arXiv subjects

Mihailo Krstić

Publications and source records attributed to Mihailo Krstić.

5 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

Algorithm with variable coefficients for computing matrix inverses

We present a general scheme for the construction of new eficient generalized Schultz iterative methods for computing the inverse matrix. These methods have the form $$ X_{k+1} = X_k(a_0^{(k)}I+a_1^{(k)}AX_k),\quad k\in\mathbb{N}, $$ where $A$ is square real matrix and $a_0^{(k)}$ and $a_0^{(k)}$ are dynamical coefficients. We are going to present basic case of the problem, while formulas are derived analogically in other cases but are more complicated. Constructed method is optimal, meaning that coefficients are chosen in optimal way in terms of Frobenius norm. We have done some numerical testing that confirm theoretical approach. Through construction and numerical testing of method we have considered numerical stability as well. In the end, constructed method in it's final form is numerically stable and optimal.

math.NA↗

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↗

arXiv2025

Extended Cauchy-Schwarz inequalities for $σ$-elementary transformers in Schatten-von Neumann norm ideals

Let $q, r, s \geqslant 1$ satisfy $\frac{1}{2q} + \frac{1}{2r} = \frac{1}{s}$ and $X \in \mathcal{C}_s(\mathcal{H})$. If $(λ_n)_{n=1}^{\infty}, (w_n)_{n=1}^{\infty}$ are sequences in $(0,+\infty)$ and $(λ_n^{(1-q)/(2q)} A_n)_{n=1}^{\infty}$, $(λ_n^{1/(2q)} A_n^*)_{n=1}^{\infty}, (w_n^{-1/(2r)} B_n)_{n=1}^{\infty}$ and $(w_n^{(r-1)/(2r)} B_n^*)_{n=1}^{\infty} $ are strongly square summable, then there exists $\sideset{^{_{\scriptstyle\,\mathcal{C}_{\large s}\!\!}}}{\phantom{}}{\textstyle\sum_{n=1}^{+\infty}}A_nXB_n$ and \begin{equation*} \begin{split} &\bigg\|\!\!\sideset{^{_{{\scriptscriptstyle\,\Large\mathcal{C}_{\!s}\!\!}}}}{\phantom{}} \sum_{\,\,\,n=1}^{\,\,\,\infty}A_nXB_n\bigg\|_s \\ &\leqslant\bigg\|\!\!\sideset{^{_{{\scriptstyle\,{s}\!}}}}{\phantom{}}\sum_{\,\,n=1}^{\,\,\infty} λ_n^{\frac{1}{q}} A_n A_n^* \bigg\|^{\!\frac{1}{2} - \frac{1}{2q}}\! \bigg\|\!\!\sideset{^{_{{\scriptstyle\,{s}\!}}}}{\phantom{}}\sum_{\,\,n=1}^{\,\,\infty}w_n^{\!-\frac{1}{r}}\! B_n^* B_n \bigg\|^{\!\frac{1}{2} - \frac{1}{2r}}\! \bigg\|\!\bigg(\!\!\!\sideset{^{_{{\scriptstyle\,{s}\!}}}}{\phantom{}}\sum_{\,\,n=1}^{\,\,\infty} λ_n^{\!\frac{1}{q}-1}\! A_n^* A_n\! \bigg)^{\!\!\frac{1}{2q}}\! X\bigg(\!\!\!\sideset{^{_{{\scriptstyle\,{s}\!}}}}{\phantom{}}\sum_{\,\,n=1}^{\,\,\infty} w_n^{1-\frac{1}{r}}\! B_n B_n^*\! \bigg)^{\!\!\frac{1}{2r}}\! \bigg\|_s\!. \end{split} \end{equation*} Equivalent inequalities are also given, together with some applications to families $(A_n)_{n=1}^{\infty}$ and $(B_n)_{n=1}^{\infty}$ in $\mathcal{B}(\mathcal{H})$ which are not double square summable. The results presented in this article significantly extends the previous results of authors related to $σ$-elementary transformers in Schatten-von Neumann ideals.

math.FA↗

arXiv2025

$q$-numerical radius of rank-one operators and the generalized Buzano inequality

Here, we study the $q$-numerical radius of rank-one operators on a Hilbert space $\mathcal{H}$. More precisely, for $q \in [0,1]$ and $a, b \in \mathcal{H}$, we establish the formula \[ ω_q(a \otimes b) = \frac{1}{2}\left(\|a\|\|b\| + q|\langle a, b \rangle| + \sqrt{1-q^2}\sqrt{\|a\|^2\|b\|^2 - |\langle a, b \rangle|^2}\right), \] which represents a generalization of the well-known formula for the numerical radius of a rank-one operator in a Hilbert space, obtained by setting $q = 1$. As a corollary, we also derive a generalization of the classical Buzano inequality.

math.FA↗