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

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 AA in B(H)B\left(H), where HH 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 ss-rr-norm and ss-rr -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).

Keywords

• power vector inequalities
• operators
• hilbert space
• mitrinović-pečarić-fink inequality
• norm inequalities
• 47a30
• 46c05
• 47a63
• 47a99