Joint Similarity and Parameterizations of Dilations of Dual \((\Omega ,\mu )\)-Frame Pairs

  • Xunxiang GuoEmail author


In this paper, we study some properties on the dilations of dual \((\Omega ,\mu )\)-frame pairs. Firstly, we introduce the concepts of joint complementary \((\Omega ,\mu )\)-frames and canonical dual \((\Omega ,\mu )\)-frame dilations of a dual \((\Omega ,\mu )\)-frame pair associated with an analysis space. Then we prove that the pairs of joint complementary \((\Omega ,\mu )\)-frames of a given dual \((\Omega ,\mu )\)-frame pair associated with an analysis space are unique in the sense of joint similarity and the set of canonical dual \((\Omega ,\mu )\)-frame dilations of a given dual \((\Omega ,\mu )\)-frame pair associated with an analysis space are parameterized by a set of invertible diagonal operators. Finally we parameterize the set of canonical dual \((\Omega ,\mu )\)-frame dilations of all dual \((\Omega ,\mu )\)-frame pairs of a given \((\Omega ,\mu )\)-frame associated with an analysis space in terms of a set of invertible lower triangular operators.


(\(\Omega , \mu \))-frame Dual (\(\Omega , \mu \))-frame pair Joint complementary (\(\Omega , \mu \))-frames Joint similarity Canonical dual (\(\Omega , \mu \))-frame dilation 

Mathematics Subject Classification

42C15 46B15 



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsSouthwestern University of Finance and EconomicsChengduPeople’s Republic of China

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