Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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C*-ALGEBRAIC SCHUR PRODUCT THEOREM, PÓLYA-SZEGŐ-RUDIN QUESTION AND NOVAK'S CONJECTURE

  • Received : 2021.10.10
  • Accepted : 2022.03.22
  • Published : 2022.07.01
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Abstract

Striking result of Vybíral [51] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vybíral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C*-algebras and derive the results of Schur and Vybíral. As an application, we state C*-algebraic version of Novak's conjecture and solve it for commutative unital C*-algebras. We formulate Pólya-Szegő-Rudin question for the C*-algebraic Schur product of positive matrices.

Keywords

Acknowledgement

Remark 4.4 is due to the reviewer and I thank the reviewer for this remark. I'm grateful to Prof. Adam H. Fuller, Department of Mathematics, Ohio University, USA for going through the arXiv version of this paper and pointing out that Theorem 2.2 is proved for certain C*-subalgebras of the concrete C*-algebra B(𝓗) (the space of all bounded linear operators on Hilbert space 𝓗) in his paper [10] (in a different way), which I was not aware (see Theorem 2.3 in [10]). In an email communication he showed me further that his arguments in can be invoked for arbitrary commutative C*-algebras. However, the definition of Schur product in this paper and in [10] differs for arbitrary C*-algebras. I thank Dr. P. Sam Johnson, Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka (NITK), Surathkal for his help and some discussions. The author is supported by Indian Statistical Institute, Bangalore, through the J. C. Bose Fellowship of Prof. B. V. Rajarama Bhat. He thanks Prof. B. V. Rajarama Bhat for the Post Doc position.

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