|Nationality||United States of America|
|Citizenship||United States of America|
Orno's theorem on regular operators on Banach lattices,|
Summability and Approximation theory in Banach spaces
|Institutions||Ohio State University|
Beginning in 1974, the fictitious Peter Orno (alternatively, Peter Ørno, P. Ørno, and P. Orno) appeared as the author of research papers in mathematics. According to Robert Phelps, the name "P. Orno" is a pseudonym that was inspired by "porno", an abbreviation for "pornography". Orno's short papers have been called "elegant" contributions to functional analysis. Orno's theorem on linear operators is important in the theory of Banach spaces. Research mathematicians have written acknowledgments that have thanked Orno for stimulating discussions and for Orno's generosity in allowing others to publish his results. The Mathematical Association of America's journals have also published more than a dozen problems whose solutions were submitted in the name of Orno.
Peter Orno appears as the author of short papers written by an anonymous mathematician; thus "Peter Orno" is a pseudonym. According to Robert R. Phelps, the name "P. Orno" was inspired by "porno", a shortening of "pornography".
Orno's papers list his affiliation as the Department of Mathematics at Ohio State University. 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. The publications list of Ohio State mathematician Gerald Edgar includes two items that were published under the name of Orno. Edgar indicates that he published them "as Peter Ørno".
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"work" on Lectures on Banach spaces of analytic functions by Aleksander Pełczyński, which includes several of Orno's unpublished results. Tomczak-Jaegermann thanked Peter Orno for his stimulating discussions.
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.
- Ørno, P. (1974). "On Banach lattices of operators". Israel Journal of Mathematics. 19 (3): 264–265. doi:10.1007/BF02757723. MR 0374859.
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).
- Ørno, P. (1976). "A note on unconditionally converging series in Lp". Proceedings of the American Mathematical Society. 59 (2): 252–254. doi:10.1090/S0002-9939-1976-0458156-7. JSTOR 2041478. MR 0458156.
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 g∈L2[0,1], and for some orthonormal sequence (wk) in L2[0,2] MR458156. Another result is what Joseph Diestel described as the "elegant proof" by Orno of a theorem of Bennet, Maurey and Nahoum.
- 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. MR 0454485.
- Ørno, P. (1991). "On J. Borwein's concept of sequentially reflexive Banach spaces". arXiv:math/9201233.
Still circulating as an "underground classic", as of October 2018[update] this paper had been cited sixteen times. 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.
Between 1976 and 1982, Peter Orno contributed problems or solutions that appeared in eighteen issues of 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:
- Quet, L.; Ørno, P. (2006). "A continued fraction related to π (Problem 11102, 2004, p. 626)". American Mathematical Monthly. 113 (6): 572–573. doi:10.2307/27641994. JSTOR 27641994.
Peter Orno is one of several pseudonymous contributors in the field of mathematics. Other pseudonymous mathematicians active in the 20th century include Nicolas Bourbaki, John Rainwater, M. G. Stanley, and H. C. Enos.
Besides connoting "pornography", the name "Ørno" features a non-standard symbol:
- ∅, which symbolizes the empty set in mathematics.
- Ø, an (archaic) English vowel, also denoted "OE", "Ö", and "Œ".
- Phelps (2002)
- 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)
- 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).
- The Garden of Constants is located at Ohio State University, according to (Ross Mathematics Program 2012, Caption "Garden of Constants at Ohio State"):
- Pietsch (2007, p. 602)
- Gerald A. Edgar, Publications, Ohio State University. Retrieved March 18, 2012; archived by WebCite at https://www.webcitation.org/66GaKYk03. Items that Edgar claims as his work, but identifies as having been attributed to "Peter Ørno", are the problem proposed in Mathematics Magazine 52 (1979), 179, and the problem solution presented in American Mathematical Monthly 113 (2006) 572–573.
- Pełczyński (1977, p. 2)
- Tomczak-Jaegermann (1979, p. 273)
- 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 978-0-444-82842-2.
- 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.
- Wickstead, A. W. (2010). "When are all bounded operators between classical Banach lattices regular?" (PDF).
- Meyer-Nieberg, P. (1991). Banach Lattices. Universitext. Springer-Verlag. ISBN 3-540-54201-9. MR 1128093.
- 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.
- 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. MR 0781131.
- 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. MR 1082571.
- Diestel (1984, p. 190)
- "On J. Borwein's concept of sequentially reflexive Banach spaces". Retrieved Oct 9, 2018 – via Google Scholar.
- "Problems" sections of Mathematics Magazine in which Peter Orno is one of the contributing authors are: Vol. 49, No. 3 (May 1976), pp. 149–154; Vol. 49, No. 4 (September 1976), pp. 211–218; Vol. 50, No. 1 (January 1977), pp. 46–53; Vol. 50, No. 4 (September 1977), pp. 211–216; Vol. 51, No. 2 (March 1978), pp. 127–132; Vol. 51, No. 3 (May 1978), pp. 193–201; Vol. 51, No. 4 (September 1978), pp. 245–249; Vol. 52, No. 1 (January 1979), pp. 46–55; Vol. 52, No. 2 (March 1979), pp. 113–118; Vol. 52, No. 3 (May 1979), pp. 179–184; Vol. 53, No. 1 (January 1980), pp. 49–54; Vol. 53, No. 2 (March 1980), pp. 112–117; Vol. 53, No. 3 (May 1980), pp. 180–186; Vol. 53, No. 4 (September 1980), pp. 244–251; Vol. 54, No. 2 (March 1981), pp. 84–87; Vol. 54, No. 4 (September 1981), pp. 211–214; Vol. 54, No. 5 (November 1981), pp. 270–274; and Vol. 55, No. 3 (May 1982), pp. 177–183.
- Diestel, J. (1984). "X Grothendieck's inequality and the Grothendieck-Lindenstrauss-Pelczynski [Pełczyński] cycle of ideas (Notes and remarks, pp. 187–191)". Sequences and series in Banach spaces. Graduate Texts in Mathematics. 92. Springer-Verlag. ISBN 0-387-90859-5. MR 0737004.
- Pełczyński, A. (1977). Banach spaces of analytic functions and absolutely summing operators. Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics. 30. American Mathematical Society. p. 2. ISBN 0-8218-1680-2. MR 0511811.
- Phelps, R. R. (2002). "Biography of John Rainwater". Topological Commentary. 7 (2).
- Pietsch, A. (2007). History of Banach Spaces and Linear Operators. Birkhäuser Verlag. ISBN 978-0-8176-4367-6. MR 2300779.
- Tomczak-Jaegermann, N. (1979). "Computing 2-summing norm with few vectors". Arkiv för Matematik. 17 (1): 273–277. Bibcode:1979ArM....17..273T. doi:10.1007/BF02385473. MR 0608320.