Inner product inequalities for two equivalent norms and applications (Q991563)

The aim of the paper is to compare \(\frac{|\langle x,y\rangle_i|}{\|x\|_i\|y\|_i}\) (resp., \(\frac{\text{Re}\langle x,y\rangle_i}{\|x\|_i\|y\|_i})\) \((i=1,2)\) whenever \(\langle \cdot,\cdot\rangle_i\) are two inner products on the same linear space \(H\) that generate two equivalent norms \(\|\cdot\|_i\) \((i=1,2)\). It is remarkable to note that in such a situation there is a positive invertible operator \(K\in B(H)\) such that \(\langle x,y\rangle_2=\langle Kx,y\rangle_1\). Thus, Theorem 1 and Theorem 2 in the paper are reduced to Corollary 2 and Corollary 4, respectively. By the way, Corollary 1 and Corollary 3 are special cases of Corollary 2 and Corollary 4, respectively.