inverses of both sides. The theorem is proved.
Next result proves the equivalence of H¨
older-McCarthy type inequality and Young type inequality.

M. B. Huban et al. /Filomat 35:7 (2021), 2165–2173 2168
Theorem 2.2. For a positive operator A ∈ B(H)and µ∈[0,1]the following inequalities are equivalent:
H¨older-McCarthy type inequality:
e
A(λ)µ>f
Aµ(λ)for all λ∈Ω.(2)
Young type inequality:
µA+I−µ∼>f
Aµ.(3)
Proof. Let us define a scalar function
f(t):=µt+1−µ−tµ
for positive numbers tand µ∈[0,1]. Then it is easy to see that f(t)is a nonnegative convex function with
the minimum value f(1)=0,so we have
µa+1−µ>aµ(4)
for positive aand µ∈[0,1].
(2) ⇒(3).Replacing aby e
A(λ)>0 and µ∈[0,1]in (4),we obtain
µe
A(λ)+1−µ>A(λ)µ>f
Aµ(λ)by (2),
so we have (3).
(3) ⇒(2).We may assume µ∈(0,1].In (3),replace Aby k1
µAfor a positive number k,then
µk1
µe
A(λ)+1−µ>kf
Aµ(λ)(5)
for λ∈Ωby (3).We put k=e
A(λ)−µin (5) if e
A(λ),0,then we have
µe
A(λ)−1e
A(λ)+1−µ>e
A(λ)−µf
Aµ(λ),
that is A(λ)µ>f
Aµ(λ)for all λ∈Ωand we get (2).If e
A(λ)=0,then it means that A1
2b
kλ=0,so Aµ
b
kλ=0 for
µ∈(0,1]by the induction and continuity of A, and thus we have (2).The theorem is proved.
Proposition 2.3. Let A ∈ B(H)be a positive invertible operator and B ∈ B(H)be an invertible operator. Then for
any real number µ, we have
ber (BAB∗)µ=ber BA1
2A1
2B∗BA1
2µ−1A1
2B∗.(6)
Proof. Let BA1
2=UP be the polar decomposition of BA 1
2, where Uis unitary and P=BA 1
2.Then it is easy
to see that:
(BAB∗)µ=UP2U∗µ=BA 1
2P−1P2µP−1A1
2B∗
=BA1
2A1
2B∗BA1
2µ−1A1
2B∗.
Now (6) is immediate from this equality.

M. B. Huban et al. /Filomat 35:7 (2021), 2165–2173 2169
3. Paranormal operators and related problems
Recall that an operator Aon a Hilbert space His called paranormal if 

A2x

≥kAxk2for every unit vector
x∈H.
Definition 3.1. We will say that A is a quasi-paranormal operator on a RKHS H=H(Ω),if 


A2
b
kλ


≥


A
b
kλ



2for
any λ∈Ω.
Definition 3.2. An operator T belongs to class e
Aif g
T2≥g
|T|2.
Definition 3.3. For each k >0,an operator T is absolute-k-quasi-paranormal if



|T|kTb
kλ


≥


Tb
kλ



k+1(7)
for every λ∈Ω.
It follows from these definitions that:
(a) If Ais quasi-paranormal, then
ber A22≥ber |A|22;
(b) If Abelongs to class e
A, then
ber A2≥ber |A|2;
(c) If Ais absolute-k-quasi-paranormal, then
ber |A|kA2≥ber (|A|)k+1.
In this section, to prove some inequalities for the Berezin number of such operators, we need to other
properties of these operators.
Proposition 3.4. Every operator in e
Ais a quasi-paranormal operator on a RKHS.
Proof. Suppose A∈e
A, i.e.,
g
A2≥g
|A|2.(8)
Then for every λ∈Ω,we have g
A2(λ)≥g
|A|2(λ),and therefore it follows from the proof of Theorem 2.1
that 


A2
b
kλ



2=DA2
b
kλ,A2
b
kλE=DA2∗A2
b
kλ,b
kλE
=A22b
kλ,b
kλ
≥DA2b
kλ,b
kλE2(see the proof of Theorem 2.1, 1))
≥D|A|2b
kλ,b
kλE2(by (8))
=


A
b
kλ



4.
Hence



A2
b
kλ


≥


A
b
kλ



2
for every λ∈Ω,so that Ais quasi-paranormal, which proves the proposition.

M. B. Huban et al. /Filomat 35:7 (2021), 2165–2173 2170
Definition 3.5. For each k >0,we say that an operator A belongs to class e
A(k)if
A∗|A|2kA1
k+1∼
≥g
|A|2.
The proof of Theorem 2.1 allows us also prove the following.
Proposition 3.6. (a) Every quasi-paranormal operator on a RKHS H=H(Ω)is an absolute-k-quasi-paranormal
operator for k ≥1.
(b) For each k >0,every class e
A(k)operator is an absolute-k-quasi-paranormal operator.
Proof. (a) Suppose that Ais a quasi-paranormal operator on a RKHS H=H(Ω).Then, for any λ∈Ωand
k≥1,we have



|A|kA
b
kλ



2=D|A|2kA
b
kλ,A
b
kλE
≥D|A|2A
b
kλ,A
b
kλEk


A
b
kλ



2(1−k)(see the proof of Theorem 2.1, 1))
=


A2
b
kλ



2k


A
b
kλ



2(1−k)
≥


A
b
kλ



4k


A
b
kλ



2(1−k)(by quasi-paranormality of A)
≥


A
b
kλ



2(k+1),
and hence



|A|kA
b
kλ


≥


A
b
kλ



k+1
for all λ∈Ωand k≥1,so that Ais absolute-k-quasi-paranormal operator for k≥1.
(b) Let A∈e
A(k)for k>0,that is
A∗|A|2kA1
k+1∼
≥g
|A|2for k>0.(9)
Then for any λ∈Ω,



|A|kA
b
kλ



2=DA∗|A|2kA
b
kλ,b
kλE
≥A∗|A|2kA1
k+1b
kλ,b
kλk+1
≥D|A|2b
kλ,b
kλEk+1(by (9))
=


A
b
kλ



2(k+1),
from which



|A|kA
b
kλ


≥


A
b
kλ



k+1for all λ∈Ω,
so that Ais absolute-k-quasi-paranormal operator for k>0.This completes the proof.
As further extension of previous results, we prove the following result.

M. B. Huban et al. /Filomat 35:7 (2021), 2165–2173 2171
Theorem 3.7. Let A ∈ B (H(Ω)) be an absolute-k-quasi-paranormal operator for k >0.Then for every λ∈Ω,
F(`)=


|A|`A
b
kλ



1
`+1
is increasing for ` > k>0,and the following inequality holds:
F(`)≥


A
b
kλ


,
i.e., A is absolute-`-quasi-paranormal operator for `≥k>0.
Proof. Assume that Ais an absolute-k-quasi-paranormal operator on H=H(Ω)for k>0,i.e.,



|A|kA
b
kλ


≥


A
b
kλ



k+1(10)
for every λ∈Ω.Clearly, (10)holds if and only if
F(k)=


|A|kA
b
kλ



1
k+1≥


A
b
kλ



for any λ∈Ω.Then for every λ∈Ωand any `such that `≥k>0,we have
F(`)=


|A|`A
b
kλ



1
`+1=D|A|2`A
b
kλ,b
kλE1
2(`+1)
≥(D|A|2kA
b
kλ,A
b
kλE1
k


A
b
kλ



2(1−1
k))1
2(`+1)
≥(


A
b
kλ



2`(k+1)
k


A
b
kλ



2(1−1
k))1
2(`+1)
(by (10))
=


A
b
kλ


,
and hence
F(`)=


|A|`A
b
kλ



1
`+1≥


A
b
kλ


(11)
for every λ∈Ωand `≥k,so that Ais absolute-`-quasi-paranormal for `≥k>0.
Now we prove that, F(`)is increasing for `≥k>0.Indeed, for any λ∈Ω,mand `such that
m≥`≥k>0,we have:
F(m)=


|A|mA
b
kλ



1
m+1=D|A|2mA
b
kλ,A
b
kλE1
2(m+1)
=(D|A|2`A
b
kλ,A
b
kλEm
`


A
b
kλ



2(1−m
`))1
2(m+1)
=(


|A|`A
b
kλ



2m
`


A
b
kλ



2(1−m
`))1
2(m+1)
≥(


|A|`A
b
kλ



2m
`


|A|`A
b
kλ



2
`+1(1−m
`))1
2(m+1)
(by (11))
=


|A|`A
b
kλ



1
`+1=F(`),
hence F(m)≥F(`),that is F(`)is increasing for `≥k>0.This proves the theorem.

M. B. Huban et al. /Filomat 35:7 (2021), 2165–2173 2172
Corollary 3.8. F(`)≥qber |A|2for `≥k>0.
The following lemma is well known (see, for instance, [8]).
Lemma 3.9. Let a and b be positive real numbers. Then,
aλbµ≤λa+µb
holds for λ > 0and µ > 0such that λ+µ=1.
Our next result characterizes absolute-k-quasi-paranormal operators Aon the RKHS H=H(Ω).
Theorem 3.10. For each k >0,an operator A on His absolute-k-quasi-paranormal if and only if
A∗|A|2kA−(k+1)αk|A|2+kαk+1∼≥0
holds for all α > 0.
Proof. ⇒.Suppose that Ais absolute-k-quasi-paranormal for k>0,i.e.,



|A|kA
b
kλ


≥


A
b
kλ



k+1(12)
for every λ∈Ω.Inequality (12)holds if and only if


|A|kAkλ

1
k+1kkλkk
k+1≥kAkλk
for all λ∈Ω,or equivalently
DA∗|A|2kAkλ,kλE1
k+1hkλ,kλik
k+1≥D|A|2kλ,kλE
for all λ∈Ω.By Lemma 3.9, we have:
DA∗|A|2kAkλ,kλE1
k+1hkλ,kλik
k+1
=(1
αkDA∗|A|2kAkλ,kλE)1
k+1
{αhkλ,kλi} k
k+1(13)
≤1
k+1
1
αkDA∗|A|2kAkλ,kλE+k
k+1αhkλ,kλi
for all λ∈Ωand α > 0,so that (12)ensures the following inequality by (13):
1
k+1
1
αkDA∗|A|2kAkλ,kλE+k
k+1αhkλ,kλi≥D|A|2kλ,kλE(14)
for all λ∈Ωand α > 0.
Conversely, (14)implies (12)by putting α=hA∗|A|2kAkλ,kλi
hkλ,kλi1
k+1; in case DA∗|A|2kAkλ,kλE=0,let α→0.
Hence (14)holds if and only if
A∗|A|2kA−(k+1)αk|A|2+kαk+1∼≥0
holds for all α > 0,which completes the proof of the theorem.
Since absolute-1-quasi-paranormal is quasi-paranormal, the following is immediate from Theorem 3.10.

M. B. Huban et al. /Filomat 35:7 (2021), 2165–2173 2173
Corollary 3.11. An operator A is quasi-paranormal if and only if
A∗2A2−2αA∗A+α2∼≥0
holds for all α > 0.
Acknowledgement
The authors would like to express their hearty thanks to the anonymous reviewer for his/her valuable
comments.
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