Strongly Integrable Operator-Valued Functions, Generated Vector Measures and Compactness of Integrals
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| Format: | Preprint |
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2026
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| _version_ | 1866913135792750592 |
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| author | Arsenović, Miloš Krstić, Mihailo Milović, Matija Milošević, Stefan |
| author_facet | Arsenović, Miloš Krstić, Mihailo Milović, Matija Milošević, Stefan |
| contents | 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 $L_{s}^1(Ω,μ,\mathcal{B}(X, Y))$ 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 $$r\left(\int_Ω\mathscr{A} \,dμ\right)\leqslant\int_Ωr(\mathscr{A}_t)\,dμ(t)$$ for a mutually commuting family $\mathscr{A}$ in $L_s^1(Ω,μ,\mathcal{B}(X))$, which generalizes a recent result obtained under a stronger assumption of Bochner integrability. We prove also approximation results in $L_s^1(Ω,μ,\mathcal{B}(X))$ 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 $L_{s}^1(Ω,μ, \mathcal{B}(X, Y))$ generates a countably additive, in operator norm, $\mathcal{B}(X, Y)$-valued measure whenever $X^*$ does not contain an isomorphic copy of $c_0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_12454 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strongly Integrable Operator-Valued Functions, Generated Vector Measures and Compactness of Integrals Arsenović, Miloš Krstić, Mihailo Milović, Matija Milošević, Stefan Functional Analysis Primary 46G10, 46B20, Secondary 47B07, 47A10, 46B15 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 $L_{s}^1(Ω,μ,\mathcal{B}(X, Y))$ 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 $$r\left(\int_Ω\mathscr{A} \,dμ\right)\leqslant\int_Ωr(\mathscr{A}_t)\,dμ(t)$$ for a mutually commuting family $\mathscr{A}$ in $L_s^1(Ω,μ,\mathcal{B}(X))$, which generalizes a recent result obtained under a stronger assumption of Bochner integrability. We prove also approximation results in $L_s^1(Ω,μ,\mathcal{B}(X))$ 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 $L_{s}^1(Ω,μ, \mathcal{B}(X, Y))$ generates a countably additive, in operator norm, $\mathcal{B}(X, Y)$-valued measure whenever $X^*$ does not contain an isomorphic copy of $c_0$. |
| title | Strongly Integrable Operator-Valued Functions, Generated Vector Measures and Compactness of Integrals |
| topic | Functional Analysis Primary 46G10, 46B20, Secondary 47B07, 47A10, 46B15 |
| url | https://arxiv.org/abs/2605.12454 |