Weak* integration of functions with values in the set of Hilbert space operators

Abstract

In this work, we consider weakly* measurable operator-valued functions $\O\ni t\mapsto A_t\in\mathcalb B\left(\HH\right)$ and $\O\ni t\mapsto\bigoplus_{n=1}^{+\infty}A_t^{(n)}\in\mathcalb{B}\left(\bigoplus_{n=1}^{+\infty}\HH_n\right)$. Also, we will consider such functions as the elements of the normed space $L_G^1(\O,\mu,\BH)$ and we show that this space is not complete in general with respect to the norm $\|(A_t)_{t\in\O}\|_G=\sup_{\|f\|=\|g\|=1}\int_\O|\scal{Af}{g}|\,d\mu(t)$. Extensions of previous results are given and new problems related to this topic are discussed. For the families $(f_n(t)A_t)_{t\in\O}$ and $(f(t)A_t)_{t\in\O}$ in $\BH$, we have proved $$\lim_{n\to\infty}\left\|\int_\O f_n(t)A_t\,d\mu(t)-\int_\O f(t)A_t\,d\mu(t)\right\|=0,$$ under some additional conditions. Furthermore, necessary and sufficient conditions are discussed for the mapping $\O\ni t\mapsto\bigoplus_{n=1}^{+\infty}A_t^{(n)}\in\mathcalb B\left(\bigoplus_{n=1}^{+\infty}\HH_n\right)$ to belongs to the space $L_G^1\left(\O,\mu,\mathcalb{B}\left(\bigoplus_{n=1}^{+\infty}\HH_n\right)\right)$. If these conditions are satisfied we proved that the operator $\bigoplus_{n=1}^{+\infty}\int_{\Omega}A_t^{(n)}d\mu(t)$ exists in $\mathcalb{B}\left(\bigoplus_{n=1}^{+\infty}\HH_n\right)$ and we have
$$\int_{\Omega}\bigoplus_{n=1}^{+\infty}A_t^{(n)}d\mu(t)=\bigoplus_{n=1}^{+\infty}\int_{\Omega}A_t^{(n)}d\mu(t).$$
We also consider the vector measures $v:\M\to\BH$ and $v:\M\to\mathcalb{B}\left(\bigoplus_{n=1}^{+\infty}\HH_n\right)$ associated to the families from the spaces $L_G^1(\O,\mu,\BH)$ and $L_G^1\left(\O,\mu,\mathcalb{B}\left(\bigoplus_{n=1}^{+\infty}\HH_n\right)\right)$ respectively. New results related to operator-valued measures are obtained. We also give a formula that connects the weak* integral and the spectral integral, when the family of operators $(A_t)_{t\in\O}$ arises from the same spectral measure. Throughout the paper, specific examples of operator-valued functions are given to illustrate the general results, and in these examples we can see how weak* integrals $\gint$ can be computed in concrete cases.