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A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (31 May 2022) | Viewed by 13802

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Special Issue Editors

Prof. Dr. Vladimir Rakocevic

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Guest Editor

Department of Mathematics, Faculty of Sciences and Mathematics, Višegradska 33, Niš, 18000, Serbia
Interests: metric space; fixed point theory; operator theory; summability and matrix transformations
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Eberhard Malkowsky

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Guest Editor

Department of Mathematics, State University Novi Pazar, Vuka Karadžića bb, Novi Pazar 36300, Serbia
Interests: metric space; operator theory; summability and matrix transformations; measures of noncompactness
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Functional analysis and operator theory are widely used in the description, understanding, and control of dynamical systems and natural processes in physics, chemistry, medicine, and the engineering sciences.

This Special Issue is focused on the most recent advances in operator theory for theoretical and applied sciences arising in all fields of science, engineering applications, and other applied fields. The topics of the special issue include but are not limited to:

Bounded and unbounded operators between Banach spaces;
Matrix transformations on sequence spaces;
Fine spectrum;
Essential spectrum and Fredholm theory;
Measures of noncompactness of operators;
Generalized inverses;
Operators in fixed point theory;
Perov type contractions.

Prof. Dr. Vladimir Rakocevic
Prof. Dr. Eberhard Malkowsky
Guest Editors

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Published Papers (6 papers)

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Research

Jump to: Review

66 pages, 702 KiB

Open AccessArticle

A New Survey of Measures of Noncompactness and Their Applications

by Moosa Gabeleh, Eberhard Malkowsky, Mohammad Mursaleen and Vladimir Rakočević

Axioms 2022, 11(6), 299; https://doi.org/10.3390/axioms11060299 - 20 Jun 2022

Cited by 12 | Viewed by 2840

Abstract

We present a survey of the theory of measures of noncompactness and discuss some fixed point theorems of Darbo’s type. We apply the technique of measures of noncompactness to the characterization of classes of compact operators between certain sequence spaces, in solving infinite systems of integral equations in some sequence spaces. We also present some recent results related to the existence of best proximity points (pairs) for some classes of cyclic and noncyclic condensing operators in Banach spaces equipped with a suitable measure of noncompactness. Finally, we discuss the existence of an optimal solution for systems of integro–differentials. Full article

(This article belongs to the Special Issue Operator Theory and Its Applications)

9 pages, 274 KiB

Open AccessArticle

The Space of Functions with Tempered Increments on a Locally Compact and Countable at Infinity Metric Space

by Józef Banaś and Rafał Nalepa

Axioms 2022, 11(1), 11; https://doi.org/10.3390/axioms11010011 - 25 Dec 2021

Viewed by 2390

Abstract

The aim of the paper is to introduce the Banach space consisting of real functions defined on a locally compact and countable at infinity metric space and having increments tempered by a modulus of continuity. We are going to provide a condition that is sufficient for the relative compactness in the Banach space in question. A few particular cases of that Banach space will be discussed. Full article

(This article belongs to the Special Issue Operator Theory and Its Applications)

14 pages, 314 KiB

Open AccessFeature PaperArticle

Structure of Iso-Symmetric Operators

by Bhagwati Prashad Duggal and In-Hyoun Kim

Axioms 2021, 10(4), 256; https://doi.org/10.3390/axioms10040256 - 14 Oct 2021

Cited by 3 | Viewed by 2003

Abstract

For a Hilbert space operator

T \in B (H)

, let

L_{T}

and

R_{T} \in B (B (H))

denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m [...] Read more.

For a Hilbert space operator

T \in B (H)

, let

L_{T}

and

R_{T} \in B (B (H))

denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m and n, let

▵_{T^{*}, T}^{m} (I) = {(L_{T^{*}} R_{T} - I)}^{m} (I)

and

δ_{T^{*}, T}^{n} (I) = {(L_{T^{*}} - R_{T})}^{m} (I)

. The operator T is said to be

(m, n)

-isosymmetric if

▵_{T^{*}, T}^{m} (δ_{T^{*}, T}^{n}, (I)) = 0

. Power bounded

(m, n)

-isosymmetric operators

T \in B (H)

have an upper triangular matrix representation

T = (\begin{matrix} T_{1} & T_{3} \\ 0 & T_{2} \end{matrix}) \in B (H_{1} \oplus H_{2})

such that

T_{1} \in B (H_{1})

is a

C_{0 .}

-operator which satisfies

δ_{T_{1}^{*}, T_{1}}^{n} (I |_{H_{1}}) = 0

and

T_{2} \in B (H_{2})

is a

C_{1 .}

-operator which satisfies

A T_{2} = (V_{u} \oplus V_{b}) |_{H_{2}} A

A = {lim}_{t \to \infty} T_{2}^{* t} T_{2}^{t}

V_{u}

is a unitary and

V_{b}

is a bilateral shift. If, in particular, T is cohyponormal, then T is the direct sum of a unitary with a

C_{00}

-contraction. Full article

(This article belongs to the Special Issue Operator Theory and Its Applications)

12 pages, 326 KiB

Open AccessArticle

Calculations on Matrix Transformations Involving an Infinite Tridiagonal Matrix

by Ali Fares, Ali Ayad and Bruno de Malafosse

Axioms 2021, 10(3), 218; https://doi.org/10.3390/axioms10030218 - 8 Sep 2021

Viewed by 1736

Abstract

Given any sequence

z = {(z_{n})}_{n \geq 1}

of positive real numbers and any set E of complex sequences, we write

E_{z}

for the set of all sequences

y = {(y_{n})}_{n \geq 1}

such that [...] Read more.

Given any sequence

z = {(z_{n})}_{n \geq 1}

of positive real numbers and any set E of complex sequences, we write

E_{z}

for the set of all sequences

y = {(y_{n})}_{n \geq 1}

such that

y / z = {(y_{n} / z_{n})}_{n \geq 1} \in E

; in particular,

s_{z}^{0}

denotes the set of all sequences y such that

y / z

tends to zero. Here, we consider the infinite tridiagonal matrix

\tilde{B (r, s, t)}

, obtained from the triangle

B (r, s, t)

, by deleting its first row. Then we determine the sets of all positive sequences

a = {(a_{n})}_{n \geq 1}

such that

{(E_{a})}_{\tilde{B (r, s, t)}} \subset E_{a}

, where

E = ℓ_{\infty}

c_{0}

, or c. These results extend some recent results. Full article

(This article belongs to the Special Issue Operator Theory and Its Applications)

11 pages, 328 KiB

Open AccessArticle

New Results on the SSIE with an Operator of the form F_Δ ⊂ Ɛ +

F_{x}^{'}

Involving the Spaces of Strongly Summable and Convergent Sequences Using the Cesàro Method

by Bruno de Malafosse

Axioms 2021, 10(3), 157; https://doi.org/10.3390/axioms10030157 - 19 Jul 2021

Cited by 2 | Viewed by 1482

Abstract

Given any sequence

a = {(a_{n})}_{n \geq 1}

of positive real numbers and any set E of complex sequences, we can use

E_{a}

to represent the set of all sequences [...] Read more.

Given any sequence

a = {(a_{n})}_{n \geq 1}

of positive real numbers and any set E of complex sequences, we can use

E_{a}

to represent the set of all sequences

y = {(y_{n})}_{n \geq 1}

such that

y / a = {(y_{n} / a_{n})}_{n \geq 1} \in E

. In this paper, we use the spaces

w_{\infty}

w_{0}

and w of strongly bounded, summable to zero and summable sequences, which are the sets of all sequences y such that

{(n^{- 1} \sum_{k = 1}^{n} |y_{k}|)}_{n}

is bounded and tends to zero, and such that

y - l e \in w_{0}

, for some scalarl. These sets were used in the statistical convergence. Then we deal with the solvability of each of the SSIE

F_{Δ} \subset Ɛ + F_{x}^{'}

, where

Ɛ

is a linear space of sequences,

F = c_{0}

, c,

ℓ_{\infty}

w_{0}

, w or

w_{\infty}

, and

F^{'} =

c_{0}

, c or

ℓ_{\infty}

. For instance, the solvability of the SSIE

w_{Δ} \subset w_{0} + s_{x}^{(c)}

relies on determining the set of all sequences

x = {(x_{n})}_{n \geq 1} \in U^{+}

that satisfy the following statement. For every sequence y that satisfies the condition

{lim}_{n \to \infty} n^{- 1} \sum_{k = 1}^{n} |y_{k} - y_{k - 1} - l| = 0

, there are two sequences u and v, with

y = u + v

such that

{lim}_{n \to \infty} n^{- 1} \sum_{k = 1}^{n} |u_{k}| = 0

and

{lim}_{n \to \infty} (v_{n} / x_{n}) = L

for some scalars l and L. Full article

(This article belongs to the Special Issue Operator Theory and Its Applications)

Review

Jump to: Research

16 pages, 334 KiB

Open AccessReview

Ideals, Nonnegative Summability Matrices and Corresponding Convergence Notions: A Short Survey of Recent Advancements

by Pratulananda Das

Axioms 2022, 11(1), 1; https://doi.org/10.3390/axioms11010001 - 21 Dec 2021

Viewed by 2103

Abstract

In this survey article, we look into some recent results concerning summability matrices, both regular as well as those which are not regular (called semi-regular) and generated matrix ideals as the overall view of the inter relationship between the notions of ideal convergence [...] Read more.

(This article belongs to the Special Issue Operator Theory and Its Applications)

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