76 research outputs found
Complex interpolation of compact operators mapping into lattice couples
Suppose that (A_0,A_1) and (B_0,B_1) are Banach couples, and that T is a
linear operator which maps A_0 compactly into B_0 and A_1 boundedly (or even
compactly) into B_1. Does this imply that T maps [A_0,A_1]_s to [B_0,B_1]_s
compactly for 0<s<1 ? (Here, as usual, [A_0,A_1]_s denotes the complex
interpolation space of Alberto Calderon.) This question has been open for 44
years. Affirmative answers are known for it in many special cases. We answer it
affirmatively in the case where (A_0,A_1) is arbitrary and (B_0,B_1) is a
couple of Banach lattices having absolutely continuous norms or the Fatou
property. Our result has some overlap with a recent result by Evgeniy
Pustylnik.Comment: 14 pages. (Page 13 contains routine and standard material which you
quite probably will not need or want to print.) The only changes in this new
version are the correction of small typographic errors on the first page and
the updating of a reference
Lecture notes on duality and interpolation spaces
Known or essentially known results about duals of interpolation spaces are
presented, taking a point of view sometimes slightly different from the usual
one. Particular emphasis is placed on Alberto Calderon's theorem describing the
duals of his complex interpolation spaces [A_0,A_1]_\theta. The pace is slow,
since these notes are intended for graduate students who have just begun to
study interpolation spaces. This second version corrects some small misprints.
It also draws attention to a convenient norming subspace of the dual of a
complex interpolation space, and to the slight difference between the spaces
\mathcal{G}(X_0,X_1) introduced by Calderon and by Stafney.Comment: 36 page
Lecture notes about a simpler approach to Riemann integration
It is of course well known that the usual definitions of Riemann integration
and Riemann integrals are equivalent to simpler definitions which can be
expressed in terms of just one sequence of partitions, using dyadic intervals
or dyadic squares or dyadic cubes for univariate, double, or triple integrals
respectively. These lecture notes, intended mainly for undergraduates, present
and prove some basic standard results of Riemann integration in detail, taking
advantage of this simpler definition. The last section of the notes provides a
proof of the equivalence of this definition with the classical one. But the
implicit suggestion is that the classical definition need be the concern of
specialists only, and that regular students can probably do just about
everything that they need to do with Riemann integration by working only with
the simpler dyadic definition. This is a preliminary version of these notes
which will surely be updated later. It is very difficult to believe that there
do not exist any other documents which give a systematic treatment of Riemann
integration using this approach. The author would be very grateful for any
information about any such documents.Comment: 23 page
Some alternative definitions for the "plus-minus" interpolation spaces of Jaak Peetre
The Peetre "plus-minus" interpolation spaces are defined variously via conditions about
the unconditional convergence of certain Banach space valued series whose terms
have coefficients which are powers of 2 or, alternatively, powers of . It
may seem intuitively obvious that using powers of 2, or of , or powers of
some other constant number greater than 1 in such definitions should produce
the same space to within equivalence of norms. To allay any doubts, we here
offer an explicit proof of this fact, via a "continuous" definition of the same
spaces where integrals replace the above mentioned series. This apparently new
definition, which is also in some sense a "limiting case" of the above
mentioned "discrete" definitions, may be relevant in the study of the
connection between the Peetre "plus-minus" interpolation spaces and Calderon
complex interpolation spaces when both the spaces of the underlying couple are
are Banach lattices on the same measure space. Related results can probably be
obtained for the Gustavsson-Peetre variant of the "plus-minus" spaces.Comment: 17 page
A quick description for engineering students of distributions (generalized functions) and their Fourier transforms
These brief lecture notes are intended mainly for undergraduate students in
engineering or physics or mathematics who have met or will soon be meeting the
Dirac delta function and some other objects related to it. These students might
have already felt - or might in the near future feel - not entirely comfortable
with the usual intuitive explanations about how to "integrate" or
"differentiate" or take the "Fourier transform" of these objects.
These notes will reveal to these students that there is a precise and
rigorous way, and this also means a more useful and reliable way, to define
these objects and the operations performed upon them. This can be done without
any prior knowledge of functional analysis or of Lebesgue integration. Readers
of these notes are assumed to only have studied basic courses in linear
algebra, and calculus of functions of one and two variables, and an
introductory course about the Fourier transform of functions of one variable.
Most of the results and proofs presented here are in the setting of the space
of tempered distributions introduced by Laurent Schwartz. But there are also
some very brief mentions of other approaches to distributions or generalized
functions.Comment: 22 pages This second version has an additional reference and some
slight modifications in one of its appendice
K-divisibility constants for some special couples
We prove new estimates of the -divisibility constants for some special
Banach couples. In particular, we prove that the -divisibility constant for
a couple of the form where and are non-trivial Hilbert
spaces equals . We also prove estimates for the -divisibility
constant of the two-dimensional version of the couple , proving
in particular that this couple is not exactly -divisible. There are also
several auxiliary results, including some estimates for relative Calder\'on
constants for finite dimensional couples
Calderon couples of p-convexified Banach lattices
This paper updates the previous version in the following ways: 1. The main
result is extended from the case of sequence spaces to the case of Dedekind
complete Banach lattices. 2. A new appendix is added to mention some sufficient
and necessary conditions (which are probably already known) for lattices to be
Dedekind complete.
We deal with the question of whether or not the p-convexified couple
(X_0^{(p)},X_1^{(p)}) is a Calderon couple under the assumption that (X_0,X_1)
is a Calderon couple of Banach lattices on some measure space. We find that the
answer is affirmative, not only in the case of sequence spaces treated in the
previous version, but also in the case where X_0 and X_1 are Dedekind complete
Banach lattices (and provided the same additional "positivity" assumption is
imposed regarding (X_0,X_1)). We also prove a quantitative version of the
result with appropriate norm estimates.Comment: 11 page
Estimates for covering numbers in Schauder's theorem about adjoints of compact operators
Let T:X --> Y be a bounded linear map between Banach spaces X and Y. Let S:Y'
--> X' be its adjoint. Let B(X) and B(Y') be the closed unit balls of X and Y'
respectively. We obtain apparently new estimates for the covering numbers of
the set S(B(Y')). These are expressed in terms of the covering numbers of
T(B(X)), or, more generally, in terms of the covering numbers of a
"significant" subset of T(B(X)). The latter more general estimates are best
possible. These estimates follow from our new quantitative version of an
abstract compactness result which generalizes classical theorems of
Arzela-Ascoli and of Schauder. Analogous estimates also hold for the covering
numbers of T(B(X)), in terms of the covering numbers of S(B(Y')) or in terms of
a suitable "significant" subset of S(B(Y')).Comment: 13 page
Interpolation of compact operators by the methods of Calder\'on and Gustavsson-Peetre
Let and be Banach couples and suppose is a linear operator such that is compact. We consider the
question whether the operator is
compact and show a positive answer under a variety of conditions. For example
it suffices that be a UMD-space or that is reflexive and there is a
Banach space so that for some $0<\alpha<1.
An alternative characterization of normed interpolation spaces between and
Given a constant , we study the following property of a
normed sequence space :
=====================
If is an element of and if
is an element of such that
and if the nonincreasing rearrangements of these two
sequences satisfy
for all , then
and for some
constant which depends only on .
=====================
We show that this property is very close to characterizing the normed
interpolation spaces between and . More specificially, we
first show that every space which is a normed interpolation space with respect
to the couple for some has the
above mentioned property. Then we show, conversely, that if has the above
mentioned property, and also has the Fatou property, and is contained in
, then it is a normed interpolation space with respect to the couple
. These results are our response to a
conjecture of Galina Levitina, Fedor Sukochev and Dmitriy Zanin in
arXiv:1703.04254v1 [math.OA].Comment: 33 page
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