Nonsurjective zero product preservers between matrix spaces over an arbitrary field

Abstract

AbstractA map Φ between matrices is said to be zero product preserving if Φ(A)Φ(B)=0wheneverAB=0.In this paper, we give concrete descriptions of an additive/linear zero product preserver Φ:Mn(F)→Mr(F) between matrix algebras of different dimensions over an arbitrary field F, and n≥2. In particular, we show that if Φ is linear and preserves zero products then Φ(A)=S(R1⊗A00Φ0(A))S−1,for some invertible matrices R1 in Mk(F), S in Mr(F) and a zero product preserving linear map Φ0:Mn(F)→Mr−nk(F) into nilpotent matrices. If Φ(In) is invertible, then Φ0 is vacuous. In general, the structure of Φ0 could be quite arbitrary, especially when Φ0(Mn(F)) has trivial multiplication, i.e. Φ0(X)Φ0(Y)=0 for all X, Y in Mn(F). We show that if Φ0(In)=0 or r−nk≤n+1, then Φ0(Mn(F)) indeed has trivial multiplication. More generally, we characterize subspaces V of square matrices satisfying XY = 0 for any X,Y∈V. Similar results for double zero product preserving maps are obtained.COMMUNICATED BY: Y.-T. PoonKeywords: Zero product preserversdouble zero preserversmatrix algebras2000 Mathematics Subject Classifications: 08A3515A8647B48 AcknowledgementsThe authors would like to express their appreciation to the referees for their useful comments and suggestions.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingC.-K. Li is an affiliate member of the Institute for Quantum Computing, University of Waterloo. His research is supported by the Simons Foundation [grant number 851334]. This research started during his academic visit to Taiwan in 2018, which was supported by grants from Taiwan MOST. He would like to express his gratitude to the hospitality of several institutions there, including the Academia Sinica, National Chung Hsing University, National Sun Yat-sen University, and National Taipei University of Technology. M.-C. Tsai, Y.-S. Wang and N.-C. Wong are supported by the Ministry of Science and Technology (MOST), Taiwan [grant numbers 110-2115-M-027-002-MY2, 111-2115-M-005-001-MY2 and 110-2115-M-110-002-MY2, respectively.

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• Linear and Multilinear Algebra - View
• arXiv (Cornell University) - View - PDF