Peter Orno
Encyclopedia
According to Mathematical Reviews
, this paper proves the following theorem, which has come to be known as "Orno's theorem": Suppose that E and F are Banach lattices, where F is an infinite-dimensional vector space that contains no Riesz subspace
that is uniformly isomorphic
to the sequence space
equipped with the supremum norm. If each linear operator in the uniform closure of the finite-rank operators from E to F has a Riesz decomposition as the difference of two positive operators, then E can be renormed so that it is an L-space (in the sense of Kakutani and Birkhoff).
According to Mathematical Reviews
, Orno proved the following theorem: The series ∑fk unconditionally converges in the Lebesgue space of absolutely integrable functions L1[0,1] if and only if, for each k and every t, we have fk(t)=akg(t)wk(t), for some sequence (ak)∈l2, some function g∈L2[0,1], and for some orthonormal sequence (wk) in L2[0,2] . Another result is the "elegant proof" by Orno of a theorem of Bennet, Maurey and Nahoum.
In this paper, Orno solves an eight-year old problem posed by Ivan Singer, according to Mathematical Reviews .
Still circulating as an "underground classic", this paper has been cited eleven times according to Google Scholar. In it, Orno solved a problem posed by Jonathan M. Borwein. Orno characterized sequentially reflexive
Banach space
s in terms of their lacking bad subspaces: Orno's theorem states that a Banach space X is sequentially reflexive if and only if the space of absolutely summable sequences ℓ1 is not isomorphic to a subspace of X.
, which is published by the Mathematical Association of America (MAA). In 2006, Orno solved a problem in the American Mathematical Monthly
, another peer-reviewed journal of the MAA.
Mathematical Reviews
Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses of many articles in mathematics, statistics and theoretical computer science.- Reviews :...
, this paper proves the following theorem, which has come to be known as "Orno's theorem": Suppose that E and F are Banach lattices, where F is an infinite-dimensional vector space that contains no Riesz subspace
Riesz space
In mathematics a Riesz space, lattice-ordered vector space or vector lattice is an ordered vector space where the order structure is a lattice....
that is uniformly isomorphic
Continuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....
to the sequence space
Sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers...
equipped with the supremum norm. If each linear operator in the uniform closure of the finite-rank operators from E to F has a Riesz decomposition as the difference of two positive operators, then E can be renormed so that it is an L-space (in the sense of Kakutani and Birkhoff).
According to Mathematical Reviews
Mathematical Reviews
Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses of many articles in mathematics, statistics and theoretical computer science.- Reviews :...
, Orno proved the following theorem: The series ∑fk unconditionally converges in the Lebesgue space of absolutely integrable functions L1[0,1] if and only if, for each k and every t, we have fk(t)=akg(t)wk(t), for some sequence (ak)∈l2, some function g∈L2[0,1], and for some orthonormal sequence (wk) in L2[0,2] . Another result is the "elegant proof" by Orno of a theorem of Bennet, Maurey and Nahoum.
In this paper, Orno solves an eight-year old problem posed by Ivan Singer, according to Mathematical Reviews .
Still circulating as an "underground classic", this paper has been cited eleven times according to Google Scholar. In it, Orno solved a problem posed by Jonathan M. Borwein. Orno characterized sequentially reflexive
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s in terms of their lacking bad subspaces: Orno's theorem states that a Banach space X is sequentially reflexive if and only if the space of absolutely summable sequences ℓ1 is not isomorphic to a subspace of X.
Problem-solving
By 2011, Peter Orno had eighteen publications in Mathematics MagazineMathematics Magazine
Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy...
, which is published by the Mathematical Association of America (MAA). In 2006, Orno solved a problem in the American Mathematical Monthly
American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....
, another peer-reviewed journal of the MAA.