Strict s-numbers of the Volterra operator

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by  Özlem Bakşi

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STRICT S-NUMBERS OF THE VOLTERRA OPERATOR

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O¨ZLEMBAKS¸I1,2,TAQSEERKHAN1,3,JANLANG1,ANDV´ITMUSIL1,4
F
.
h
t Abstract. For Volterra operator V: L1(0,1) → C[0,1] and summation opera-
a tor σ: ℓ1 → c, we obtain exact values of Approximation, Gelfand, Kolmogorov,
m MityaginandIsomorphismnumbers.
[

2
v
1 1. introduction and main results
1
Compactoperatorsandtheirsub-classes(nuclearoperators,Hilbert-Schmidtopera-
1
1tors, etc.) play a crucialrole in many different areas of Mathematics. These operators
0are studied extensively but somehow less attention is devoted to operators which are
.non-compactbutcloseto the classofcompactoperators. Inthis workwe willfocuson
1
0two such operators.
7First, consider the Volterra operator V, given by
1
t
v: Vf(t)= f(s)ds, (0≤t≤1), for f ∈L1(0,1). (1.1)
Z
i 0
X
When V is regarded as an operator from Lp into Lq, (1 < p,q < ∞), it is a compact
aroperator,howeverinthelimitingcase,whenV mapsL1 intothespaceC ofcontinuous
functions on the closedunit interval, the operatoris bounded, with the operatornorm
kVk = 1, but non-compact. It is worth mentioning that, despite being non-compact
or even weakly non-compact, this operator possesses some good properties as being
strictlysingular(followsfrom[BG89],orsee[Lef17]). ThismakesVolterraoperator,in
the above-mentioned limiting case, an interesting example of a non-compact operator
“close” to the class of compact operators. The focus of our paper will be on obtaining
exact values of strict s-numbers for this operator.
Volterraoperatorwasalreadyextensivelystudied. Letus brieflyrecallthose results
relatedto ourwork. The firstcreditgoesto V.I.Levin[Lev38], whocomputed explic-
itly the norm of V between two Lp spaces (1 < p < ∞) and described the extremal
functionwhichisconnectedwiththefunctionsin . Lateron,E.Schmidtin[Sch40]ex-
p
tendedthisresultforV : Lp →Lq,where1<p,q <∞. Thisoperatorwasalsostudied
in the context of Approximation theory [Pin85b, Kol36, Tih60, TB67, BS67, Pin85a].
Date:February28,2017.
2010 Mathematics Subject Classification. Primary47B06, Secondary47G10.
Keywords and phrases. Integral operator,summationoperator,s-numbers.
ThisresearchwaspartlysupportedbytheUnitedStates–IndiaEducationalFoundation(USIEF)
andbythegrantP201-13-14743S oftheGrantAgencyoftheCzechRepublic.
1
2 O¨.BAKS¸I,T.KHAN,J.LANG,ANDV.MUSIL
Later a weighted version of this operator was studied in connection with Brownian
motion [LL02], Spectral theory [EE04, EE87] and Approximation theory [EL11].
Recently, sharp estimates for Bernstein-numbers of V in the limiting case were
obtained in [Lef17, Theorem 2.2], more specifically,
1
b (V)= for n∈N, (1.2)
n
2n−1
andalsotheestimatesfortheessentialnormcanbefoundinrecentpreprint[AHLM16].
Inourpaper,wewillcomputeexactvaluesofalltheremainingstricts-numbers,i.e.,
Approximation,Gelfand,Kolmogorov,MityaginandIsomorphismnumbersdenotedby
a , c , d , m and i respectively (for the exact definitions see Section 2). Our main
n n n n n
result reads as follows.
Theorem 1.1. Let V : L1(0,1)→C[0,1] be defined as in (1.1). Then
1
a (V)=c (V)=d (V)= for n≥2 (1.3)
n n n
2
and
1
m (V)=i (V)= for n∈N. (1.4)
n n
2n−1
If we also include the result (1.2) concerning the Bernstein numbers, we see that
all the strict s-numbers of V split between two groups. The upper half (1.3) remains
bounded from below while the lower half (1.4) converges to zero. This phenomenon
for this operator was already observed; for instance compare [BS67] and [BG89], and
for weighted version see [EL07] and [EL06].
The similar results continue to hold for the sequence spaces and for the discrete
analogue of V, namely for the operator σ: ℓ1 →c, defined as
k
σ(x) = x , (k ∈N), for x∈ℓ1, (1.5)
k j
Xj=1
where we denoted x={x }∞ for brevity. The operator is well-defined and bounded
j j=1
with the operator norm kσk=1. It is shown in [Lef17, Theorem 3.2] that
1
b (σ)= for n∈N.
n
2n−1
We have the next result.
Theorem 1.2. Let σ: ℓ1 →c be the operator from (1.5). Then
1
a (σ)=c (σ)=d (σ)= for n≥2
n n n
2
and
1
m (σ)=i (σ)= for n∈N.
n n
2n−1
The proofs are provided at the end of Section 3.
STRICT S-NUMBERS OF THE VOLTERRA OPERATOR 3
2. background material
We shall fix the notation in this section, although we mostly work with standard
notions from functional analysis.
2.1. Normed linear spaces. For normed linear spaces X and Y, we denote by
B(X,Y) the set of all bounded linear operators acting between X and Y. For any
T ∈ B(X,Y), we use just kTk for its operator norm, since the domain and target
spaces are always clear from the context. By B , we mean the closed unit ball of X
X
and, similarly, S stands for the unit sphere of X. It is well-known fact that B is
X X
compact if and only if X is finite-dimensional.
Let Z be a closed subspace of the normed space X. The quotient space X/Z is the
collection of the sets [x]=x+Z ={x+z; z ∈Z} equipped with the norm
k[x]k =inf{kx−zk ; z ∈Z}.
X/Z X
Wesometimesadoptthenotationkxk whennoconfusionislikelytohappen. Recall
X/Z
the notion of canonical map Q : X →X/Z, given by Q (x)=[x].
Z Z
By the Lebesgue space L1, we mean the set of all real-valued, Lebesgue integrable
functions on (0,1) identified almost everywhere and equipped with the norm
1
kfk = |f(s)|ds.
1
Z
0
Thespaceofreal-valued,continuousfunctionson[0,1],denotedbyC,enjoysthenorm
kfk = sup |f(t)|.
∞
0≤t≤1
The discrete counterpart to L1 is the space of all summable sequences, ℓ1, where
∞
kxk = |x |
1 j
Xj=1
and, similarly to the space C, we denote by c the space of all convergent sequences
endowed with the norm
kxk =sup|x |.
∞ j
j∈N
Here and in the latter, we use the abbreviation x={x }∞ for the sequences and we
j j=1
write them in bold font. Note that we also consider only real-valued sequence spaces.
All the above-mentionedspaces are complete, i.e., they form Banach spaces.
2.2. s-numbers. Let X and Y be Banach spaces. To every operator T ∈ B(X,Y),
one can attach a sequence of non-negative numbers s (T) satisfying for every n ∈ N
n
the following conditions
(S1) kTk=s (T)≥s (T)≥···≥0,
1 2
(S2) s (T +S)≤s (T)+kSk for every S ∈B(X,Y),
n n
(S3) s (B◦T ◦A)≤kBks (T)kAk for every A∈B(X ,X) and B ∈B(Y,Y ),
n n 1 1
(S4) s (Id: ℓ2 →ℓ2)=1,
n n n
(S5) s (T)=0 whenever rankT <n.
n
4 O¨.BAKS¸I,T.KHAN,J.LANG,ANDV.MUSIL
The number s (T) is then called the n-th s-number of the operator T. When (S4) is
n
replaced by a stronger condition
(S6) s (Id: E →E)=1 for every Banach space E, dimE =n,
n
we say that s (T) is the n-th strict s-number of T.
n
Note that the original definition of s-numbers, which was introduced by Pietsch
in [Pie74], uses the condition (S6) which was later modified to accommodate wider
class of s-numbers (like Weyl, Chang and Hilbert numbers). For a detailed account of
s-numbers, one is referred for instance to [Pie07], [CS90] or [EL11].
We shall briefly recall some particular strict s-numbers. Let T ∈ B(X,Y) and
n ∈ N. Then the n-th Approximation, Gelfand, Kolmogorov, Isomorphism, Mityagin
and Bernstein numbers of T are defined by
a (T)= inf kT −Fk,
n
F∈B(X,Y)
rankF<n
c (T)= inf sup kTxk ,
n Y
M⊆X x∈BM
codimM<n
d (T)= inf sup kTxk ,
n Y/N
N⊆Y x∈BX
dimN<n
i (T)=supkAk−1kBk−1,
n
where the supremum is taken over all Banach spaces E with dimE ≥ n and A ∈
B(Y,E), B ∈B(E,X) such that A◦T ◦B is the identity map on E,
m (T)=sup sup Q TB ⊇ρB ,
n N X Y/N
ρ≥0 N⊆Y
codimN≥n
and
b (T)= sup inf kTxk ,
n Y
M⊆X x∈SM
dimM≥n
respectively.
3. Proofs
Lemma 3.1. Let n∈N. Then we have the following lower bounds of the Isomorphism
numbers of V and σ.
(i)
1
i (V)≥ ;
n
2n−1
(ii)
1
i (σ)≥ .
n
2n−1
STRICT S-NUMBERS OF THE VOLTERRA OPERATOR 5
Proof. (i) Let n ∈ N be fixed. We shall construct a pair of maps A and B such that
the chain
ℓ1 −B→L1 −V→C −→A ℓ1
w,n w,n
forms the identity on ℓ1 . Here ℓ1 is the n-dimensional weighted space ℓ1 with the
w,n w,n
norm given by
n
kxkℓ1 = wk|xk|.
w,n
kX=1
For the purpose of this proof, we choose w =2 for 1≤k≤n−1 and w =1.
k n
Now, define A: C →ℓ1 by
w,n
2k−1
(Af) =(2n−1)f , (1≤k ≤n), for f ∈C.
k
(cid:18)2n−1(cid:19)
Obviously, A is bounded with the operator norm
kAk=(2n−1)2.
Inorderto constructthe mappingB, considerthe partitionofthe unitintervalinto
subintervals I , I , ..., I of the same length, i.e.
1 2 2n−1
k−1 k
I = , for 1≤k≤2n−1,
k
(cid:20)2n−1 2n−1(cid:21)
and define
n−1
B(x)= x χ −χ +x χ .
k I2k−1 I2k n I2n−1
Xk=1 (cid:0) (cid:1)
Clearly B(x) is integrable for every x ∈ ℓ1 , B is bounded and the operator norm
w,n
satisfies
1
kBk= .
2n−1
OnecanobservethatthecompositionA◦V ◦B istheidentitymappingonℓ1 and
w,n
by the very definition of the n-th Isomorphism number we have
1
i (V)≥kAk−1kBk−1 =
n
2n−1
which completes the proof.
As for the discrete case (ii), we consider the chain
ℓ∞ −B→ℓ1 −→σ c−→A ℓ∞
n n
where ℓ∞ stands for the n-dimensional space ℓ∞, A is given by
n
A(x) =x , (1≤k ≤n), for x∈c
k 2k−1
and B satisfies
B(y)=(y ,−y ,...,y ,−y ,y ,0,0,...) for y∈ℓ∞.
1 1 n−1 n−1 n n
6 O¨.BAKS¸I,T.KHAN,J.LANG,ANDV.MUSIL
Both A and B are bounded, the composition A◦σ◦B forms the identity on ℓ∞ and
n
thus
1
i (σ)≥kAk−1kBk−1 = ,
n
2n−1
since kAk=1 and kBk=2n−1. (cid:3)
Lemma 3.2. Let n∈N then we have the following estimates of the Mityagin numbers
of V and σ.
(i)
1
m (V)≤ ;
n
2n−1
(ii)
1
m (σ)≤ .
n
2n−1
Proof. (i) Fix some n ∈ N and any 0 < ̺ < m (V). By the definition of the n-th
n
Mityagin number, there is a subspace of C, say N, such that codimN ≥n and
QNVBL1 ⊇̺BC/N. (3.1)
Define
E = f ∈L1; span{Vf}∩N ={0}
and observe that, since V is(cid:8)injective, E is a subspace of d(cid:9)imension at most n and
satisfies
QNVBL1 = [Vf]; f ∈BE . (3.2)
Thus, by (3.1) and (3.2), we have (cid:8) (cid:9)
kVfk ≥kVfk ≥̺ for f ∈S ,
∞ C/N E
and hence
kVfk ≥̺kfk for f ∈E.
∞ 1
Therefore, thanks to [Lef17, Lemma 2.4],
1
̺≤ (3.3)
2n−1
and the lemma follows by taking the limit ̺→m (V).
n
(ii) The exact analogy holds in the discrete case, as for every 0 < ̺ < m (σ) one
n
can find a n-th dimensional subspace E in ℓ1, such that
kσ(x)k ≥̺kxk for x∈E.
∞ 1
Theassertion(3.3)thenalsoholdsby[Lef17,Lemma2.4]andits discretemodification
in the proof of [Lef17, Theorem 3.2]. (cid:3)
Lemma 3.3. Let n ≥ 2 then the following lower bounds of the Kolmogorov numbers
of V and σ hold.
(i)
1
d (V)≥ ;
n
2
STRICT S-NUMBERS OF THE VOLTERRA OPERATOR 7
(ii)
1
d (σ)≥ .
n
2
Proof. (i) Fix arbitrary n ≥ 2 and ε > 0. By the very definition of the n-th Kol-
mogorov number, there exists a subspace of C, say N, such that dimN <n and
d (V)+ε≥ sup kVfk . (3.4)
n C/N
f∈BL1
Let us define the trial functions
fk =2k+1χ(2−k−1,2−k), (k ∈N). (3.5)
There is kf k = 1 for every k ∈ N. Now, by the definition of the quotient norm, to
k 1
every Vf one can attach a function g ∈N in a way that
k k
kVf −g k ≤kVf k +ε. (3.6)
k k ∞ k C/N
Observe that the set of all the functions g is bounded in N. Indeed, by (3.4) and
k
(3.6),
kg k ≤kVf −g k +kVf k ≤kVf k +ε+kVkkf k ≤d (V)+2ε+1
k ∞ k k ∞ k ∞ k C/N k 1 n
foreveryk ∈N. Thus,since N is finite-dimensional,there isaconvergentsubsequence
of {g } which we denote {g } again. Hence g converges to, say, g ∈ N, i.e. there is
k k k
an index k such that kg −gk < ε for every k ≥ k . The limiting function g then
0 k ∞ 0
satisfies
kVf −gk ≤kVf −g k +kg −gk ≤kVf k +2ε
k ∞ k k ∞ k ∞ k C/N
for k ≥k and thus
0
sup kVf −gk ≤d (V)+3ε. (3.7)
k ∞ n
k≥k0
Next, we estimate the left hand side of (3.7) by taking the value attained at zero, i.e.
sup kVf −gk ≥ sup |Vf (0)−g(0)|=|g(0)|, (3.8)
k ∞ k
k≥k0 k≥k0
or at the points 2−k, i.e.
sup kVf −gk ≥ sup |Vf (2−k)−g(2−k)|≥|1−g(0)|, (3.9)
k ∞ k
k≥k0 k≥k0
where we used that g is continuous. Combining (3.7), (3.8) and (3.9), we get
1
d (V)+3ε≥max |g(0)|,1−|g(0)| ≥ .
n
2
(cid:8) (cid:9)
Therefore d (V)≥1/2 by the arbitrariness of ε.
n
(ii) The proof needs just slight modifications here. Instead of functions f , one can
k
use the canonical vectors ek =(0,...,0,1,0,...), where the element 1 is placed at the
k-th coordinate. If we follow the previous lines, we find close points yk ∈ N, their
limit y in c and we end up with
sup kσ(ek)−yk ≤d (σ)+3ε. (3.10)
∞ n
k≥k0
8 O¨.BAKS¸I,T.KHAN,J.LANG,ANDV.MUSIL
Now, we estimate the supremum in (3.10) by taking the k-th coordinate, i.e.
sup kσ(ek)−yk ≥ sup |σ(ek) −y |≥|1− lim y |,
∞ k k k
k≥k0 k≥k0 k→∞
or by taking the coordinate k−1, i.e.
sup kσ(ek)−yk ≥ sup |σ(ek) −y |≥| lim y |,
∞ k−1 k−1 k
k≥k0 k≥k0 k→∞
and the conclusion follows. (cid:3)
Lemma 3.4. Let n≥2 then the estimates of Gelfand numbers of V and σ read as
(i)
1
c (V)≥ ;
n
2
(ii)
1
c (σ)≥ .
n
2
Proof. (i)Letn≥2andε>0befixed. Bythedefinitionofthen-thGelfandnumber,
we can find a subspace M in L1 having codimM <n and satisfying
c (V)+ε≥ sup kVfk . (3.11)
n ∞
f∈BM
The proof will be finished once we show that the supremum in (3.11) is at least one
half. We make use the step functions f , (k ∈ N), defined by (3.5) in the proof of
k
Lemma 3.3. Recall that kf k = 1, kf −f k = 2 and kVf −Vf k = 1 for k 6= l.
k 1 k l 1 k l ∞
Note that the quotient space L1/M is of finite dimension thus, the projected sequence
{[f ]} is bounded and hence there is a Cauchy subsequence, which we denote {[f ]}
k k
again. Now, let η >0 be fixed. We have
kfk−flkL1/M <η (3.12)
for k and l sufficiently large. Let us denote f = 1(f −f ) for these k and l. Thanks
2 k l
to (3.12) and the definition of quotient norm, one canfind a function g ∈M such that
kf −gk ≤η.
1
On setting
g
h= ,
1+η
we have
1
khk ≤ kfk +kf −gk ≤1,
1 1 1
1+η
(cid:0) (cid:1)
STRICT S-NUMBERS OF THE VOLTERRA OPERATOR 9
whence h∈B . Next
M
1
kVhk ≥ kVfk −kV(f −g)k
∞ ∞ ∞
1+η
(cid:0) (cid:1)
1 1
≥ −kVkkf −gk
1
1+η(cid:18)2 (cid:19)
1 1
≥ −η ,
1+η(cid:18)2 (cid:19)
thus
1 1
sup kVfk ≥ −η
∞
1+η(cid:18)2 (cid:19)
f∈BM
and the lemma follows, since η >0 was arbitrarily chosen.
The proof of the discrete counterpart(ii) is completely analogous,once we consider
the canonical vectors ek instead of f , hence we omit it. (cid:3)
k
Lemma 3.5. Let n≥2. Then we have the following upper bounds of the Approxima-
tion numbers of V and σ.
(i)
1
a (V)≤ ;
n
2
(ii)
1
a (σ)≤ .
n
2
Proof. (i) Consider the one-dimensional operator F: L1 →C given by
1 1
Ff(t)= f(s)ds, (0≤t≤1), for f ∈L1.
2Z
0
Then F is a sufficient approximationof V. Indeed,
t 1 1
kVf −Ffk = sup f(s)ds− f(s)ds
∞
0≤t≤1(cid:12)(cid:12)Z0 2Z0 (cid:12)(cid:12)
(cid:12)(cid:12)1 t 1 1 (cid:12)(cid:12)
= sup f(s)ds− f(s)ds
0≤t≤1(cid:12)(cid:12)2Z0 2Zt (cid:12)(cid:12)
(cid:12)(cid:12)1 t 1 1 (cid:12)(cid:12)
≤ sup |f(s)|ds+ |f(s)|ds
0≤t≤12Z0 2Zt
1
= kfk
1
2
and therefore
1
a (V)≤kV −Fk≤ .
n
2
In order to show (ii), choose the operator
∞
1
̺(x) = x , (k ∈N), for x∈ℓ1.
k j
2
Xj=1
10 O¨.BAKS¸I,T.KHAN,J.LANG,ANDV.MUSIL
This is a well-defined one-dimensional operator and, by the calculations similar to
above, kσ−̺k≤1/2. The proof is complete. (cid:3)
Now, we are at the position to prove the main results.
Proof of Theorem 1.1. Let n ≥ 2 be fixed. Since a (V) is the largest between all
n
s-numbers,weimmediatelyobtaintheinequalitiesa (V)≥c (V)anda (V)≥d (V).
n n n n
Using Lemma 3.5 with Lemma 3.4 and Lemma 3.3 we obtain
1 ≥a (V)≥c (V)≥ 1 and 1 ≥a (V)≥d (V)≥ 1
2 n n 2 2 n n 2
respectively. This gives (1.3).
Next, let n be arbitrary. Due to i (V) being the smallest strict s-number, we have
n
thati (V)≤b (V)andalsoi (V)≤m (V). Forthelowerbound,weuseLemma3.1,
n n n n
while fortheupper, wemakeuse ofLemma3.2andthe resultofLef`evre,[Lef17]. This
gives (1.4). (cid:3)
Proof of Theorem 1.2. The proof follows along exactly the same lines as that of
Theorem 1.1 and hence omitted. (cid:3)
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