arXiv subjects

Eliahu Levy

Publications and source records attributed to Eliahu Levy.

27 records · Page 2Linked to original sources

arXiv2008

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')).

math.FA↗

arXiv2008

Weakly Compact "Matrices", Fubini-Like Property and Extension of Densely Defined Semigroups of Operators

Taking matrix as a synonym for a numerical function on the Cartesian product of two (in general, infinite) sets, a simple purely algebraic "reciprocity property" says that the set of rows spans a finite-dim space iff the set of columns does so. Similar topological reciprocity properties serve to define strongly compact and weakly compact matrices, featured in the well-known basic facts about almost periodic functions and about compact operators. Some properties, especially for the weak compact case, are investigated, such as the connection with the matrix having a Fubini-like property for general means. These are applied to prove possibility of extension to the entire semigroup of bounded densely defined semigroups of operators in a Banach space with weak continuity properties.

math.FA↗

arXiv2008

Continuous extension of a densely parameterized semigroup

Let S be a dense sub-semigroup of the positive real numbers, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over S can be extended to a weakly continuous semigroup over the positive real numbers. We obtain similar results for non-linear, non-expansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over the positive real numbers.

math.FA↗

arXiv2006

A note about entropy

A mathematical interpretation of the usual definition of entropy (for a discrete probability distribution or a trace 1 positive operator) is given. This formulation makes some properties of entropy immediate.

math.GM↗

arXiv2006

Kac's formula, vertices of chains and equidecomposability of functions and enhanced functions (PHD thesis)

This PHD thesis hinges on the terms mentioned in the title. It introduces a formalism which allows one to find equalities generalizing the formula of Mark Kac which deals with a measure-preserving transformation. The formalism is meaningful for general measure-preserving actions of groups ("discrete", "continuous", "multidimensional"). Some aspects of the notions introduced are investigated.

math.DS↗

arXiv2006

Topological interpretation of function spaces stable under a general operation (tentative version)

Function (linear) spaces on which an arbitrary function operates (i.e. the space is stable w.r.t. the pointwise unary operation defined by the function) were investigated, for continuous real or complex operations, by deLeeuw-Katznelson, Sternfeld and Weit. They showed that, with suitable assumptions, a real or complex function space on which a non-affine continuous function operates is like an algebra. In this note, however, the point of view is somewhat different: the scalars are any field K, and from the outset no topology or continuity are assumed. Rather, a function space operated by a non-additively-affine (multivariate) function induces a topology on the set on which it is defined and the family of functions which operate on it induces topologies on K^n. These are used to derive a density property and to investigate "homomorphisms" of such spaces w.r.t. the "operations".

math.GN↗

arXiv2006

Why do partitions occur in Faa di Bruno's chain rule for higher derivatives?

It is well-known that the coefficients in Faa di Bruno's chain rule for higher derivatives can be expressed via numeration of partitions. It turns out that this has a natural form as a formula for the vector case. To this formula two proofs are presented, both "explaining" its form involving partitions: one as a purely algebraic fact, and one "from first principles" for the case of Frechet derivatives of mappings between Banach spaces.

math.GM↗

arXiv2006

Differentiability of scalar functions applied to Hermitian operators - a Fourier transform approach

Let g be a (say, sufficiently differentiable) function on the reals. One knows how to apply g to Hermitian elements A of a C* algebra. Yet the question of differentiability of the mapping A to g(A) is not trivial, since in general "A and dA do not commute". However, since this mapping depends linearly on g, one can, via Fourier transform, reduce the case of general g to the case of the exponential function. For the latter one has an explicit formula for the n-th derivative (more complicated than in the scalar case - still "A and dA do not commute"). In this way one bounds the norm of the n-th derivative of (A to g(A)) on the r-ball by a Sobolev norm involving the (n+1)-th derivative of g on [-r,r].

math.OA↗

arXiv2006

How to axiomatize school geometry

This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system); to be appropriate to the way geometry is done in science and engineering - not to conceal its algebraic nature; to respond to the desire that one would accept intuitively/empirically that the axioms are valid in our physical everyday world (or rather in the idealization that geometry is) - that seemingly disfavoring taking the theorem of Pythagoras as an axiom; to have accessible the rigor and standards of "pure" mathematics. The style in this note is that of usual mathematical writings - for an unsophisticated audience the style of the presentation must surely be quite different.

math.HO↗