Let \(\tilde{A}=(A_{t})_{t\in T}\) and \(\tilde{B}=(B_{t})_{t\in T}\) be two continuous fields of strictly positive operators on a Hilbert space \(\mathcal {H}\) and \(f:[0,\infty )\rightarrow \mathbb {R}\) a continuous function. We introduce the notion of noncommutative \(\lambda \) -perspective with \(\lambda \in \left[ 0,1\right] \) and the Csiszár \((f,\lambda )\) -divergence operator mapping by setting \(\begin{aligned} P_{f,\lambda }(A_{t},B_{t}):=A_{t}^{1/2}f(\lambda A_{t}^{-1/2}B_{t}A_{t}^{-1/2}+1-\lambda )A_{t}^{1/2} \end{aligned}\) and \(\begin{aligned} \textbf{I}_{f,\lambda }(\tilde{A},\tilde{B})=\int _{T}P_{f,\lambda }(A_{t},B_{t})d\mu (t), \end{aligned}\) respectively. We also consider the (HH)-f-divergence operator mapping by setting \(\begin{aligned} \textbf{I}_{HH}^{f}(\tilde{A},\tilde{B})=\int _{0}^{1}\int _{T}P_{f,\lambda }(A_{t},B_{t})d\mu (t)d\lambda . \end{aligned}\) In this paper, we investigate some fundamental properties of (HH)-f-divergence operator. Some upper and lower bounds of interest are also provided.
