On Schwarz type inequalities

We show Schwarz type inequalities and consider their converses. A continuous function f : [ 0 , ∞ ) → [ 0 , ∞ ) f : [0, \infty ) \rightarrow [0, \infty ) is said to be semi-operator monotone on ( a , b ) (a,b) if { f ( t 1 2 ) } 2 \{f( t^{\frac {1}{2}} ) \}^{2} is operator monotone on ( a 2 , b 2 ) (a^{2},b^{2}) . Let T T be a bounded linear operator on a complex Hilbert space H {\mathcal H} and T = U | T | T = U \vert T \vert be the polar decomposition of T T . Let 0 ≤ A , B ∈ B ( H ) 0 \leq A, B \in B( {\mathcal H}) and ‖ T x ‖ ≤ ‖ A x ‖ , ‖ T ∗ y ‖ ≤ ‖ B y ‖ \Vert Tx \Vert \leq \Vert Ax\Vert , \Vert T^{*} y \Vert \leq \Vert By \Vert for x , y ∈ H x, y \in {\mathcal H} . (1) If a non-zero function f f is semi-operator monotone on ( 0 , ∞ ) (0, \infty ) , then | ⟨ T x , y ⟩ | ≤ ‖ f ( A ) x ‖ ‖ g ( B ) y ‖ \vert \langle Tx, y \rangle \vert \leq \Vert f(A) x \Vert \Vert g(B) y \Vert for x , y ∈ H x, y \in {\mathcal H} , where g ( t ) = t / f ( t ) g(t) = t/f(t) . (2) If f , g f, g are semi-operator monotone on ( 0 , ∞ ) (0, \infty ) , then | ⟨ U f ( | T | ) g ( | T | ) x , y ⟩ | ≤ ‖ f ( A ) x ‖ ‖ g ( B ) y ‖ \vert \langle U f(\vert T \vert )g(\vert T \vert )x, y \rangle \vert \leq \Vert f(A) x \Vert \Vert g(B) y \Vert for x , y ∈ H x, y \in {\mathcal H} . Also, we show converses of these inequalities, which imply that semi-operator monotonicity is necessary.