On subsequences of the Haar system inL p [0, 1], (1&lt;p&lt;∞)

Gamlen, John L. B.; Gaudet, Rachelle

doi:10.1007/bf02757079

Cited by 37 publications

(29 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Clearly, there exists an infinite subset N 1 ⊆ N and a number μ ε such that Let Z = span{a n,i : n ∈ N 1 , i ∈ σ(n)}. Using a result of Gamlen and Gaudet (see [7]), we have that Z L p and clearly Z is complemented in L p . Now note that…”

Section: 2mentioning

confidence: 99%

Commutators on 𝐿_{𝑝}, 1≤𝑝<∞

Dosev

Johnson

Schechtman

2012

J. Amer. Math. Soc.

View full text Add to dashboard Cite

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 102 DETELIN DOSEV, WILLIAM B. JOHNSON, AND GIDEON SCHECHTMAN in [4] a complete classification of the commutators on 1 , which, as one may expect, is the same as the classification of the commutators on 2 . A common feature of all the spaces X = p , 1 ≤ p < ∞ and X = c 0 is that the ideal of compact operators K(X ) on X is the largest nontrivial ideal in L(X ). The situation for X = ∞ is different. Recall that an operator T : X → Y is strictly singular provided the restriction of T to any infinite-dimensional subspace of X is not an isomorphism. On p , 1 ≤ p < ∞, and on c o , every strictly singular operator is compact, but on L( ∞ ), the ideal of strictly singular operators contains noncompact operators (and, incidentally, agrees with the ideal of weakly compact operators). In L( ∞ ), the ideal of strictly singular operators is the largest ideal, and it was proved in [5] that all operators on ∞ that are not commutators have the form λI + S, where λ = 0 and S is strictly singular.The classification of the commutators on p , 1 ≤ p ≤ ∞, and on c 0 , as well as partial results on other spaces, suggest the following:Licensed to Univ of Georgia. Prepared on Mon Jul 6 22:15:33 EDT 2015 for download from IP 128.192.Proof. As we already mentioned, we only need to consider the case 1 ≤ p < 2, and the case 2 < p < ∞ will follow by a duality argument.If T is a commutator, from the remarks we made in the introduction it follows that T − λI cannot be in M for any λ = 0. For proving the other direction we have to consider two cases:Case I. If T ∈ M (λ = 0), we first apply Theorem 1.3 to obtain a complemented subspace X ⊂ L p such that T |X is a compact operator and then apply [4, Corollary 12], which gives us the desired result.Case II. If T − λI / ∈ M for any λ ∈ C, we are in position to apply Theorem 1.2, which combined with Theorem 1.1 implies that T is a commutator.The rest of this paper is devoted to the proofs of Theorems 1.2 and 1.3. We consider the case L 1 separately since some of the ideas and methods used in this case are quite different from those used for the case L p , 1 < p < ∞. Notation and basic resultsThroughout this manuscript, if X is a Banach space and X ⊆ X is complemented, by P X we denote a projection from X onto X. For any two subspaces (possibly not closed) X and Y of a Banach space Z letA well-known consequence of the open mapping theorem is that for any two closed subspaces X and Y of Z, if X ∩ Y = {0}, then X + Y is a closed subspace of Z if and only if d(X, Y ) > 0. Note also that 2d(X, Y ) ≥ d(Y, X) ≥ 1/2d(X, Y ); thus d(X, Y ) and d(Y, X) are equivalent up to a constant factor of 2. The following proposition was proved in [5] and will allow us later to consider only isomorphisms instead of arbitrary operators on L p . Proposition 2.1 ([5, Proposition 2.1]). Let X be a Banach space and T ∈ L(X ) be such that there exists a subspace Y ⊂ X for which T is an isomorphism on Y and d(Y, T Y ) > 0. ...

show abstract

Section: 2mentioning

confidence: 99%

Commutators on 𝐿_{𝑝}, 1≤𝑝<∞

Dosev

Johnson

Schechtman

2012

J. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…It is a well-known theorem of Gamlen and Gaudet [3] that the closed linear span of any subsequence of the Haar system is isomorphic to L p [0, 1] or p . We shall construct a subsequence of the levels of the Haar system such that the norm of the linear span of a collection of Haar functions from these levels behaves like the unit vector basis of p when the number of functions from each Haar level is proportional to the total number of Haar functions from the previous level in the subsequence.…”

Section: Now We Consider Subsequences Of the Haar System Inmentioning

confidence: 99%

“…We need the following 'uniform' version of a theorem of Gamlen and Gaudet [3]. We have not been able to find an explicit statement of this result in the literature, but the proof is essentially contained in [7].…”

Section: Non-equivalent Greedy and Almost Greedy Bases In Pmentioning

confidence: 99%

Non‐equivalent greedy and almost greedy bases in l_p

Dilworth

Hoffmann

Kutzarova

2006

Journal of Function Spaces

View full text Add to dashboard Cite

Abstract. For 1 < p < ∞ and p = 2 we construct a family of mutually non-equivalent greedy bases in p having the cardinality of the continuum. In fact, no basis from this family is equivalent to a rearranged subsequence of any other basis thereof. We are able to extend this statement to the spaces Lp and H1 . Moreover, the technique used in the proof adapts to the setting of almost greedy bases where similar results are obtained.

show abstract

“…The last statement of the proposition follows from Gamlen and Gaudet's theorem [3] and from the equivalence of norms given by Proposition 4.4.…”

Section: Thus We Obtainmentioning

confidence: 79%

Characterization of 1-greedy bases

Albiac

Wojtaszczyk

2006

Journal of Approximation Theory

View full text Add to dashboard Cite

A basis for a Banach space X is greedy if and only if the greedy algorithm provides, up to a constant C depending only on X, the best m-term approximation for each element of the space. It is known that the Haar (or good wavelet) basis is a greedy basis in L p (0, 1) for 1 < p < ∞ [V.N. Temlyakov, The best m-term approximation and greedy algorithms, Adv. in Comp. Math. 8 (1998) 249-265]. In this particular example, unfortunately, the constant of greediness C = C(p) is strictly bigger than 1 unless p = 2. Our goal is to investigate 1-greedy bases, i.e., bases for which the greedy algorithm provides the best m-term approximation. We find a characterization of 1-greediness, study how 1-greedy bases relate to symmetric bases, and show that 1-greediness does not imply 1-symmetry, answering thus two questions raised in [P.

show abstract

On subsequences of the Haar system inL p [0, 1], (1<p<∞)

Cited by 37 publications

References 9 publications

Commutators on 𝐿_{𝑝}, 1≤𝑝<∞

Commutators on 𝐿_{𝑝}, 1≤𝑝<∞

Non‐equivalent greedy and almost greedy bases in l_p

Characterization of 1-greedy bases

Contact Info

Product

Resources

About

On subsequences of the Haar system inL p [0, 1], (1<p<∞)

Cited by 37 publications

References 9 publications

Commutators on 𝐿_{𝑝}, 1≤𝑝<∞

Commutators on 𝐿_{𝑝}, 1≤𝑝<∞

Non‐equivalent greedy and almost greedy bases in lp

Characterization of 1-greedy bases

Contact Info

Product

Resources

About

Non‐equivalent greedy and almost greedy bases in l_p