Peter Orno

Peter Ørno
Born 1974
Columbus, Ohio
Residence Columbus, Ohio
Citizenship United States of America
Nationality United States of America
Fields Functional analysis
Institutions Ohio State University
Known for Orno's theorem on regular operators on Banach lattices,
Summability and Approximation theory in Banach spaces
Influenced Aleksander Pełczyński
NicoleTomczak-Jaegermann

Peter Orno (alternatively, Peter Ørno, P. Ørno, and P. Orno) is the pseudonym of a fictitious mathematician, who appears as the author of short papers by one or more mathematicians. According to Robert R. Phelps (2002), the name "P. Orno" was inspired by "porno", an abbreviation for "pornography".[1][2] Orno is renowned for his elegant papers in functional analysis, and for Orno's theorem on linear operators on Banach spaces. In the community of mathematicians, his peers prize Orno's stimulating discussions and generosity in allowing publication of Orno's results.

Contents

Biography

Peter Orno is the pseudonym of a fictitious mathematician, who appears as the author of short papers by one or more mathematicians. According to Robert R. Phelps (2002), the name "P. Orno" was inspired by "porno", a shortening of "pornography".[1][2]

Orno's papers list his affiliation as the Department of Mathematics at Ohio State University, and this affiliation is confirmed in the description of Orno as a "special creation" at Ohio State in Pietsch's History of Banach spaces and linear operators.[3]

Research

His papers feature "surprisingly simple" proofs and solutions to open problems in functional analysis and approximation theory, according to reviewers from Mathematical Reviews: In one case, Orno's "elegant" approach was contrasted with the previously known "elementary, but masochistic" approach. Peter Orno's "permanent interest and sharp criticism stimulated" the "work" on Lectures on Banach spaces of analytic functions by Aleksander Pełczyński, which includes several of Orno's unpublished results.[4] Tomczak-Jaegermann thanked Peter Orno for his stimulating discussions.[5]

Selected publications

Peter Orno has published in research journals and in collections; his papers have always been short, having lengths between one and three pages. Orno has also established himself as a formidable solver of mathematical problems in peer-reviewed journals published by the Mathematical Association of America.

Research papers

According to Mathematical Reviews (MR374859), 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).[6][7][8][9][10][11][12]

According to Mathematical Reviews (MR458156), 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 gL2[0,1], and for some orthonormal sequence (wk) in L2[0,2] MR458156. Another result is the "elegant proof" by Orno of a theorem of Bennet, Maurey and Nahoum.[13]

  • Ørno, P. (1977). "A separable reflexive Banach space having no finite dimensional Čebyšev subspaces". In Baker, J.; Cleaver, C.; Diestel, J.. Banach Spaces of Analytic Functions: Proceedings of the Pelczynski Conference Held at Kent State University, Kent, Ohio, July 12–17, 1976. Lecture Notes in Mathematics. 604. Springer. pp. 73–75. doi:10.1007/BFb0069208. MR454485. 

In this paper, Orno solves an eight-year old problem posed by Ivan Singer, according to Mathematical Reviews (MR454485).

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 spaces 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 Magazine, 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.

Evaluation

Among pseudonymous mathematicians, Orno is not as old or as renowned as Nicolas Bourbaki or even John Rainwater. However, he is comparable to M. G. Stanley and H. C. Enos, according to Robert R. Phelps.[1]

Notes

  1. ^ a b c Another pseudonymous mathematician, John Rainwater, "is not as old or famous as N. Bourbaki (who may still be alive) but he is clearly older than Peter Orno .... (At least one of his authors had an interest in pornography, hence P. Orno.) He is also older than M. G. Stanley (with four papers) and H. C. Enoses [sic.] (with only two)." (Phelps 2002)
  2. ^ a b In the index to his Sequences and series in Banach spaces, Joseph Diestel places Peter Orno under the letter "p" as "P. ORNO", with all-capital letters in Diestel's original. (Diestel 1984, p. 259).
  3. ^ Pietsch (2007, p. 602)
  4. ^ Pełczyński (1977, p. 2)
  5. ^ Tomczak-Jaegermann (1979, p. 273)
  6. ^ Abramovich, Y. A.; Aliprantis, C. D. (2001). "Positive Operators". In Johnson, W. B.; Lindenstrauss, J.. Handbook of the Geometry of Banach Spaces. Handbook of the Geometry of Banach Spaces. 1. Elsevier Science B. V.. pp. 85–122. doi:10.1016/S1874-5849(01)80004-8. ISBN 9780444828422. 
  7. ^ Yanovskii, L. P. (1979). "Summing and serially summing operators and characterization of AL-spaces". Siberian Mathematical Journal 20 (2): 287–292. doi:10.1007/BF00970037. 
  8. ^ Wickstead, A. W. (2010). When are all bounded operators between classical Banach lattices regular?. http://www.qub.ac.uk/puremaths/Preprints/5_2010.pdf. 
  9. ^ Meyer-Nieberg, P. (1991). Banach Lattices. Universitext. Springer-Verlag. ISBN 3-540-54201-9. MR1128093. 
  10. ^ In MR763464, Manfred Wulff noted that Orno's theorem implies several propositions in the following paper: Xiong, H. Y. (1984). "On whether or not L(E,F) = Lr(E,F) for some classical Banach lattices E and F". Nederl. Akad. Wetensch. Indag. Math. 46 (3): 267–282. 
  11. ^ In MR763464, Manfred Wolff noted that Orno's theorem has a good exposition and proof in the following textbook: Schwarz, H.-U. (1984). Banach Lattices and Operators. Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. 71. BSB B. G. Teubner Verlagsgesellschaft. p. 208. MR781131. 
  12. ^ Abramovich, Y. A. (1990). "When each continuous operator is regular". In Leifman, L. J.. Functional Analysis, Optimization, and Mathematical economics. Clarendon Press. pp. 133–140. ISBN 0-19-505729-5. MR1082571. 
  13. ^ Diestel (1984, p. 190) Diestel's "An elegant proof of this was uncovered by P. Orno (1976)" is visible at Google Books.

References

External resources


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