Some vector inequalities for continuous functions of self-adjoint operators in Hilbert spaces (Q535527)

Let \(H\) be a complex Hilbert space and \(A\) be a selfadjoint bounded linear operator on \(H\) with the spectrum \(Sp(A)\in [m,M]\) for some real numbers \(m\) and \(M\), \(m<M\). Upper bounds of \(|\langle [f(M)1_{H}-f(A)][f(A)-f(m)1_{H}]x,y\rangle|\) for \(x,y\in H\) and \(f:[m,M]\to \mathbb{C}\) are given under the following conditions: (1) \(f\) is a continuous function of bounded variation, (2) \(f\) is a Lipschitzian function and (3) \(f:[m,M]\to \mathbb{R}\) is a monotonic nondecreasing function. As concrete examples of the results, the author gives some inequalities by putting \(f(t)=t^{p}\) for \(p\in \mathbb{R}\backslash \{0\}\) and \(f(t)=\log t\) into the results.