Rate of Decay of the Bernstein Numbers
We show that if a Banach space X contains uniformly complemented l₂ⁿ 's then there exists a universal constant b = b(X) > 0 such that for each Banach space Y, and any sequence dn ↓ 0 there is a bounded linear operator T : X → Y with the Bernstein numbers bn(T) of T satisfying b⁻¹dn ≤ bn(T) ≤...
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nasplib_isofts_kiev_ua-123456789-1067372025-02-09T17:38:54Z Rate of Decay of the Bernstein Numbers Plichko, A. We show that if a Banach space X contains uniformly complemented l₂ⁿ 's then there exists a universal constant b = b(X) > 0 such that for each Banach space Y, and any sequence dn ↓ 0 there is a bounded linear operator T : X → Y with the Bernstein numbers bn(T) of T satisfying b⁻¹dn ≤ bn(T) ≤ bdn for all n. Показано, что для B-выпуклого сепарабельного пространства X, произвольного банахова пространства Y и любой последовательности dn ↓ 0 существует такой ограниченный линейный оператор T : X → Y и b > 0, что для всех чисел Бернштейна bn(T) оператора T имеем для любого n b⁻¹dn ≤ bn(T) ≤ bdn. The author express his thanks to T. Oikhberg and M. Popov for valuable consultations. 2013 Article Rate of Decay of the Bernstein Numbers / A. Plichko // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 1. — С. 59-72. — Бібліогр.: 26 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106737 en Журнал математической физики, анализа, геометрии application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We show that if a Banach space X contains uniformly complemented l₂ⁿ 's then there exists a universal constant b = b(X) > 0 such that for each Banach space Y, and any sequence dn ↓ 0 there is a bounded linear operator T : X → Y with the Bernstein numbers bn(T) of T satisfying b⁻¹dn ≤ bn(T) ≤ bdn for all n. |
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Plichko, A. Rate of Decay of the Bernstein Numbers Журнал математической физики, анализа, геометрии |
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Plichko, A. |
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Plichko, A. |
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Rate of Decay of the Bernstein Numbers |
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Rate of Decay of the Bernstein Numbers |
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Rate of Decay of the Bernstein Numbers |
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Rate of Decay of the Bernstein Numbers |
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Rate of Decay of the Bernstein Numbers |
| title_sort |
rate of decay of the bernstein numbers |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Rate of Decay of the Bernstein Numbers / A. Plichko // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 1. — С. 59-72. — Бібліогр.: 26 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT plichkoa rateofdecayofthebernsteinnumbers |
| first_indexed |
2025-11-28T19:35:12Z |
| last_indexed |
2025-11-28T19:35:12Z |
| _version_ |
1850064024113250304 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 1, pp. 59–72
Rate of Decay of the Bernstein Numbers
A. Plichko
Department of Mathematics, Cracow University of Technology
Cracow, Poland
E-mail: aplichko@pk.edu.pl
Received August 2, 2012
We show that if a Banach space X contains uniformly complemented
`n
2 ’s then there exists a universal constant b = b(X) > 0 such that for
each Banach space Y , and any sequence dn ↓ 0 there is a bounded linear
operator T : X → Y with the Bernstein numbers bn(T ) of T satisfying
b−1dn ≤ bn(T ) ≤ bdn for all n.
Key words: B-convex space, Bernstein numbers, Bernstein pair, uni-
formly complemented `n
2 , superstrictly singular operator.
Mathematics Subject Classification 2010: 47B06, 47B10.
To the memory of M.I. Kadets
1. Introduction and Main Result
Let X, Y be Banach spaces and let L(X,Y ) be the space of all bounded
linear operators from X to Y . Notationally, all spaces are infinite dimensional
real Banach spaces unless otherwise specified.
Definition 1. An operator T ∈ L(X, Y ) is called superstrictly singular (SSS
for short; finitely strictly singular in other terminology) if there are no number
c > 0 and no sequence of subspaces En ⊂ X, dimEn = n, such that
‖Tx‖ ≥ c‖x‖ for all x in ∪n En . (1)
Put for an operator T
bn(T ) = sup min
x∈SE
‖Tx‖ , (2)
where supremum is taken over all n-dimensional subspaces E ⊂ X and SE is the
unit sphere of E. Evidently,
‖T‖ = b1(T ) ≥ b2(T ) ≥ · · · ≥ 0 ,
c© A. Plichko, 2013
A. Plichko
T is SSS if and only if
bn(T ) → 0 as n →∞
and the greatest constant c for which (1) is satisfied, is equal to limn→∞ bn(T ) .
Obviously, every compact operator is SSS and T has finite rank if and only
if bn(T ) = 0 beginning with some integer n. Observe, that if T has infinite rank
then for each n the set In(T ) of all n-dimensional subspaces E such that T |E are
injective, is dense in the set of all n-dimensional subspaces. Then the formula (2)
turns into the following one
bn(T ) = sup
E∈In(T )
1
‖(T |E)−1‖ . (3)
The bn(T ), which are called the Bernstein numbers, were considered in Ap-
proximation and Operator Theory. The constants bn(T ) show how small is the
T -image of the unit sphere SX . For a compact operator T in a Hilbert space H
they coincide with s-numbers which are defined as eigenvalues of the operator
(T ∗T )1/2. There are several generalizations of s-numbers to Banach spaces (see
below for details).
The Bernstein numbers take origin (see Whitley [26]) in the following classical
inequalities:
If pn is a polynomial of degree at most n, then for its derivative
‖p′n‖ ≤ n2‖pn‖ ,
the norm being the supremum norm on [−1, 1] (Markov [13]).
If qn is a complex trigonometric polynomial of degree at most n, then
‖q′n‖ ≤ n‖qn‖ ,
the norm being the supremum norm on the unite circle (Bernstein [2]).
Both of these inequalities have the same form: A Banach space, a derivation
operator D and an (n + 1)-dimensional subspace F are given. The conclusion
estimates the value of ‖D|F ‖. From this point of view it is natural to ask to what
extent the norm depend on F . In particular, what improvement is possible, i.e.
what is the best possible constant
inf{‖D|F ‖ : dimF = n} ?
It appears that this constant is equal to n [26]. Considering the inverse of D
we arrive to the notion of the Bernstein numbers. We find bn(T ) as far as in
(Krein/Krasnoselskĭı/Milman [11]). After (Mitiagin/Henkin [16]), SSS operators
60 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1
Rate of Decay of the Bernstein Numbers
were introduced implicitly by Mitiagin and PeÃlczyński [17] and explicitly, under
the name “operators of the class C∗
0”, by Milman [15].
The important role has been played by Pietsch’s paper [19] where systematic
theory of abstract s-numbers in Banach spaces was developed (see also [20]). In
particular, Pietsch noted the importance of duality and of the principle of local
reflexivity. The term “superstrictly singular operator” was introduced in (Hin-
richs/Pietsch [7]), where this class was investigated by machinery of superideals,
and by Mascioni [14]. For further progress in the theory of SSS operators in
general Banach spaces see e.g. (Plichko [24]) and (Flores/Hernández/Raynaud
[6]).
As we noted, an operator T is SSS if and only if bn(T ) ↓ 0. One can pose
an “inverse” problem. Let X, Y be Banach spaces and dn ↓ 0. Does there exist
T ∈ L(X,Y ) such that bn(T ) = dn for every n? We have a little chance to obtain
a positive answer. So, we will consider a weaker question which is natural in a
more general setting.
According to Pietsch [21], a map s which assigns to each bounded linear ope-
rator T between Banach spaces a unique sequence (sn(T )), is called an s-function
if for all Banach spaces W,X, Y, Z:
1. ‖T‖ = s1(T ) ≥ s2(T ) ≥ · · · ≥ 0 for all T ∈ L(X, Y ).
2. sn(S + T ) ≤ sn(S) + ‖T‖ for all S, T ∈ L(X,Y ) and all n.
3. sn(RST )≤ ‖R‖sn(S)‖T‖ for all T ∈ L(X,Y ), S∈ L(Y,Z) and R∈ L(Z, W ).
4. If T ∈ L(X,Y ) and rank T < n, then sn(T ) = 0.
5. sn(I) = 1 for all n, where I is the identity map of `n
2 .
The scalar sn(T ) is called the nth s-number of the operator T . The Bernstein
numbers are s-numbers. Another example of s-numbers are the approximation
numbers defined by the formula
an(T ) = inf{‖T − L‖ : L ∈ L(X,Y ), rank L < n}.
These numbers are connected with the well known approximation property of
Banach spaces and characterize the ideal of approximable operators: an(T ) → 0
if and only if T is approximable. The approximation numbers are the largest
s-numbers [19].
Aksoy and Lewicki [1] have introduced the following general concept.
Definition 2. Banach spaces X and Y are said to form a Bernstein pair with
respect to s-numbers sn if for any sequence dn ↓ 0, there exists T ∈ L(X, Y ) such
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 61
A. Plichko
that (sn(T )) is equivalent to (dn), i.e. there is a constant b depending only on T
such that for every n
b−1dn ≤ sn(T ) ≤ bdn .
This definition was motivated by well known Bernstein’s “lethargy” theorem
[4] and is a generalization of Bernstein pair with respect to the approximation
numbers (see Hutton/Morell/Retherforsd [8, 9]). Note that Hutton, Morell and
Retherforsd implicitly refereed Bernstein’s lethargy theorem to [3]. In [8, 9] it was
proved that many pairs of classical Banach spaces form the Bernstein pair with
respect to the approximation numbers. The authors advanced a hypothesis that
all couples of Banach spaces form Bernstein pairs (with respect to approximation
numbers). Aksoy and Lewicki [1] showed that many classical Banach spaces form
Bernstein pairs with respect to all s-numbers. Detailed investigations of “rate
of decay” of many s-numbers (Kolmogorov, Gelfand, Weyl, Hilbert,. . . numbers)
was carried out by Oikhberg [18]. We consider a similar question for the Bernstein
numbers. Ideal properties of the Bernstein numbers was considered by Samarskĭı
[25] and Pietsch [21].
First, we present simple examples of pairs (X,Y ) which are not Bernstein
with respect to the Bernstein numbers. They are, in fact, well known (see e.g.
Mitiagin/PeÃlczyński [17]).
For a subspace E of a Banach space X denote by λ(E, X) the relative projec-
tion constant
λ(E, X) = inf ‖P‖ ,
where inf is taken over all projections P of X onto E. Given a Banach space X
put
pn(X) = inf{λ(E,X) : E ⊂ X , dimE = n}.
Note that one can take infimum here only over a dense subset of all
n-dimensional subspaces.
Proposition 1. Let T ∈ L(X,H), where H is a Hilbert space and dimT (X)
= ∞. Then for every n
bn(T ) ≤ 1
pn(X)
‖T‖ .
P r o o f. Let b > 1 and Eb ∈ In(T ) be such that ‖(T |Eb
)−1‖ < bbn(T ) (see
(3)). Take the orthogonal projection Q of H onto T (Eb). Then P = (T |Eb
)−1QT
is a projection of X onto Eb. So
λ(Eb, X) ≤ ‖P‖ ≤ ‖(T |Eb
)−1‖ · ‖Q‖ · ‖T‖ < bbn(T )−1‖T‖.
Hence
pn(X) = inf
dimE=n
λ(E,X) ≤ λ(Eb, X) ≤ bbn(T )−1‖T‖.
Since b > 1 is arbitrary, this implies Proposition 1.
62 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1
Rate of Decay of the Bernstein Numbers
Corollary 1. Let an operator T ∈ L(X, Y ) can be factored through a Hilbert
space H : T = RS, R ∈ L(X, H), S ∈ L(H,Y ) and dimT (X) = ∞. Then for
every n
bn(T ) ≤ 1
pn(X)
‖R‖‖S‖ .
P r o o f. Indeed, by Proposition 1,
bn(T ) ≤ bn(R)‖S‖ ≤ 1
pn(X)
‖R‖‖S‖ .
For operators, factored through Hilbert spaces see [12].
Definition 3. We say that a Banach space X contains no uniformly comple-
mented finite-dimensional subspaces if pn(X) →∞ as n →∞.
The well known Pisier space P [22, 23] contains no uniformly complemented
finite-dimensional subspaces. Moreover, there exists λ > 0 such that pn(P) ≥
λ
√
n for all n.
Corollary 2. Every operator from a Banach space X, containing no uniformly
complemented finite-dimensional subspaces, into a Hilbert space H is SSS.
Corollary 3. There is λ > 0 such that for every operator T ∈ L(P,H) and
every n
bn(T ) ≤ 1
λ
√
n
‖T‖ .
R e m a r k 1. Since every n-dimensional subspace E ⊂ X is a range of a
projection P : X → E with ‖P‖ ≤ √
n (Kadets/Snobar [10]), one cannot obtain
a better estimation of bn(T ) with using of projections. A similar estimation for
operators from C(K) into H, but with constants 4
√
n instead of
√
n, was noted
in [17].
Proposition 1 implies
Corollary 4. Assume X contains no uniformly complemented finite-dimensional
subspaces and H is a Hilbert space. Then the pair (X,H) is not Bernstein with
respect to the Bernstein numbers.
P r o o f. Indeed, by Proposition 1, for every T ∈ L(X, H) the sequence
bn(T ) cannot go to 0 “more slowly” than 1/pn(c).
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 63
A. Plichko
An n-dimensional normed space (E, ‖ ‖) is said to be a-isomorphic to `n
2
(write E
a∼ `n
2 ), a > 1, if there exists an Euclidean norm ‖ ‖2 on E such that for
every e ∈ E
a−1‖e‖ ≤ ‖e‖2 ≤ a‖e‖.
If in this definition the constants a and n are inessential, we say simply about
almost Euclidean subspaces.
R e m a r k 2. If E
a∼ `n
2 then for every subspace F ⊂ E there is a projection
P : E → F with ‖P‖ ≤ a2.
If E
a∼ `n
2 then it have an a-orthonormal basis, i.e. a system (ei)n
1 such that
‖ei‖ = 1 for all i and for all scalars (ai)
a−1
(∑n
1
a2
i
)1/2
≤
∥∥∥
∑n
1
aiei
∥∥∥ ≤ a
(∑n
1
a2
i
)1/2
.
R e m a r k 3. For an a-orthonormal basis, the norm of each projection Pi ,
i < n, of E onto lin(ej)i
1 along to lin(ej)n
i+1 is not greater than a2.
Definition 4. (see e.g. [22, p. 215]). A Banach space X contains uniformly
complemented `n
2 ’s if there is a constant d such that for every ε > 0 and for each
n there is a subspace E ⊂ X and a projection P : X → E such that E
1+ε∼ `n
2 and
‖P‖ < d.
Note that by Dvoretzky’s theorem, if this holds for some ε, then it automat-
ically holds for all ε.
We will show that uniformly complemented almost Euclidean subspaces play
a crucial role in constructing of Bernstein pairs.
Theorem 1. If a Banach space X contains uniformly complemented `n
2 ’s then
there exists a universal constant b = b(X) > 0 such that for each Banach space
Y , and any sequence dn ↓ 0 there exist a bounded linear operator T : X → Y such
that for all n
b−1dn ≤ bn(T ) ≤ bdn .
Corollary 5. Let a Banach space X contain uniformly complemented `n
2 ’s.
Then for every Banach space Y the pair (X, Y ) is Bernstein with respect to the
Bernstein numbers.
A Banach space X is B-convex if it does not contain `n
1 ’s uniformly. Since
every B-convex Banach space contains uniformly complemented `n
2 ’s (see e.g. [22,
pp. 208, 215]), we have
64 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1
Rate of Decay of the Bernstein Numbers
Corollary 6. Let X be a B-convex Banach space. Then for every Banach
space Y the pair (X, Y ) is Bernstein with respect to the Bernstein numbers.
This corollary recalls us the well known Davis–Johnson compact non-nuclear
operator in a B-convex Banach space [5].
Problem. Does (X, X) form a Bernstein pair with respect to the Bernstein
numbers for every Banach space X?
2. Proof of the Main Result
To prove Theorem 1 we construct a “bounded minimal system” consisting of
almost Euclidean subspaces of arbitrary large dimensions in an arbitrary Banach
space containing uniformly complemented `n
2 ’s.
Lemma 1. Let X contain uniformly complemented `n
2 ’s, with corresponding
ε and d and let d′ > (1 + ε)4d. Then for each finite codimensional subspace
X ′ ⊂ X, each finite dimensional subspace E ⊂ X and each m there exists a
subspace E′ ⊂ X ′ , E′ 1+ε∼ `m
2 and a projection P ′ : X → E′ with ‖P ′‖ < d′ and
kerP ′ ⊃ E.
P r o o f. By definition, one can find an almost Euclidean subspace E0 ⊂ X,
dimE0 > m + dimE + dimX/X0 and a projection P0 : X → E0 with ‖P0‖ < d.
Since E0 is almost Euclidean, by Remark 2, there exists a projection Q0 : E0 →
E1 := E0 ∩ X0 with ‖Q0‖ ≤ (1 + ε)2. Obviously, dimE1 ≥ m + dimE. Put
P1 = Q0P0. Then P1 is a projection of X onto E1 and ‖P1‖ ≤ (1 + ε)2d.
Since E1 is almost Euclidean, by Remark 2, there exists a subspace E′ ⊂ E1 ,
dimE′ = m, and a projection Q1 : E1 → E′ with ‖Q1‖ ≤ (1 + ε)2 and kerQ1 ⊃
P (E). Then P ′ = Q1P1 is the desired projection.
Lemma 2. Let X contain uniformly complemented `n
2 ’s, with corresponding ε
and d. Then for any subsequence (mk)∞k=1 of integers there are subspaces Ek ⊂ X,
each Ek
1+ε∼ `mk
2 , with projections Pk : X → Ek , ‖Pk‖ ≤ d, such that each Ei,
i 6= k, belongs to kerPk.
P r o o f. Of course, one must write ‖Pk‖ ≤ d′, where d′ is from the previous
lemma, but the exact value of the constant d is non-essential here. We present a
construction only.
Take, by definition, a subspace E1 ⊂ X, E1
1+ε∼ `m1
2 , and a projection
P1 : X → E1 with ‖P1‖ ≤ d.
Then take, by Lemma 1, a subspace E2 ⊂ kerP1, E2
1+ε∼ `m2
2 , and a projection
P2 : X → E2 with ‖P2‖ ≤ d and kerP2 ⊃ E1.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 65
A. Plichko
Next take, by Lemma 1, a subspace E3 ⊂ kerP1 ∩ kerP2, E3
1+ε∼ `m3
2 , and a
projection P3 : X → E3 with ‖P3‖ ≤ d and kerP3 ⊃ (E1 ∪ E2), and so on.
R e m a r k 4. Let (Ek) be subspaces from Lemma 2. Then for every k ≥ 1
X = E1 ⊕ E2 ⊕ · · ·Ek ⊕ (∩k
i=1 kerPi).
Next, using the Dvoretzky theorem, we construct in an arbitrary Banach
space a subspace with “bounded minimal system” consisting of almost Euclidean
subspaces of arbitrary large dimensions. Denote by [A] the closed linear span of
the set A.
Lemma 3. Let Y be a Banach space, ε > 0, and (mk)∞k=1 be a sequence of
integers. Then there exist subspaces Fk ⊂ Y , each Fk
1+ε∼ `mk
2 , and projections
Qk : [Fi]∞1 → lin(Fi)k
1 along [Fi]∞k+1 with ‖Qk‖ ≤ 1 + ε.
P r o o f. Lemma 3 is a standard combination of the Dvoretzky and Mazur
theorems. We present a construction only. Recall that a subset Φ ⊂ Y ∗ λ-norms
a subspace F ⊂ Y if for every y ∈ SF there is ϕ ∈ Φ such that ϕ(y) ≥ λ. For each
finite-dimensional subspace F ⊂ Y and 0 < λ < 1 there is a finite set Φ ⊂ SY ∗
which λ-norms F .
So, take a subspace F1 ⊂ Y , F1
1+ε∼ `m1
2 , and a finite subset Φ1 ⊂ SX∗ which
(1 + ε)−1-norms F1.
Then take a subspace
F2 ⊂ Φ>1 := {y ∈ Y : ϕ(y) = 0 for all ϕ ∈ Φ1},
F2
1+ε∼ `m2
2 , and a finite subset Φ2 ⊂ SX∗ which (1 + ε)−1-norms F1 + F2.
Next, take a subspace F3 ⊂ Φ>2 , F3
1+ε∼ `m3
2 , and a finite subset Φ3 ⊂ SX∗
which (1 + ε)−1-norms F1 + F2 + F3, and so on.
In the proof we use diagonal operators in Euclidean spaces whose Bernstein
numbers are well known.
Definition 5. Let E and F be linear spaces with bases (en)m
1 and (fn)m
1 . Let
(dn)m
1 be scalars. A map
D
(∑m
1
anen
)
=
∑m
1
dnanfn
is called the diagonal operator corresponding to (en), (fn) and (dn).
66 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1
Rate of Decay of the Bernstein Numbers
Proposition 2. (sf. [19, Th. 7.1]). Let (en)m
1 be the standard basis of `m
2 ,
d1 ≥ d2 ≥ · · · ≥ dm ≥ 0 and
D be the diagonal operator in `m
2 corresponding to (en)m
1 and (dn)m
1 .
Then for all n ≤ m
min{‖Dx‖ : x ∈ lin(ej)n
1 , ‖x‖ = 1} = dn and
max{‖Dx‖ : x ∈ lin(ej)m
n , ‖x‖ = 1} = dn .
Corollary 7. Assume m-dimensional normed spaces E and F have
a-orthonormal bases (en)m
1 and (fn)m
1 ,
d1 ≥ d2 ≥ · · · ≥ dm ≥ 0 and
D is the diagonal operator corresponding to (en), (fn), (dn).
Then there is c > 1, depending only on a, such that for all n ≤ m
min{‖Dx‖ : x ∈ lin(ej)n
1 , ‖x‖ = 1} ≥ dn
c
and
max{‖Dx‖ : x ∈ lin(ej)m
n , ‖x‖ = 1} ≤ cdn .
P r o o f of Theorem 1. Let dn ↓ 0. Take a subsequence (nk)∞k=1 of integers
which approach to ∞ so quickly that for all k ≥ 1
dnk+1
<
1
4
dnk
. (4)
Hence, for every k ≥ 1
∞∑
i=k+1
dni <
1
2
dnk
. (5)
Let 0 < ε < 1 , Ek be subspaces from Lemma 2 and Fk , k ≥ 1, be subspaces
from Lemma 3 with mk := nk − nk−1 (and n0 = 0). Take in each Ek and each
Fk some (1 + ε)-orthonormal bases. Rearrange these bases in the natural way,
putting first the basis e1, . . . , en1 of E1, then the basis en1+1, . . . , en2 of E2 and
so on; and similarly for Y . We obtain systems (en)∞1 in X and (fn)∞1 in Y .
Put Nk = {n : nk−1 < n ≤ nk}. Using Corollary 7, (with c from this
corollary) we construct for every k ≥ 1 the diagonal operator Dk : Ek → Fk
corresponding to the bases (en), (fn) and scalars (dn) , n ∈ Nk, such that for all
n ∈ Nk
min
{
‖Dkx‖ : x ∈ [ej ]nnk−1+1 , ‖x‖ = 1
}
≥ dn
c
and (6)
max {‖Dkx‖ : x ∈ [ej ]nk
n , ‖x‖ = 1} ≤ cdn . (7)
Let Pk be the projections from Lemma 2. For every x ∈ X put
Tx =
∞∑
i=1
DiPix (8)
(bellow we will show that the series (8) converges for each x ∈ X).
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 67
A. Plichko
We make forth estimations. Let d be from Lemma 2 and c be from Corollary 7.
1. For every k ≥ 1 and x ∈ X, ‖x‖ = 1,
∞∑
i=k+1
‖DiPix‖ < 2cddnk
.
Indeed, Pix ∈ Ei and ‖Pix‖ ≤ ‖Pi‖‖x‖ ≤ d for all i, so
∞∑
i=k+1
‖DiPix‖ ≤ by (7) ≤ cddnk+1 +
∞∑
i=k+2
cddni+1
≤ by (5) ≤ cddnk
+
c
2
ddnk+1
< 2cddnk
.
In particular, this inequality shows that series (8) converges for each x ∈ X,
so T is well defined.
2. For every k ≥ 1 and n ∈ Nk
sup
{
‖Tx‖ : x ∈ [ej ]nk
n ⊕ ∩k
i=1 kerPi , ‖x‖ = 1
}
≤ 3cddn
(by Remark 4, the sum here is direct).
Indeed, take x ∈ [ej ]nk
n ⊕ ∩k
i=1 kerPi , ‖x‖ = 1. Then, by definition of Pi,
Tx =
∑∞
i=k DiPix, so
‖Tx‖ ≤ ‖DkPkx‖+
∞∑
i=k+1
‖DiPix‖ ≤ (by 1) ≤ ‖DkPkx‖+ 2cddnk
≤ (since ‖Pk‖ ≤ d, by (7)) ≤ cddn + 2cddn = 3cddn .
3. For every k ≥ 1 and x ∈ lin(Ei)k
1 , ‖x‖ = 1,
‖Tx‖ ≥ 1
4(1 + ε)
· dnk
c
.
We prove estimation 3 by induction. For k = 1, Tx = D1x, so 3 is followed
from (6) if we take in (6) n = n1. Suppose k > 1, estimation 3 is proved for k−1,
and x ∈ lin(Ei)k
1 , ‖x‖ = 1. Then
x = x1 + x2 , x1 ∈ lin(Ei)k−1
1 , x2 ∈ Ek ,
and, by the construction of Pi,
Tx1 =
k−1∑
i=1
DiPix1 and Tx2 = DkPkx2.
68 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1
Rate of Decay of the Bernstein Numbers
Hence, by the construction of Di,
Tx1 ∈ lin(Fi)k−1
1 and Tx2 ∈ Fk .
By the construction of projections Qi from Lemma 3,
Qk−1Tx = Qk−1Tx1 + Qk−1Tx2 = Tx1
and
(Qk −Qk−1)Tx = (Qk −Qk−1)Tx1 + (Qk −Qk−1)Tx2 = Tx2 .
Since ‖Qi‖ ≤ 1 + ε, hence ‖Qi −Qi−1‖ ≤ 2(1 + ε). So,
‖Tx‖ ≥ 1
1 + ε
‖Qk−1Tx‖ =
1
1 + ε
‖Tx1‖ (9)
and
‖Tx‖ ≥ 1
2(1 + ε)
‖(Qk −Qk−1)Tx‖ =
1
2(1 + ε)
‖Tx2‖. (10)
Since ‖x‖ = 1, we have that
either ‖x1‖ ≥ 1
2
or ‖x2‖ ≥ 1
2
.
If ‖x1‖ ≥ 1
2 , then by the induction assumption
‖Tx‖
by (9)
≥ 1
1 + ε
‖Tx1‖ ≥ 1
1 + ε
· 1
2
· 1
4(1 + ε)
· dnk−1
c
by (4)
≥ 1
2(1 + ε)2
· 1
4
· 4dnk
c
since ε < 1≥ 1
4(1 + ε)
· dnk
c
.
If ‖x2‖ ≥ 1
2 , then
‖Tx‖
by (10)
≥ 1
2(1 + ε)
‖Tx2‖
by (6)
≥ 1
2(1 + ε)
· 1
2
· dnk
c
=
1
4(1 + ε)
· dnk
c
.
Therefore, 3 is proved.
4. For every k ≥ 1 and n ∈ Nk
min{‖Tx‖ : x ∈ lin(ej)n
1 , ‖x‖ = 1} ≥ 1
4(1 + ε)
· dn
c
.
Indeed, take x ∈ lin(ej)n
1 , ‖x‖ = 1, where n ∈ Nk. Then, as in 3, x = x1+x2,
x1 ∈ lin(Ei)k−1
1 , x2 ∈ Ek; either ‖x1‖ ≥ 1
2 or ‖x2‖ ≥ 1
2 ; Tx1 ∈ lin(Fi)k−1
1 ,
Tx2 ∈ Fk, and the inequalities (9), (10) hold.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 1 69
A. Plichko
If ‖x1‖ ≥ 1
2 , then
‖Tx‖ ≥ by 3 ≥ 1
4(1 + ε)
· dnk−1
c
≥ 1
4(1 + ε)
· dn
c
.
If ‖x2‖ ≥ 1
2 , then
‖Tx‖
by (10)
≥ 1
2(1 + ε)
‖Tx2‖ =
1
2(1 + ε)
‖Dkx2‖
by (6)
≥ 1
4(1 + ε)
· dn
c
.
Therefore, 4 is proved.
Put b = max{3cd, 4(1 + ε)c}. Inequality 4 shows that for all n
bn(T ) ≥ b−1dn.
Let G ⊂ X be an n-dimensional subspace and n ∈ Nk. Then, by Remark 4,
G ∩
(
[ej ]nk
n ⊕ ∩k
i=1 kerPi
)
6= 0.
So, the inequality 4 confirms that for all n
min
x∈SG
‖Tx‖ ≤ bdn ,
i.e.
bn(T ) ≤ bdn .
Acknowledgements. The author express his thanks to T. Oikhberg and
M. Popov for valuable consultations.
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