Strongly Integrable Operator-Valued Functions, Generated Vector Measures and Compactness of Integrals

Abstract

Gel'fand integral of a family of compact operators on a Hilbert space is not always compact, even with additional property of positivity and commutativity. We prove that integrals of a family, consisting of compact operators, in the space $L_s^1(\hat{I}\text{c◯},\hat{I}¼,\mathcal{B}(X,Y))$ of strongly integrable families are compact whenever $X$ does not contain an isomorphic copy of ${\ell }^1$. In addition, we prove an integral inequality for spectral radius $$r\left({\int }_{\hat{I}}\text{c◯}\mathcal{A}d\hat{I}¼\right)⩽{\int }_{\hat{I}}\text{c◯}r({\mathcal{A}}_t)d\hat{I}¼(t)$$ for a mutually commuting family $\mathcal{A}$ in $L_s^1(\hat{I}\text{c◯},\hat{I}¼,\mathcal{B}(X))$, which generalizes a recent result obtained under a stronger assumption of Bochner integrability. We prove also approximation results in $L_s^1(\hat{I}\text{c◯},\hat{I}¼,\mathcal{B}(X))$ in the case $X$ has finite dimensional Schauder decomposition. All these results are based on a key theorem of this paper which states that every function in $L_s^1(\hat{I}\text{c◯},\hat{I}¼,\mathcal{B}(X,Y))$ generates a countably additive, in operator norm, $\mathcal{B}(X,Y)$-valued measure whenever $X^{\ast }$ does not contain an isomorphic copy of $c_0$.