保持算子束部分等距的映射

Abstract

Abstract:

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H, PI(H) the set of all partial isometries in B(H). It is proved that a surjective map Φ on B(H) preserves partial isometries of pencils of operators, that is, A-λB∈PI(H)⇔Φ(A)-λΦ(B)∈PI(H) if and only if there are two unitary operators U and V on H such that Φ(X)=UXV for all X in B(H) or there are two anti-unitary operators U and V on H such that Φ(X)=UX^*V for all X in B(H).

Key words: Partial isometry, Partial ordering, Surjective map, Operator pencil

CLC Number:

O177.1

Wei Yanan, Ji Guoxing. Maps Preserving Partial Isometries of Operator Pencils[J].Acta mathematica scientia,Series A, 2016, 36(3): 413-424.

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References

[1] 侯晋川, 崔建莲. 算子代数上线性映射引论. 北京:科学出版社, 2002 Hou J C, Cui J L. Introduction to the Linear Maps on Operator Algebras. Beijing:Science Press, 2002
[2] Ji G X, Gao Y L. Maps preserving operator pairs whose products are projections. Linear Algebra and Its Applications, 2010, 433(7):1348-1364
[3] 吉国兴, 曲凡连. 保持算子乘积投影的线性映射. 数学学报, 2010, 52(2):315-322Ji G X, Qu F L. Linear maps preserving projections of products of operators. Acta Math Sin, 2010, 52(2):315-322
[4] Petek Tatjana. Mappings preserving the idempotency of products of operators. Linear Multilinear Algebra, 2010, 58(7/8):903-925
[5] Fang L, Ji G X, Pang Y F. Maps preserving the idempotency of products of operators. Linear Algebra Appl, 2007, 426(1):40-52
[6] Gil' Michael. Norm estimate for the inverse of a linear pencil with Hilbert-Schmidt operators. Applicable Analysis, 2015, 94(2):409-418
[7] Shkalikov A A, Tretter C. Spectral analysis for linear pencils N-λP of ordinary differential operators. Math Nachr, 1996, 179:275-305
[8] Grunbaum F A, Vinet L, Zhedanov A. Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials. Journal of Physics A:Mathematical and General, 2004, 37(31):7711-7725
[9] Tretter C. Linear operator pencils A-B with discrete spectrum. Integral Equations Operator Theory, 2000, 37(3):357-373
[10] Dolinar G. Maps on matrix algebras preserving idempotents. Linear Algebra Appl, 2003, 371(15):287-300
[11] Dolinar G. Maps on B(H) preserving idempotents. Linear Multilinear Algebra, 2004, 52(5):335-347
[12] 李晓梅, 张建华. B(H)上保投影的映射. 数学进展, 2014, 43(3):425-428 Li X M, Zhang J H. Maps on B(H) preserving projections. Advances in Mathematics, 2014, 43(3):425-428
[13] Molnar L. On certain antomorphisms of sets of partial isometries. Arch Math, 2002, 78:43-50
[14] 石薇薇, 吉国兴. B(H)上双边保持部分等距的可加映射. 陕西师范大学学报, 2015, 43(5):14-17Shi W W, Ji G X. Additive maps presevuing partial isometries on B(H). Journal of Shaanxi Normal University, 2015, 43(5):14-17
[15] Pearcy C, Topping D. Sums of small numbers of idempotents. Michigan Math J, 1967, 14(4):453-465

Maps Preserving Partial Isometries of Operator Pencils

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