arXiv subjects

Matija Milović

Publications and source records attributed to Matija Milović.

3 recordsLinked to original sources

arXiv2026

Pettis integrability of functions with values in separable symmetrically-normed ideals and related norm estimates

In this paper we will investigate Pettis integrability of $\mathcal{C}^{\circ}_Φ(\mathcal{H})$-valued functions. We will study weakly$^*$ integrable $\mathcal{B}(\mathcal{H})$-valued functions and establish sufficient conditions for such functions to be Pettis integrable as $\mathcal{C}^{\circ}_Φ(\mathcal{H})$-valued functions. In addition, we prove the inequality $$\left\|\int_E\mathscr{A}^*\mathscr{B}dμ\right\|_{Φ^{(p)}} \leqslant \|\mathscr{A}\|_{L^q_s}\cdot\left\|\sqrt[p]{\int_E|\mathscr{B}|^pdμ}\right\|_{Φ^{(p)}},$$ where $Φ^{(p)}$ is $p$-modification of the function $Φ$ and the functions $\mathscr{A}$ and $\mathscr{B}$ belong to the suitable spaces of operator-valued functions. Finally, under some additional integrability assumptions on $\mathscr{B}$ we provide similar estimates of the Pettis norm.

math.FA↗

arXiv2026

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

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 $c_0$.

math.FA↗

arXiv2026

Gel'fand Integration of B(E, F*)-Valued Functions With Emphasis on (q, p)-Summing Operators

We generalize results concerning Gel'fand integration of functions taking values in the space of operators on Hilbert spaces to certain Banach spaces. Building on ideas from \cite{M24} we provide sufficient conditions for the Gel'fand integral to be $(q,p)$-summing and we use the developed techniques to answer a question posed in the mentioned article. Applications to positive operator-valued functions between certain function spaces are also given.

math.FA↗