Selberg type inequalities in a Hilbert \(C^\ast\)-module and its applications (Q2815566)

The classical Selberg inequality, as a joint extension of the Bessel and the Cauchy-Schwarz inequalities, states that if \((H,\langle\cdot,\cdot\rangle)\) is a Hilbert space, then for non-zero vectors \(x, y_1, \dots, y_n\in H\), the inequality \(\sum_{i=1}^n\frac{|\langle y_i,x\rangle|^2}{\sum_{j=1}^n|\langle y_j,y_i\rangle|}\leq \|x\|^2\) holds. Moreover, \textit{T. Furuta} [Nihonkai Math. J. 2, No. 1, 25--29 (1991; Zbl 0956.15500)] posed some conditions enjoying the equality.NEWLINENEWLINEThe authors of the present paper give a Selberg-type inequality in the setting of Hilbert \(C^*\)-modules. Their main result says that if \(X\) is an inner product \(C^*\)-module over a unital \(C^*\)-algebra \(A\), elements \(a, y_1, \dots, y_n\in X\) are non-zero and \(y_1, \dots, y_n\) are non-singular, then NEWLINE\[NEWLINE\sum_{i=1}^n\langle x,y_i\rangle\left(\sum_{j=1}^n|\langle y_j,y_i\rangle|\right)^{-1}\langle y_i,x\rangle \leq \langle x,x\rangle.NEWLINE\]NEWLINE This is simultaneously an extension of the Bessel inequality due to \textit{N. Bounader} and \textit{A. Chahbi} [Int. J. Math. Anal., Ruse 7, No. 5--8, 385--391 (2013; Zbl 1288.46035)] and the Cauchy-Schwarz inequality due to \textit{S. S. Dragomir} et al. [Linear Multilinear Algebra 58, No. 7--8, 967--975 (2010; Zbl 1210.46043)] in the framework of Hilbert \(C^*\)-modules.