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Funktsional. Anal. i Prilozhen., 2002, Volume 36, Issue 3, Pages 20–35 (Mi faa201)  

This article is cited in 38 scientific papers (total in 38 papers)

On Sums of Projections

S. A. Kruglyak, V. I. Rabanovich, Yu. S. Samoilenko

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: In the paper, for all $n\in\mathbb{N}$, we describe the set $\Sigma_n$ of all real numbers $\alpha$ admitting a collection of projections $P_1,…,P_n$ on a Hilbert space $H$ such that $\sum_{k=1}^n P_k=\alpha I$ ($I$ is the identity operator on $H$) and study the problem to find all collections of this kind for a given $\alpha\in\Sigma_n$.

Keywords: algebra, representation, operator, matrix, projection, identity

DOI: https://doi.org/10.4213/faa201

Full text: PDF file (257 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2006, 36:3, 182–195

Bibliographic databases:

UDC: 517.98
Received: 08.11.2001

Citation: S. A. Kruglyak, V. I. Rabanovich, Yu. S. Samoilenko, “On Sums of Projections”, Funktsional. Anal. i Prilozhen., 36:3 (2002), 20–35; Funct. Anal. Appl., 36:3 (2006), 182–195

Citation in format AMSBIB
\Bibitem{KruRabSam02}
\by S.~A.~Kruglyak, V.~I.~Rabanovich, Yu.~S.~Samoilenko
\paper On Sums of Projections
\jour Funktsional. Anal. i Prilozhen.
\yr 2002
\vol 36
\issue 3
\pages 20--35
\mathnet{http://mi.mathnet.ru/faa201}
\crossref{https://doi.org/10.4213/faa201}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1935900}
\zmath{https://zbmath.org/?q=an:1038.47001}
\transl
\jour Funct. Anal. Appl.
\yr 2006
\vol 36
\issue 3
\pages 182--195
\crossref{https://doi.org/10.1023/A:1020193804109}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0036377329}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. T. V. Shulman, “On Sums of Projections in $C^*$-Algebras”, Funct. Anal. Appl., 37:4 (2003), 316–317  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Kruglyak S., Rabanovich V., Samoilenko Yu., “Decomposition of a scalar matrix into a sum of orthogonal projections”, Linear Algebra Appl., 370 (2003), 217–225  crossref  mathscinet  zmath  isi  scopus
    3. A. S. Mellit, V. I. Rabanovich, Yu. S. Samoilenko, “When Is a Sum of Partial Reflections Equal to a Scalar Operator?”, Funct. Anal. Appl., 38:2 (2004), 157–160  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. Samoilenko Yu.S., “When is a sum of projections equal to a scalar operator?”, J. Nonlinear Math. Phys., 11, suppl. (2004), 92–103  crossref  mathscinet  adsnasa  isi  scopus
    5. S. A. Kruglyak, A. V. Roiter, “Locally Scalar Graph Representations in the Category of Hilbert Spaces”, Funct. Anal. Appl., 39:2 (2005), 91–105  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. M. A. Vlasenko, A. S. Mellit, Yu. S. Samoilenko, “Algebras Generated by Linearly Dependent Elements with Prescribed Spectra”, Funct. Anal. Appl., 39:3 (2005), 175–186  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. S. A. Kruglyak, L. A. Nazarova, A. V. Roiter, “Orthoscalar representations of quivers in the category of Hilbert spaces”, J. Math. Sci. (N. Y.), 145:1 (2007), 4793–4804  mathnet  crossref  mathscinet  zmath  elib
    8. I. K. Redchuk, “Separating functions and their applications”, J. Math. Sci. (N. Y.), 145:1 (2007), 4805–4810  mathnet  crossref  mathscinet  zmath  elib
    9. Yuliya P. Moskaleva, Yurii S. Samoilenko, “On Transitive Systems of Subspaces in a Hilbert Space”, SIGMA, 2 (2006), 042, 19 pp.  mathnet  crossref  mathscinet  zmath
    10. Enomoto M., Watatani Y., “Relative position of four subspaces in a Hilbert space”, Adv. Math., 201:2 (2006), 263–317  crossref  mathscinet  zmath  isi  scopus
    11. Albeverio S., Ostrovskyi V., Samoilenko Yu., “On functions on graphs and representations of a certain class of $*$-algebras”, J. Algebra, 308:2 (2007), 567–582  crossref  mathscinet  zmath  isi  scopus
    12. Hrushevoi R.V., “On conditions under which the sum of selfadjoint operators with given integer spectra is a scalar operator”, Ukrainian Math. J., 60:4 (2008), 540–550  crossref  mathscinet  isi  scopus
    13. Yusenko A.A., “Quintuplets of orthoprojectors associated by a linear relation”, Ukrainian Math. J., 61:5 (2009), 834–846  crossref  mathscinet  zmath  isi  scopus
    14. Enomoto M., Watatani Y., “Indecomposable representations of quivers on infinite-dimensional Hilbert spaces”, J. Funct. Anal., 256:4 (2009), 959–991  crossref  mathscinet  zmath  isi  scopus
    15. Shulman T., “On universal $C^*$-algebras generated by $n$ projections with scalar sum”, Proc. Amer. Math. Soc., 137:1 (2009), 115–122  crossref  mathscinet  zmath  isi  scopus
    16. Samoilenko Yu.S., Strelets A.V., “On Simple N-Tuples of Subspaces of a Hilbert Space”, Ukrainian Math J, 61:12 (2009), 1956–1994  crossref  mathscinet  isi  scopus
    17. Omel'chenko P.V., “On Reduction of Block Matrices in a Hilbert Space”, Ukrainian Math J, 61:10 (2009), 1578–1588  crossref  mathscinet  zmath  isi  scopus
    18. Yakimenko D.Yu., “UNITARIZATION OF REPRESENTATIONS OF A PARTIALLY ORDERED SET ASSOCIATED WITH A GRAPH (E)over-tilde(6)”, Ukrainian Math J, 61:10 (2009), 1672–1683  crossref  mathscinet  zmath  isi  scopus
    19. Bottcher A., Spitkovsky I.M., “A gentle guide to the basics of two projections theory”, Linear Algebra Appl., 432:6 (2010), 1412–1459  crossref  mathscinet  zmath  isi  elib  scopus
    20. S. A. Kruglyak, L. A. Nazarova, A. V. Roiter, “Orthoscalar quiver representations corresponding to extended Dynkin graphs in the category of Hilbert spaces”, Funct. Anal. Appl., 44:2 (2010), 125–138  mathnet  crossref  crossref  mathscinet  zmath  isi
    21. Yakimenko D.Yu., “UNITARIZATION OF SCHUR REPRESENTATIONS OF A PARTIALLY ORDERED SET ASSOCIATED WITH (E)over-tilde(7)”, Ukrainian Math J, 62:6 (2010), 982–988  crossref  mathscinet  zmath  isi  scopus
    22. Yusenko A.A., “Quadruples of Orthoprojectors Connected By a Linear Relationship”, Ukrainian Math J, 62:2 (2010), 289–301  crossref  mathscinet  zmath  isi  scopus
    23. Albeverio S., Rabanovich S., “Decomposition of a scalar operator into a product of unitary operators with two points in spectrum”, Linear Algebra Appl, 433:6 (2010), 1127–1137  crossref  mathscinet  zmath  isi  scopus
    24. Kruhlyak S.A., Livins'kyi I.V., “REGULAR ORTHOSCALAR REPRESENTATIONS OF THE EXTENDED DYNKIN GRAPH (E)over-tilde(8) AND *-ALGEBRA ASSOCIATED WITH IT”, Ukrainian Math J, 62:8 (2011), 1213–1233  crossref  mathscinet  isi  scopus
    25. Kaftal V., Ng PingWong, Zhang Shuang, “Positive combinations of projections in von Neumann algebras and purely infinite simple C*-algebras”, Science China-Mathematics, 54:11 (2011), 2383–2393  crossref  mathscinet  zmath  adsnasa  isi  scopus
    26. Kaftal V., Ng P.W., Zhang Sh., “Positive Combinations and Sums of Projections in Purely Infinite Simple C*-Algebras and their Multiplier Algebras”, Proc Amer Math Soc, 139:8 (2011), 2735–2746  crossref  mathscinet  zmath  isi  scopus
    27. S. A. Kruglyak, I. V. Livinsky, “Orthoscalar representations of the partially ordered set $(N, 4)$”, Algebra Discrete Math., 14:2 (2012), 217–229  mathnet  mathscinet  zmath
    28. Samoilenko Yu.S., Yakimenko D.Yu., “Scalar Operators Equal to the Product of Unitary Roots of the Identity Operator”, Ukr. Math. J., 64:6 (2012), 938–947  crossref  mathscinet  zmath  isi  scopus
    29. Kaftal V., Ng P.W., Zhang Sh., “Finite Sums of Projections in Purely Infinite Simple $C^*$-Algebras with Torsion K-0”, Proc. Amer. Math. Soc., 140:9 (2012), 3219–3227  crossref  mathscinet  zmath  isi  scopus
    30. Feshchenko I.S., “On Closeness of the Sum of Subspaces of a Hilbert Space”, Ukr. Math. J., 63:10 (2012), 1566–1622  crossref  mathscinet  zmath  isi  scopus
    31. Halpern H., Kaftal V., Ng P.W., Zhang Sh., “Finite Sums of Projections in Von Neumann Algebras”, Trans. Am. Math. Soc., 365:5 (2013), 2409–2445  crossref  mathscinet  zmath  isi  scopus
    32. Livins'kyi I.V., Radchenko D.V., “Representations of Algebras Defined by a Multiplicative Relation and Corresponding to the Extended Dynkin Graphs (D)Over-Tilde(4), (E)Over-Tilde(6), (E)Over-Tilde(7), and (E)Over-Tilde(8)”, Ukr. Math. J., 64:12 (2013), 1865–1892  crossref  mathscinet  zmath  isi  scopus
    33. Choi M.-D., Wu P.Yu., “Sums of Orthogonal Projections”, J. Funct. Anal., 267:2 (2014), 384–404  crossref  mathscinet  zmath  isi  scopus
    34. Enomoto M. Watatani Ya., “Strongly Irreducible Operators and Indecomposable Representations of Quivers on Infinite-Dimensional Hilbert Spaces”, Integr. Equ. Oper. Theory, 83:4 (2015), 563–587  crossref  mathscinet  zmath  isi  scopus
    35. Rabanovych V.I., “on Decompositions of a Scalar Operator Into a Sum of Self-Adjoint Operators With Finite Spectrum”, Ukr. Math. J., 67:5 (2015), 795–813  crossref  mathscinet  zmath  isi  scopus
    36. Bownik M., Luoto K., Richmond E., “a Combinatorial Characterization of Tight Fusion Frames”, Pac. J. Math., 275:2 (2015), 257–294  crossref  mathscinet  zmath  isi  scopus
    37. Dragan C., Kaftal V., “Sums of Equivalent Sequences of Positive Operators in Von Neumann Factors”, Houst. J. Math., 42:3 (2016), 991–1017  mathscinet  zmath  isi  elib
    38. Dykema K., Paulsen V.I., Prakash J., “Non-Closure of the Set of Quantum Correlations Via Graphs”, Commun. Math. Phys., 365:3 (2019), 1125–1142  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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