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Strongly integrable operator functions generate norm-countably additive measures
The key theorem requires only that X* contain no copy of c0 and yields compactness and spectral-radius results for integrals.
abstract click to expand
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 $L_{s}^1(\Omega,\mu,\mathcal{B}(X, Y))$ 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 $$r\left(\int_\Omega\mathscr{A} \,d\mu\right)\leqslant\int_\Omega r(\mathscr{A}_t)\,d\mu(t)$$ for a mutually commuting family $\mathscr{A}$ in $L_s^1(\Omega,\mu,\mathcal{B}(X))$, which generalizes a recent result obtained under a stronger assumption of Bochner integrability. We prove also approximation results in $L_s^1(\Omega,\mu,\mathcal{B}(X))$ 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 $L_{s}^1(\Omega,\mu, \mathcal{B}(X, Y))$ generates a countably additive, in operator norm, $\mathcal{B}(X, Y)$-valued measure whenever $X^*$ does not contain an isomorphic copy of $c_0$.
