A sufficient condition for the sum of complemented subspaces to be complemented
We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be complemented. Under this condition, the formula for a projection onto the sum is given. The condition is sharp (in a certain sense). As an application, we provide a sufficient conditi...
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irk-123456789-1504622019-04-08T01:25:32Z A sufficient condition for the sum of complemented subspaces to be complemented Feshchenko, I.S. Математика We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be complemented. Under this condition, the formula for a projection onto the sum is given. The condition is sharp (in a certain sense). As an application, we provide a sufficient condition for the complementability of the sum of marginal subspaces in L^p. Наведено достатню умову для того, щоб сума скінченного числа доповнювальних підпросторів банахового простору була доповнювальною. За цієї умови отримано формулу для проектора на цю суму підпросторів. Ця умова є точною (в певному сенсі). Як застосування наведено достатню умову для доповнювальності суми маргінальних підпросторів у просторі L^p. Ключові слова: сума підпросторів, доповнювальний підпростір, замкнений підпростір, маргінальний підпростір, проектор. Приведено достаточное условие для того, чтобы сумма конечного числа дополняемых подпространств банахова пространства была дополняема. При этом условии получена формула для проектора на эту сумму подпространств. Это условие является точным (в определенном смысле). В качестве применения приведено достаточное условие для дополняемости суммы маргинальных подпространств в пространстве L^p. 2019 Article A sufficient condition for the sum of complemented subspaces to be complemented / I.S. Feshchenko // Доповіді Національної академії наук України. — 2019. — № 1. — С. 10-15. — Бібліогр.: 13 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2019.01.010 http://dspace.nbuv.gov.ua/handle/123456789/150462 517.982.22 en Доповіді НАН України Видавничий дім "Академперіодика" НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Математика Математика |
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Математика Математика Feshchenko, I.S. A sufficient condition for the sum of complemented subspaces to be complemented Доповіді НАН України |
| description |
We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be
complemented. Under this condition, the formula for a projection onto the sum is given. The condition is sharp (in a
certain sense). As an application, we provide a sufficient condition for the complementability of the sum of marginal
subspaces in L^p. |
| format |
Article |
| author |
Feshchenko, I.S. |
| author_facet |
Feshchenko, I.S. |
| author_sort |
Feshchenko, I.S. |
| title |
A sufficient condition for the sum of complemented subspaces to be complemented |
| title_short |
A sufficient condition for the sum of complemented subspaces to be complemented |
| title_full |
A sufficient condition for the sum of complemented subspaces to be complemented |
| title_fullStr |
A sufficient condition for the sum of complemented subspaces to be complemented |
| title_full_unstemmed |
A sufficient condition for the sum of complemented subspaces to be complemented |
| title_sort |
sufficient condition for the sum of complemented subspaces to be complemented |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2019 |
| topic_facet |
Математика |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/150462 |
| citation_txt |
A sufficient condition for the sum of complemented subspaces to be complemented / I.S. Feshchenko // Доповіді Національної академії наук України. — 2019. — № 1. — С. 10-15. — Бібліогр.: 13 назв. — англ. |
| series |
Доповіді НАН України |
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AT feshchenkois asufficientconditionforthesumofcomplementedsubspacestobecomplemented AT feshchenkois sufficientconditionforthesumofcomplementedsubspacestobecomplemented |
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2025-07-13T00:10:02Z |
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2025-07-13T00:10:02Z |
| _version_ |
1837488316417572864 |
| fulltext |
10 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2019. № 1
1. Complemented subspaces in Banach spaces. Let X be a (complex or real) Banach space. By a
subspace of X , we will mean a linear subset of X . Let M be a subspace of X . M is said to be
complemented in X if there exists a continuous linear projection onto M , i.e., a continuous lin-
ear operator →:P X X such that ∈Px M for all ∈x X and =Px x for ∈x M . It is easily seen
that each complemented subspace is closed. Note that one can give the following (equivalent)
definition of complementability: a subspace M is said to be complemented in X if M is closed
and there exists a closed subspace N (a complement) such that ∩ = {0}M N and + =M N X .
If X is a Hilbert space, then each closed subspace M of X is complemented in X (one can
consider the orthogonal projection onto M ). Of course, this is true if X is isomorphic to a Hilbert
space. But if X is not isomorphic to a Hilbert space, then, by the Lindenstrauss—Tzafriri theorem,
X contains a closed subspace which is not complemented in X .
For the further information on complemented and uncomplemented subspaces in Banach
spa ces and, in particular, various examples of uncomplemented closed subspaces see, e.g., [1, 2]
and the references therein.
2. Formulations of problems. Let X be a Banach space and …1, , nX X be complemented
subspaces of X . Define the sum of …1, , nX X in the natural way, namely,
1 1 1 1: { | , , }.n n n nX X x x x X x X+ + = + + ∈ ∈… … …
The natural question arises:
© I.S. Feshchenko, 2019
doi: https://doi.org/10.15407/dopovidi2019.01.010
UDC 517.982.22
I.S. Feshchenko
Taras Shevchenko National University of Kiev
E-mail: ivanmath007@gmail.com
A sufficient condition for the sum
of complemented subspaces to be complemented
Presented by Academician of the NAS of Ukraine Yu.S. Samoilenko
We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be
complemented. Under this condition, the formula for a projection onto the sum is given. The condition is sharp (in a
certain sense). As an application, we provide a sufficient condition for the complementability of the sum of marginal
subspaces in Lp.
Keywords: sum of subspaces, complemented subspace, closed subspace, marginal subspace, projection.
11ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 1
A sufficient condition for the sum of complemented subspaces to be complemented
Question 1: Is + +…1 nX X complemented in X ?
Note that Question 1 makes sense, since the sum of two complemented subspaces may be
uncomplemented and even nonclosed. A simple example is as follows: if X is a Hilbert space, then
a subspace is complemented if and only if it is closed, and there are well-known simple examples
of two closed subspaces with nonclosed sum.
If Question 1 has a positive answer, then the next natural question arises:
Question 2: Suppose that we know some (continuous linear) projections …1, , nP P onto
…1, , nX X , respectively. Is there a formula for a projection onto + +…1 nX X (in terms of …1, , nP P )
(of course, under certain conditions)?
Since each complemented subspace is closed, Question 1 is closely related to the following
Question 3: Is + +…1 nX X closed in X ?
It is worth mentioning that if X is a Hilbert space, then Question 1 coincides with Question 3.
Systems of subspaces …1, , nX X , for which Question 3 is very important, arise in various
branches of mathematics, for example, in theoretical tomography and the theory of ridge func-
tions (plane waves) (see, e.g., [3, Introduction, Chapter 7 and the references therein]), theory
of wavelets and multiresolution analysis (see, e.g., [4] and references therein), statistics (see,
e.g., [5]), approximation algorithms in Hilbert and Banach spaces and, in particular, methods of
alternating projections (see, e.g., [3, Chapter 9 and the bibliography therein]) and others.
3. Linear independence. Another property of systems of subspaces, which will be of inte-
rest to us, is the linear independence of subspaces. A system of subspaces …1, , nX X is said to
be linearly independent if the equality + + =…1 0nx x , where ∈ ∈…1 1, , n nx X x X , implies that
= = =…1 0nx x .
4. Notation. Throughout the paper, X is a real or complex Banach space with norm ⋅ .
The identity operator on X is denoted by I . By a projection we always mean a continuous li-
near projection. The kernel of an operator T will be denoted by ker( )T . All vectors are vector-
columns; the letter “t” means the transpose.
5. Known results. Let X be a Banach space, …1, , nX X be complemented subspaces of X ,
and …1, , nP P be projections onto …1, , nX X , respectively.
For = 2n sufficient conditions for +1 2X X to be complemented in X can be found in [6— 9].
As an example, we present a result from [9]: if the restriction of the operator − 2 1I P P to its in-
variant subspace 2X is Fredholm, then +1 2X X is complemented in X . Concerning Question 2,
a few formulas for a projection onto +1 2X X (under certain conditions) can be found in [7].
For arbitrary n each of the following conditions is sufficient for + +…1 nX X to be comp-
lemented in X :
1. ([6, Corollary]) …1, , nX X are pairwise totally incomparable;
2. ([7, Corollary 2.9]) i jP P is compact for every pair ≠i j , ∈ …, {1, , }i j n . Moreover, under
this condition, there exists a projection P onto + +…1 nX X such that P equals + +…1 nP P mod-
ulo compact operators.
6. Our results. We will provide a new sufficient condition for + +…1 nX X to be comple-
mented in X . Under the condition, a formula for a projection onto the sum will be given.
We begin with a simple observation on Questions 1 and 2. The observation was used by many
authors. If =| 0i X j
P for all ≠i j , ∈ …, {1, , }i j n , then …1, , nX X are linearly independent, their
sum is complemented in X , and = + +…1 nP P P is a projection onto + +…1 nX X .
12 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2019. № 1
I.S. Feshchenko
Our result can be regarded as a strengthening of the observation.
Suppose that nonnegative numbers εij , ≠i j , ∈ …, {1, , }i j n are such that
ε ∈,i ij jP x x x X�
for every ≠i j , ∈ …, {1, , }i j n .
Define the ×n n matrix = ( )ijE e by
=⎧⎪= ⎨ε ≠⎪⎩
0, ;
, .ij
ij
if i j
e
if i j
Denote, by ( )r E , the spectral radius of E . Set = + +…1: nA P P .
Theorem 1. If <( ) 1r E , then the subspaces …1, , nX X are linearly independent, their sum is
complemented in X , and the subspace ∩ ∩…1ker( ) ker( )nP P is a complement of + +…1 nX X in
X . Moreover, the sequence of operators
− −( )NI I A
converges uniformly to the projection P onto + +…1 nX X along ∩ ∩…1ker( ) ker( )nP P as →∞N .
For practical applications, it is important to know how rapidly the sequence − −( )NI I A
converges to P . Our next result shows that the rate of convergence can be estimated from
above by αNC , where α ∈[0,1) . To formulate the result, we need the following notation: for
two vectors ∈, nu v , we write u v� if u v� coordinatewise.
Theorem 2. The following statements on the rate of convergence of − −( )NI I A to P are true.
1. Suppose a vector = …1( , , )t
nw w w with positive coordinates and a number α ∈[0,1) satisfy
αEw w� . Then
α− − − + +
−α
… …1 1 1( ) ( )max{(1/ ) , , (1/ ) }
1
N
N
n n nI I A P w w w P w P�
for each 1N � .
2. Suppose a vector = …1( , , )t
nw w w with positive coordinates and a number α ∈[0,1) satisfy
αtE w w� . Then
α− − − + +
−α
… …1 1 1( ) ( )max{(1/ ), , (1/ )}
1
N
N
n n nI I A P w P w P w w�
for each 1N � .
Remark 1. Since E is a nonnegative matrix, the existence of a vector ∈ nw with positive
coordinates and a number α ∈[0,1) such that αEw w� is equivalent to <( ) 1r E . More precisely,
if such w and α exist, then α <( ) 1r E � (see [10, Corollary 8.1.29]). Conversely, suppose
that <( ) 1r E . If E is irreducible, then one can take α to be ( )r E , and w is a Perron—Frobenius
vector of E . If E is not irreducible, then we consider the matrix = + δ′ ( )ijE e for sufficiently
small δ > 0, and take α to be ′( )r E and w a Perron—Frobenius vector of ′E .
Similarly, the existence of a vector w with positive coordinates and a number α ∈[0,1)
such that αtE w w� is equivalent to <( ) 1r E .
13ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 1
A sufficient condition for the sum of complemented subspaces to be complemented
The assumption <( ) 1r E is a sharp sufficient condition for + +…1 nX X to be comple-
mented in X .
Theorem 3. Let = ( )ijE e be an ×n n matrix with = 0iie for = …1, ,i n and 0ije � for every
pair ≠i j , ∈ …, {1, , }i j n . If =( ) 1r E , then there exist a Banach space X , complemented subspaces
…1, , nX X of X , and projections …1, , nP P onto …1, , nX X , respectively, such that
1. =i ijP x e x , ∈ jx X , for each pair ≠i j , ∈ …, {1, , }i j n ;
2. …1, , nX X are linearly independent;
3. + +…1 nX X is closed and not complemented in X .
Remark 2. In the case where >( ) 1r E the theorem can be applied to the matrix (1/ ( ))r E E .
7. Sums of marginal subspaces. As an application of Theorem 1, we provide a sufficient
condition for the complementability of the sum of marginal subspaces in pL .
Let Ω μ( , , )F be a probability space. Denote by K a base field of scalars, i.e., or . For an
F -measurable function (random variable) ξ Ω→: K we denote by ξE the expectation of ξ (if
it exists). Two random variables ξ and η are said to be equivalent if ξ ω = η ω( ) ( ) for μ -almost
all ω . For ∈ ∞ ∪ ∞[1, ) { }p denote by = Ω μ( ) ( , , )p pL LF F the set of equivalence classes of ran-
dom variables ξ Ω→: K such that ξ < ∞| |pE if ∈ ∞[1, )p , and ξ is μ -essentially bounded if
= ∞p . For ξ ∈ ( )pL F , set ξ = ξ 1/( | | )p p
p E if ∈ ∞[1, )p and ∞ξ = ξesssup | | if = ∞p . Then
( )pL F is a Banach space. For every sub-σ -algebra A of F , we define the mar ginal subspace cor-
responding to A, ( )pL A , as follows. ( )pL A consists of elements (equivalence classes) of ( )pL F
which contain at least one A-measurable random variable. Denote by 0 ( )pL A the subspace of all
ξ ∈ ( )pL A with ξ = 0E .
We study the following problem. Let …1, , nF F be sub-σ -algebras of F . Question: when is the
sum of the corresponding marginal subspaces, + +…1( ) ( )p p
nL LF F , complemented in ( )pL F ?
One can check that + +…1( ) ( )p p
nL LF F is complemented in ( )pL F if and only if + +…10 ( )pL F
+ 0 ( )p
nL F is. Since each complemented subspace is closed, the question on the complementability
of the sum of marginal subspaces is closely related to the question on the closedness of the sum
(for = 2p , these questions coincide). One can check that + +…1( ) ( )p p
nL LF F is closed in ( )pL F
if and only if + +…10 0( ) ( )p p
nL LF F is.
The question on the closedness of the sum of marginal subspaces arises, for example, in ad-
ditive modeling (see, e.g., [11, Subsection 8.1]) and the theory of ridge functions (see, e.g., [3,
Chapter 7]) (note that every subspace of ridge functions ( ; )pL a K can be considered as marginal).
The question on the closedness is not trivial; examples where +1 2( ) ( )p pL LF F is not closed
in ( )pL F can be found in [12, Proposition 4.4(a)] (for ∈ ∞[1, )p ), [11, Subsection 8.3] (for = 2p ),
[3, Section 7.2] (for ∈ ∞ ∪ ∞[1, ) { }p ).
Sufficient conditions for the sum of marginal subspaces to be closed can be found in [5,
p.1332, Proof of Lemma 1], [11, Section 8] and [3, Chapter 7]. Our result (Theorem 4) is mo-
tivated by the result of [5] and contains it as a special case.
To formulate our result on the complementability of the sum of marginal subspaces, we need
an auxiliary notion. Let Ω μ( , , )F be a probability space. For two sub-σ -algebras ,A B of F
de fine the following measure of their dependence:
⎧ ⎫μ ∩ψ = ∈ ∈ μ > μ >′ ⎨ ⎬μ μ⎩ ⎭
( )
( , ) inf | , , ( ) 0, ( ) 0 .
( ) ( )
A B
A B A B
A B
A B A B
14 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2019. № 1
I.S. Feshchenko
This measure of dependence is well known (see, e.g., [13]). It is easily seen that ψ′0 ( , ) 1A B� �
and ψ =′( , ) 1A B if and only if A and B are independent.
Let us formulate our result. Let …1, , nF F be sub-σ -algebras of F . Define the ×n n mat-
rix E = (eij)by
=⎧⎪= ⎨ −ψ ≠′⎪⎩
0, ;
1 ( , ), .ij
i j
if i j
e
if i jF F
Theorem 4. If <( ) 1r E , then the marginal subspaces 10 0( ), , ( )p p
nL L…F F are linearly in-
dependent and their sum is complemented in ( )pL F (for arbitrary ∈ ∞ ∪ ∞[1, ) { }p ).
This research was supported by the project 2017-3M from the Department of Targeted Training
of Taras Shevchenko National University of Kyiv at the NAS of Ukraine.
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2. Moslehian, M. S. (2006). A survey of the complemented subspace problem. Trends in Mathematics, In for-
mation Center for Mathematical Sciences, 9, No. 1, pp. 91-98.
3. Pinkus, A. (2015). Ridge functions. Cambridge Tracts in Mathematics. Cambridge: Cambridge University
Press.
4. Kim, H. O., Kim, R. Y. & Lim, J. K. (2006). Characterization of the closedness of the sum of two shift-inva-
riant subspaces. J. Math. Anal. Appl., 320, Iss. 1, pp. 381-395.
5. Bickel, P. J., Ritov, Y. & Wellner, J. A. (1991). Efficient estimation of linear functionals of a probability mea-
sure P with known marginal distributions. Ann. Statist., 19, No. 3, pp. 1316-1346.
6. LaVergne, A. (1979). Remark on sums of complemented subspaces. Colloq. Math., 41, No. 1, pp. 103-104.
7. Svensson, L. (1987). Sums of complemented subspaces in locally convex spaces. Ark. Mat., 25, Iss. 1, pp. 147-153.
8. Gonzalez, M. (1994). On essentially incomparable Banach spaces. Math. Z., 215, pp. 621-629.
9. Önal, S. & Yurdakul, M. (2013). On sums of complemented subspaces. In Mathematical Forum. Vol. 7. Studies
on mathematical analysis (pp. 148-152). Vladikavkaz.
10. Horn, R. A. & Johnson, C. H. (2013). Matrix analysis. 2 ed. New York: Cambridge University Press.
11. Buja, A. (1996). What criterion for a power algorithm? In Rieder, H. (Ed.). Robust Statistics, Data Analysis,
and Computer Intensive Methods. Lecture Notes in Statistics, Vol. 109 (pp. 49-61). New York: Springer.
12. Rüschendorf, L. & Thomsen, W. (1998). Closedness of sum spaces and the generalized Schrödinger problem.
Theory Probab. Appl., 42, No. 3, pp. 483-494.
13. Bradley, R. C. (2005). Basic properties of strong mixing conditions. A survey and some open questions.
Probab. Surveys, 2, pp. 107-144.
Received 02.10.2018
15ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2019. № 1
A sufficient condition for the sum of complemented subspaces to be complemented
І.С. Фещенко
Київський національний університет ім. Тараса Шевченка
E-mail: ivanmath007@gmail.com
ДОСТАТНЯ УМОВА ДЛЯ ТОГО, ЩОБ СУМА
ДОПОВНЮВАЛЬНИХ ПІДПРОСТОРІВ БУЛА ДОПОВНЮВАЛЬНОЮ
Наведено достатню умову для того, щоб сума скінченного числа доповнювальних підпросторів банахового
простору була доповнювальною. За цієї умови отримано формулу для проектора на цю суму підпросторів.
Ця умова є точною (в певному сенсі). Як застосування наведено достатню умову для доповнювальності
суми маргінальних підпросторів у просторі Lp.
Ключові слова: сума підпросторів, доповнювальний підпростір, замкнений підпростір, маргінальний під-
простір, проектор.
И.С. Фещенко
Киевский национальный университет им. Тараса Шевченко
E-mail: ivanmath007@gmail.com
ДОСТАТОЧНОЕ УСЛОВИЕ ДЛЯ ТОГО, ЧТОБЫ СУММА
ДОПОЛНЯЕМЫХ ПОДПРОСТРАНСТВ БЫЛА ДОПОЛНЯЕМА
Приведено достаточное условие для того, чтобы сумма конечного числа дополняемых подпространств
банахова пространства была дополняема. При этом условии получена формула для проектора на эту сумму
подпространств. Это условие является точным (в определенном смысле). В качестве применения при-
ведено достаточное условие для дополняемости суммы маргинальных подпространств в пространстве Lp.
Ключевые слова: сумма подпространств, дополняемое подпространство, замкнутое подпространство,
мар гинальное подпространство, проектор.
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