Power vector inequalities for operator pairs in Hilbert spaces and their applications

Abstract

Abstract This study explores the power vector inequalities for a pair of operators ( B , C ) \left(B,C) in a Hilbert space. By utilizing a Mitrinović-Pečarić-Fink-type inequality for inner products and norms, we derive various power vector inequalities. Specifically, we consider the cases where ( B , C ) \left(B,C) is equal to ( A , A * ) \left(A,{A}^{* }) or ( Re ( A ) , Im ( A ) ) (\hspace{0.1em}\text{Re}\hspace{0.1em}\left(A),\hspace{0.1em}\text{Im}\hspace{0.1em}\left(A)) for an operator A A in B ( H ) B\left(H) , where H H is a Hilbert space. This leads to the derivation of vector, norm, and numerical radius inequalities for a single operator. Furthermore, we obtain power inequalities for the s s - r r -norm and s s - r r -numerical radius of the operator pair ( B , C ) ∈ B ( H ) \left(B,C)\in B\left(H) , which generalizes the Euclidean norm and Euclidean numerical radius. Finally, we apply these results to derive the corresponding inequalities for a single operator A ∈ B ( H ) A\in B\left(H) .