On variance and covariance for bounded linear operators (Q5958936)

Let \(H\) be a complex Hilbert space. For a vector \(0\neq y\in H\) and operators \(T,S\in B(H)\), \(\text{Cov}_{y}(S,T)\) and \(\text{Var}_{y}(S)\) are defined as \[ \begin{aligned} \text{Cov}_{y}(S,T)&= \|y\|^{2}(Sx,Tx)-(Sx,y)(y,Tx)\\ \text{and}\\ \text{Var}_{y}(S)&= \|y\|^{2}\|Sx\|^{2} -|(Sx,y)|^{2},\end{aligned} \] respectively. As a relation between \(\text{Cov}_{y}(S,T)\) and \(\text{Var}_{y}(S)\), the following covariance-variance inequality holds: \[ |\text{Cov}_{y}(S,T)|^{2} \leq \text{Var}_{y}(S) \text{Var}_{y}(T). \] In this paper, the author claims that the covariance-variance inequality implies the Cauchy-Schwarz, Bernstein-type and generalized Heinz-Kato-Furuta-type inequalities, and discusses the equalities of the above inequalities.