Solution of operator equations and proof of Kannan mapping theorem and Chatterjea-type mapping theorem in C^*-algebra valued S_b-metric spaces
Subject Areas : StatisticsS. Samira Razavi 1 , Hashem Parvaneh Masiha 2
1 - Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
2 - Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
Keywords: فضای S_b-متریک C^*-جبر مقدار, نقطه ثابت, نگاشت نوع کاترجا, نگاشت نوع کانان, معادلات عملگری,
Abstract :
In this paper, based on the results and theorems of C^*-algebra valued S_b-metric spaces, we solve one type of matrix operator equations in L(H) defined by X-∑_(n=۱)^∞▒〖A_n^* XA_n=Q〗, in which H is a Hilbert space and L(H) is the set of linear and bounded operators on H. Also, we prove that the Kannan contraction mapping has a unique fixed point in C^*-algebra valued S_b-metric spaces. Furthermore, we prove that the Chatterjea-type contraction mapping has a unique fixed point in C^*-algebra valued S_b-metric spaces. Finally, by using the Banach contraction principle in C^*-algebra valued S_b-metric spaces that has previously been studied by the authors of this article and by using the results of the above theorems, we solve the above matrix operator equation in C^*-algebra valued S_b-metric spaces. As well as, we show that this matrix operator equation has a unique solution in L(H), and this solution is a Hermitian operator.
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