arXiv subjects

Nikola Sarajlija

Publications and source records attributed to Nikola Sarajlija.

9 recordsLinked to original sources

arXiv2026

Invertible completions of FLI and FRI upper triangular operator matrices

If $A\in\mathcal{B}(\mathcal{H})$ and $B\in\mathcal{B}(\mathcal{K})$ are given operators, denote by $M_C$ an operator matrix of the form $$M_C=\begin{pmatrix} A & C\\ 0 & B \end{pmatrix}\in\mathcal{B}(\mathcal{H}\oplus\mathcal{K})$$ acting on a direct sum of infinite dimensional separable Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, where $C\in\mathcal{B}(\mathcal{K},\mathcal{H})$ is unknown. In this article we solve the completion problem of $M_C$ to Fredholm left and Fredholm right invertibility, and we obtain appropriate perturbation results as consequences. We illustrate our results by solving the 'filling in holes' problem for Fredholm left and Fredholm right spectra. Finally, we consider some special classes of operators. Throughout the article, we recover some known results.

math.FA↗

arXiv2026

On the fractional relaxation equation with Scarpi derivative

In this article we solve the Cauchy problem for the relaxation equation posed in a framework of variable order fractional calculus. After introducing some general mathematical theory we establish concepts of Scarpi derivative and transition functions which represent the essentials of our problem. Next, we provide an integral representation for the solution of our initial value problem where the transition function is chosen arbitrary. After that, we find an integral representation in several special cases in which we choose the transition function concretely. Finally, we give some numerical insights which prove our theoretical results.

math.GM↗

arXiv2025

Completion problem of upper triangular $3\times3$ operator matrices on arbitrary Banach spaces

We solve the completion problem of $3\times3$ upper triangular operator matrix acting on a direct sum of Banach spaces and hence generalize the famous result of Han, Lee, Lee (Proc. Amer. Math. Soc. 128 (1) (2000), 119-123) to a greater dimension of a matrix. Our main tools are Harte's ghost of an index theorem and Banach spaces embeddings. We overcome the lack of orthogonality in Banach spaces by exploiting decomposition properties of inner regular operators, and of Fredholm regular operators when needed. Finally, we provide some necessity results related to the invertibility of $n\times n$ upper triangular operators, $n>3$.

math.FA↗

arXiv2025

Upper triangular operator matrices and stability of their various spectra

Denote by $T_n^d(A)$ an upper triangular operator matrix of dimension $n$ whose diagonal entries $D_i$ are known, where $A=(A_{ij})_{1\leq i<j\leq n}$ is an unknown tuple of operators. This article is aimed at investigation of defect spectrum $\mathcal{D}^{σ_*}=\bigcup\limits_{i=1}^nσ_*(D_i)\setminusσ_*(T_n^d(A))$ , where $σ_*$ is a spectrum corresponding to various types of invertibility: (left, right) invertibility, (left, right) Fredholm invertibility, left Weyl invertibility, right Weyl invertibility. We give characterizations for each of the previous types, and provide some sufficent conditions for the stability of certain spectrum (case $\mathcal{D}^{σ_*}=\emptyset$). Our main results hold for an arbitrary dimension $n\geq2$ in arbitrary Hilbert or Banach spaces without assuming separability, thus generalizing results from \cite{WU}, \cite{WU2}. Hence, we complete a trilogy to previous work \cite{SARAJLIJA2}, \cite{SARAJLIJA3} of the same author, whose goal was to explore basic invertibility properties of $T_n^d(A)$ that are studied in Fredholm theory. We also retrieve a result from \cite{BAI} in the case $n=2$, and we provide a precise form of the well known 'filling in holes' result from \cite{HAN}.

math.FA↗

arXiv2025

Perturbing the spectrum of operator $T_n^d(A)$

Let $T_n^d(A)$ denote a partial upper triangular operator matrix whose diagonal entries are given and the others unknown. In this article we have aim to find characterizations of (left,right) invertibility of $T_n^d(A)$ in terms of diagonal entries solely, and hence we provide statements which generalize and correct results of Zhang S., Wu Z. (2012). We pose our results without invoking separability condition, thus improving results of Zhang S., Wu Z. (2012), and we give appropriate n-dimensional analogues without assuming separability as well. We recover many perturbation results of Djordjević D. S. (2002), and obtain some results of Du H. K., Pan J. (1994) and Han J. K., Lee H. Y., Lee W. Y. (2000) in the case of the Hilbert space setting.

math.FA↗

arXiv2022

Invertibility properties of operator matrices on Hilbert spaces

Denote by $T_n^d(A)$ an upper triangular operator matrix of dimension $n$ whose diagonal entries are given and the others are unknown. In this article we provide necessary and sufficient conditions for various types of Fredholm and Weyl completions of $T_n^d(A)$. As consequences, we get corrected perturbation results of Wu et al. (2020). In the special case $n=2$, we recover many already existing known results, and specially we correct results of Zhang et al. (2012). Finally, in the case of essential Fredholm invertibility, in the special case $n=2$ we obtain some results that seem new in the literature. Our method is based on the space decomposition technique, similarly to the work of Huang et al. (2019), but our approach extends to arbitrary dimension $n\geq2$.

math.FA↗

arXiv2022

Fredholmness and Weylness of block operator matrices

This paper has aim to characterize Fredholmness and Weylness of upper triangular operator matrices having arbitrary dimension n. We present various characterization results in the setting of infinite dimensional Hilbert spaces, thus extending some known results from Cao X. et al. (2006, 2005) and Zhang et al. (2012) to the case of arbitrary dimension n. We pose our results without using separability assumption, thus improving perturbation results from Wu X. et al. (2020).

math.FA↗

arXiv2021

Algebraic properties of the group inverse

We investigate properties of the group inverse in rings with unit related to products and differences of idempotents, and thus we extend some results from \cite{DENG} to more general settings. We show that most part of \cite{DENG} is easily translated into the context of rings with unit, and that there are some difficulties in generalizing statements dealing with complementary idempotents.

math.FA↗

arXiv2021

Intersections of various spectra of operator matrices

This paper is concerned with general $n\times n$ upper triangular operator matrices with given diagonal entries. We characterize perturbations of the left (right) essential spectrum, the essential spectrum, as well as the left (right) the Weyl spectrum of such operators, thus generalizing and improving some results of \cite{CAO}, \cite{CAO2}, \cite{SUNTHEORY}, \cite{SUN}, \cite{LIandDU}, \cite{ZHANG} from $n=2$ to case of $n\geq2$. We show that related results hold in arbitrary Banach spaces without assuming separability, thus extending and improving recent results from \cite{WU}, \cite{WU2}. For the lack of adjointness and orthogonality in Banach spaces, we use an adequate alternative: the concept of inner generalized inverse.

math.FA↗