Similarity between two projections. (English) Zbl 1508.47021
Summary: Given two orthogonal projections \(P\) and \(Q\), we are interested in all unitary operators \(U\) such that \(UP=QU\) and \(UQ=PU\). Such unitaries \(U\) have previously been constructed by Wang, Du, and Dou [Y.-Q. Wang et al., Acta Math. Sin., Engl. Ser. 25, No. 4, 679–686 (2009; Zbl 1187.47015)] and also by one of the authors [B. Simon, Linear Algebra Appl. 528, 436–441 (2017; Zbl 06950136)]. One purpose of this note is to compare these constructions. Very recently, Dou, Shi, Cui, and Du [Y.-N. Dou et al., Linear Algebra Appl. 531, 575–591 (2017; Zbl 1492.47003)] described all unitaries \(U\) with the required property. Their proof is via the two projections theorem by P. R. Halmos [Trans. Am. Math. Soc. 144, 381–389 (1969; Zbl 0187.05503)]. We here give a proof based on the supersymmetric approach by Avron, Seiler, and one of the authors [J. E. Avron et al., Commun. Math. Phys. 159, No. 2, 399–422 (1994; Zbl 0822.47056)].
MSC:
| 47A62 | Equations involving linear operators, with operator unknowns |
| 46H15 | Representations of topological algebras |
| 46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |
| 47A67 | Representation theory of linear operators |
| 47C15 | Linear operators in \(C^*\)- or von Neumann algebras |
Keywords:
intertwining operators; intertwining unitaries; similar projections; two projections theoremReferences:
| [1] | Avron, J.: Barry and Pythagoras. From Mathematical Physics to Analysis: A Walk in Barry Simon’s Mathematical Garden, II. Notices Am. Math. Soc. 63, 878-889 (2016) · Zbl 1352.01037 |
| [2] | Avron, J., Seiler, R., Simon, B.: Charge deficiency, charge transport and comparison of dimensions. Commun. Math. Phys. 159, 399-422 (1994) · Zbl 0822.47056 · doi:10.1007/BF02102644 |
| [3] | Avron, J., Seiler, R., Simon, B.: The index of a pair of projections. J. Funct. Anal. 120, 220-237 (1994) · Zbl 0822.47033 · doi:10.1006/jfan.1994.1031 |
| [4] | Böttcher, A., Gohberg, I., Karlovich, YuI, Krupnik, N., Roch, S., Silbermann, B., Spitkovsky, I.M.: Banach algebras generated by N idempotents and applications. Oper. Theory Adv. Appl. 90, 19-54 (1996) · Zbl 0885.46041 |
| [5] | Böttcher, A., Spitkovsky, I.M.: A gentle guide to the basics of two projections theory. Linear Algebra Appl. 432, 1412-1459 (2010) · Zbl 1189.47073 · doi:10.1016/j.laa.2009.11.002 |
| [6] | Dou, Y.N., Shi, W.J., Cui, M.M., Du, H.K.: General explicit expressions for intertwining operators and direct rotations of two orthogonal projections, pp. 1-14. arXiv:1705.05870v1 [math.SP] (2017) · Zbl 1492.47003 |
| [7] | Giles, R., Kummer, H.: A matrix representation of a pair of projections in a Hilbert space. Can. Math. Bull. 14, 35-44 (1971) · Zbl 0209.14702 · doi:10.4153/CMB-1971-006-5 |
| [8] | Gohberg, I., Krupnik, N.: Extension theorems for invertibility symbols in Banach algebras. Integr. Equ. Oper. Theory 15, 991-1010 (1992) · Zbl 0795.46043 · doi:10.1007/BF01203124 |
| [9] | Gohberg, I., Krupnik, N.: Extension theorems for Fredholm and invertibility symbols. Integr. Equ. Oper. Theory 16, 514-529 (1993) · Zbl 0811.46044 · doi:10.1007/BF01205291 |
| [10] | Halmos, P.L.: Two subspaces. Trans. Am. Math. Soc. 144, 381-389 (1969) · Zbl 0187.05503 · doi:10.1090/S0002-9947-1969-0251519-5 |
| [11] | Kato, T.: On the perturbation theory of closed linear operators. J. Math. Soc. Jpn. 4, 323-337 (1952) · Zbl 0048.35402 · doi:10.2969/jmsj/00430323 |
| [12] | Kato, T.: Notes on projections and perturbation theory. Technical Report No. 9, University of California, Berkley (1955) · Zbl 0795.46043 |
| [13] | Mityagin, B.: Perturbation of an orthogonal projection and the intertwining unitary operator. Russ. J. Math. Phys. 12, 489-496 (2005) · Zbl 1190.47017 |
| [14] | Roch, S., Santos, P.A., Silbermann, B.: Non-commutative Gelfand Theories. Springer, London (2011) · Zbl 1209.47002 · doi:10.1007/978-0-85729-183-7 |
| [15] | Roch, S., Silbermann, B.: Algebras generated by idempotents and the symbol calculus for singular integral operators. Integr. Equ. Oper. Theory 11, 385-419 (1988) · Zbl 0658.47050 · doi:10.1007/BF01202079 |
| [16] | Simon, B.: A Comprehensive Course in Analysis, Part 4: Operator Theory. American Mathematical Society, Providence (2015) · Zbl 1334.00003 |
| [17] | Simon, B.: Unitaries permuting two orthogonal projections. Linear Algebra Appl. 528, 436-441 (2017) · Zbl 06950136 · doi:10.1016/j.laa.2017.03.026 |
| [18] | Simon, B.: Tosio Kato’s work on non-relativistic quantum mechanics 1-215. arXiv:1711.00528v1 [math-ph] (2017) · Zbl 0885.46041 |
| [19] | Spitkovsky, I.M.: Once more on algebras generated by two projections. Linear Algebra Appl. 208(209), 377-395 (1994) · Zbl 0803.46070 · doi:10.1016/0024-3795(94)90450-2 |
| [20] | Wang, Y.Q., Du, H.K., Dou, Y.N.: On the index of Fredholm pairs of idempotents. Acta Math. Sin. (Engl. Ser.) 25, 679-686 (2009) · Zbl 1187.47015 · doi:10.1007/s10114-009-7067-1 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
