21 research outputs found
Primarity of direct sums of Orlicz spaces and Marcinkiewicz spaces
Let be either an Orlicz sequence space or a Marcinkiewicz
sequence space. We take advantage of the recent advances in the theory of
factorization of the identity carried on in [R. Lechner, Subsymmetric weak*
Schauder bases and factorization of the identity, arXiv:1804.01372 [math.FA]]
to provide conditions on that ensure that, for any , the infinite direct sum of in the sense of
is a primary Banach space, enlarging this way the list of Banach spaces that
are known to be primary
Characterization of 1-quasi-greedy bases
We show that a (semi-normalized) basis in a Banach space is quasi-greedy with
quasi-greedy constant equal to 1 if and only if it is unconditional with
suppression-unconditional constant equal to 1
Optimal average approximations for functions mapping in quasi-Banach spaces
In 1994, M. M. Popov [On integrability in F-spaces, Studia Math. no 3,
205-220] showed that the fundamental theorem of calculus fails, in general, for
functions mapping from a compact interval of the real line into the lp-spaces
for 0<p<1, and the question arose whether such a significant result might hold
in some non-Banach spaces. In this article we completely settle the problem by
proving that the fundamental theorem of calculus breaks down in the context of
any non-locally convex quasi-Banach space. Our approach introduces the tool of
Riemann-integral averages of continuous functions, and uses it to bring out to
light the differences in behavior of their approximates in the lack of local
convexity. As a by-product of our work we solve a problem raised in [F. Albiac
and J.L. Ansorena, Lipschitz maps and primitives for continuous functions in
quasi-Banach space, Nonlinear Anal. 75 (2012), no. 16, 6108-6119] on the
different types of spaces of differentiable functions with values on a
quasi-Banach space.Comment: 14 page
Lipschitz free -spaces for
This paper initiates the study of the structure of a new class of -Banach
spaces, , namely the Lipschitz free -spaces (alternatively called
Arens-Eells -spaces) over -metric spaces.
We systematically develop the theory and show that some results hold as in the
case of , while some new interesting phenomena appear in the case
which have no analogue in the classical setting. For the former, we, e.g., show
that the Lipschitz free -space over a separable ultrametric space is
isomorphic to for all , or that isomorphically
embeds into for any -metric space
. On the other hand, solving a problem by the first author and N.
Kalton, there are metric spaces such that the
natural embedding from to
is not an isometry
Embeddability of and bases in Lipschitz free -spaces for
Our goal in this paper is to continue the study initiated by the authors in
[Lipschitz free -spaces for ; arXiv:1811.01265 [math.FA]] of the
geometry of the Lipschitz free -spaces over quasimetric spaces for
, denoted . Here we develop new techniques
to show that, by analogy with the case , the space embeds
isomorphically in for . Going further we
see that despite the fact that, unlike the case , this embedding need not
be complemented in general, complementability of in a Lipschitz free
-space can still be attained by imposing certain natural restrictions to
. As a by-product of our discussion on basis in , we obtain the first-known examples of -Banach spaces for
that possess a basis but fail to have an unconditional basis
Optimality of the rearrangement inequality with applications to Lorentz-type sequence spaces
We characterize the sequences of non-negative numbers
for which when runs over
all non-increasing sequences of non-negative numbers. As a by-product of our
work we settle a problem raised in [F. Albiac, Jose L. Ansorena and B. Wallis;
arXiv:1703.07772[math.FA]] and prove that Garling sequences spaces have no
symmetric basis
Lipschitz free spaces isomorphic to their infinite sums and geometric applications
We find general conditions under which Lipschitz-free spaces over metric
spaces are isomorphic to their infinite direct -sum and exhibit several
applications. As examples of such applications we have that Lipschitz-free
spaces over balls and spheres of the same finite dimensions are isomorphic,
that the Lipschitz-free space over is isomorphic to its
-sum, or that the Lipschitz-free space over any snowflake of a doubling
metric space is isomorphic to . Moreover, following new ideas from [E.
Bru\`e, S. Di Marino and F. Stra, Linear Lipschitz and extension
operators through random projection, arXiv:1801.07533] we provide an elementary
self-contained proof that Lipschitz-free spaces over doubling metric spaces are
complemented in Lipschitz-free spaces over their superspaces and they have BAP.
Everything, including the results about doubling metric spaces, is explored in
the more comprehensive setting of -Banach spaces, which allows us to
appreciate the similarities and differences of the theory between the cases
and
On the permutative equivalence of squares of unconditional bases
We prove that if the squares of two unconditional bases are equivalent up to
a permutation, then the bases themselves are permutatively equivalent. This
settles a twenty year-old question raised by Casazza and Kalton in [Uniqueness
of unconditional bases in Banach spaces, Israel J. Math. 103 (1998), 141--175].
Solving this problem provides a new paradigm to study the uniqueness of
unconditional basis in the general framework of quasi-Banach spaces. Multiple
examples are given to illustrate how to put in practice this theoretical
scheme. Among the main applications of this principle we obtain the uniqueness
of unconditional basis up to permutation of finite sums of quasi-Banach spaces
with this property
Uniqueness of unconditional basis of
We provide a new extension of Pitt's theorem for compact operators between
quasi-Banach lattices, which permits to describe unconditional bases of finite
direct sums of Banach spaces as
direct sums of unconditional bases of its summands. The general splitting
principle we obtain yields, in particular, that if each has a
unique unconditional basis (up to equivalence and permutation), then
has a unique unconditional
basis too. Among the novel applications of our techniques to the structure of
Banach and quasi-Banach spaces we have that the space has a unique unconditional basis
Greedy approximation for biorthogonal systems in quasi-Banach spaces
The general problem addressed in this work is the development of a systematic
study of the thresholding greedy algorithm for general biorthogonal systems
(also known as Markushevich bases) in quasi-Banach spaces from a
functional-analytic point of view. We revisit the concepts of greedy,
quasi-greedy, and almost greedy bases in this comprehensive framework and
provide the (nontrivial) extensions of the corresponding characterizations of
those types of bases. As a by-product of our work, new properties arise, and
the relations amongst them are carefully discussed