Tensorial and Hadamard Product Inequalities for Synchronous Functions

Abstract

Let H be a Hilbert space. In this paper we show among others that, if f, g are synchronous and continuous on I and A, B are selfadjoint with spectra Sp(A), Sp(B)⊂I, then
 
 (f(A)g(A))⊗1+1⊗(f(B)g(B))≥f(A)⊗g(B)+g(A)⊗f(B)
 
 and the inequality for Hadamard product
 
 (f(A)g(A)+f(B)g(B))∘1≥f(A)∘g(B)+f(B)∘g(A).
 
 Let either p,q∈(0,∞) or p,q∈(-∞,0). If A, B>0, then
 
 A^{p+q}⊗1+1⊗B^{p+q}≥A^{p}⊗B^{q}+A^{q}⊗B^{p},
 
 and
 
 (A^{p+q}+B^{p+q})∘1≥A^{p}∘B^{q}+A^{q}∘B^{p}.